diff --git a/-9E3T4oBgHgl3EQfSwmi/content/tmp_files/2301.04436v1.pdf.txt b/-9E3T4oBgHgl3EQfSwmi/content/tmp_files/2301.04436v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b782925c99cab108cb3ad5e0e69881a7b58f4ca --- /dev/null +++ b/-9E3T4oBgHgl3EQfSwmi/content/tmp_files/2301.04436v1.pdf.txt @@ -0,0 +1,931 @@ +arXiv:2301.04436v1 [math.CA] 11 Jan 2023 +OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS +WITH TWO VARIABLES +ISROIL A. IKROMOV, MICHAEL RUZHANSKY, AKBAR R. SAFAROV∗ +Abstract. In this paper we consider the problem of estimation of oscillatory in- +tegrals with Mittag-Leffler functions in two variables. The generalisation is that +we replace the exponential function with the Mittag-Leffler-type function, to study +oscillatory type integrals. +Contents +1. +Introduction +1 +2. +Preliminaries +2 +3. +Auxiliary statements +4 +4. +Proof of the main result +7 +Acknowledgements +9 +Data availability +9 +References +9 +1. Introduction +The function Eα(z) is named after the Swedish mathematican G¨osta Magnus +Mittag-Leffler (1846-1927) who defined it by a power series +Eα(z) = +∞ +� +k=0 +zk +Γ(αk + 1), +α ∈ C, Re(α) > 0, +(1.1) +and studied its properties in 1902-1905 in several subsequent notes [18, 19, 20, 21] in +connection with his summation method for divergent series. +A classical generalization of the Mittag-Leffler function, namely the two-parametric +Mittag-Leffler function is +Eα,β(z) = +∞ +� +k=0 +zk +Γ(αk + β), +α, β ∈ C, Re(α) > 0, +(1.2) +which was deeply investigated independently by Humbert and Agarval in 1953 ([1, +10, 11]) and by Dzherbashyan in 1954 ([4, 5, 6]) as well as in [9]. +∗Corresponding author +2010 Mathematics Subject Classification. 35D10, 42B20, 26D10. +Key words and phrases. Mittag-Leffler functions, phase function, amplitude. +All authors contributed equally to the writing of this paper. All authors read and approved the +final manuscript. +1 + +2 +I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV +It has the property that +E1,1(x) = ex, and we can refer to [23] for other properties. +(1.3) +In harmonic analysis one of the most important estimates for oscillatory integral is +van der Corput lemma [24, 25, 26, 34]. Estimates for oscillatory integrals with poly- +nomial phases can be found, for instance, in papers [2, 15, 29, 30, 31]. In the current +paper we replace the exponential function with the Mittag-Leffler-type function and +study oscillatory type integrals (2.3). In the papers [26] and [27] analogues of the van +der Corput lemmas involving Mittag-Leffler functions for one dimensional integrals +have been considered. We extend results of [26] and [27] for two-dimensional inte- +grals with phase having some simple singularities. Analogous problem on estimates +for Mittag-Leffler functions with the smooth phase functions of two variables having +simple singularities was considered in [28] and [32]. +2. Preliminaries +Definition 2.1. An oscillatory integral with phase f and amplitude a is an integral +of the form +J(λ, f, a) = +� +Rn a(x)eiλf(x)dx, +(2.1) +where a ∈ C∞ +0 (Rn) and λ ∈ R. +If the support of a lies in a sufficiently small neighborhood of the origin and f is +an analytic function at x = 0, then for λ → ∞ the following asymptotic expansion +holds ([17]): +J(λ, f, a) ≈ eiλf(0) � +s +n−1 +� +k=0 +bs,k(a)λs(ln λ)k, +(2.2) +where s belongs to a finite number of arithmetic progressions, independent of a, +composed of negative rational numbers, bs,k is a distribution with support in the +critical set {x : ∇f(x) = 0}. +Inspired by the terminology from [3], we refer to the maximal value of s, denoting +it by α in this case, as the growth index of f, or the oscillation index at the origin, +and the corresponding value of k is referred to as the multiplicity. +More precisely, the multiplicity of the oscillation index of an analytic phase at a +critical point is the maximal number k possessing the property: for any neighbour- +hood of the critical point there is an amplitude with support in this neighbourhood +for which in the asymptotic series (2.2) the coefficient bs,k(a) is not equal to zero. +The multiplicity of the oscillation index will be denoted by m (see [3]). +Let f be a smooth real-valued function defined on a neighborhood of the origin in +R2 with f(0, 0) = 0, ∇f(0, 0) = 0, and consider the associated Taylor series +f(x1, x2) ∼ +∞ +� +j,k=0 +cjkxj +1xk +2 +of f centered at the origin. The set +ℑ(f) := {(j, k) ∈ N2 : cjk = +1 +j!k!∂j +x1∂k +x2f(0, 0) ̸= 0} + +OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS +3 +is called the Taylor support of f at (0, 0). We shall always assume that +ℑ(f) ̸= ∅, +i.e., that the function f is of finite type at the origin. If f is real analytic, so that the +Taylor series converges to f near the origin, this just means that f ̸= 0. The Newton +polyhedron ℵ(f) of f at the origin is defined to be the convex hull of the union of +all the quadrants (j, k) + R2 ++, with (j, k) ∈ ℑ(f). The associated Newton diagram +ℵd(f) in the sense of Varchenko [33] is the union of all compact faces of the Newton +polyhedron; here, by a face, we mean an edge or a vertex. +We shall use coordinates (t1, t2) for points in the plane containing the Newton +polyhedron, in order to distinguish this plane from the (x1, x2) - plane. +The distance d = d(f) between the Newton polyhedron and the origin in the sense +of Varchenko is given by the coordinate d of the point (d, d) at which the bisectrix +t1 = t2 intersects the boundary of the Newton polyhedron. +The principal face π(f) of the Newton polyhedron of f is the face of minimal +dimension containing the point (d, d). Deviating from the notation in [33], we shall +call the series +fp(x1, x2) := +� +j,k∈π(f) +cjkxj +1xk +2 +the principal part of f. In the case that π(f) is compact, fπ is a mixed homogeneous +polynomial; otherwise, we shall consider fπ as a formal power series. +Note that the distance between the Newton polyhedron and the origin depends +on the chosen local coordinate system in which f is expressed. By a local analytic +(respectively smooth) coordinate system at the origin we shall mean an analytic (re- +spectively smooth) coordinate system defined near the origin which preserves 0. If +we work in the category of smooth functions f, we shall always consider smooth co- +ordinate systems, and if f is analytic, then one usually restricts oneself to analytic +coordinate systems (even though this will not really be necessary for the questions we +are going to study, as we will see). The height of the analytic (respectively smooth) +function f is defined by +h := h(f) := sup{dx}, +where the supremum is taken over all local analytic (respectively smooth) coordinate +systems x at the origin, and where dx is the distance between the Newton polyhedron +and the origin in the coordinates x. +A given coordinate system x is said to be adapted to f if h(f) = dx. +Let π be the principal face. We assume that π is a point or a compact edge, then +fπ is a weighted homogeneous polynomial. Denote by ν the maximal order of roots +of fπ on the unit circle at the origin, so +ν := max +S1 ord(fπ). +If there exists a coordinate system x such that ν = dx then we set m = 1. It can +be shown that in this case x is adapted to f (see [12]). Otherwise we take m = 0. +Following A. N. Varchenko we call m the multiplicity of the Newton polyhedron. +In the classical paper by A. N. Varchenko [33], he obtained the sharp estimates +for oscillatory integrals in terms of the height. Also in the paper [13] the height was +used to get the sharp bound for maximal operators associated to smooth surfaces in + +4 +I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV +R3. It turns out that analogous quantities can be used for oscillatory integrals with +the Mittag-Leffler function. +We consider the following integral with phase f and amplitude ψ, of the form +Iα,β = +� +U +Eα,β(iλf(x))ψ(x)dx, +(2.3) +where 0 < α < 1, β > 0, U is a sufficiently small neighborhood of the origin. We +are interested in particular in the behavior of Iα,β when λ is large, as for small λ the +integral is just bounded. In particular if α = 1 and β = 1 we have oscillatory integral +(2.1). +The main result of the work is the following. +Theorem 2.2. Let f be a smooth finite type function of two variables defined in a +sufficiently small neighborhood of the origin and let ψ ∈ C∞ +0 (U). +Let h be the height of the function f, and let m = 0, 1 be the multiplicity of its +Newton polyhedron. If 0 < α < 1, β > 0, h > 1, and λ ≫ 1 then we have the +estimate +���� +� +U +Eα,β(iλf(x1, x2))ψ(x)dx +���� ≤ +C| ln λ|m∥ψ∥L∞(U) +λ +1 +h +. +(2.4) +If 0 < α < 1, β > 0, h = 1 and λ ≫ 1, then we have following estimate +���� +� +U +Eα,β(iλf(x1, x2))ψ(x)dx +���� ≤ +C| ln λ|2∥ψ∥L∞(U) +λ +, +(2.5) +where the constants C are independent of the phase, amplitude and λ. +3. Auxiliary statements +We first recall some useful properties. +Proposition 3.1. If 0 < α < 2, β is an arbitrary real number, µ is such that πα/2 < +µ < min{π, πα}, then there is C > 0, such that we have +|Eα,β(z)| ≤ +C +1 + |z|, z ∈ C, µ ≤ | arg(z)| ≤ π. +(3.1) +See [4], [9], [23]. +Proposition 3.2. Let Ω be an open, bounded subset of +R2, and let f : Ω → R be a +measurable function such that for all λ ≫ 1 and for some positive δ ̸= 1, we have +���� +� +Ω +eiλf(x)dx +���� ≤ C|λ|−δ| ln λ|m, +(3.2) +with m ≥ 0. Then, we have +��x ∈ Ω : |f(x)| ≤ ε| ≤ Cδεδ| ln ε|m, for δ < 1, +for 0 < ε ≪ 1, and for δ > 1, |x ∈ Ω : |f(x)| ≤ ε| ≤ Cδε , +for δ = 1, +��x ∈ Ω : |f(x)| ≤ ε| ≤ Cδε| ln ε|m+1, +where Cδ depends only on δ, |A| means the Lebesgue measure of a set A. See [7]. + +OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS +5 +Proof. For the convenience of the reader we give an independent proof of Proposition +3.2. We consider an even non-negative smooth function +ω(x) = +� +1, +when |x| ≤ 1, +0, +when |x| ≥ 2. +For the characteristic function of Ω with Ω ⊂ U, the following inequality holds true +|x ∈ Ω : |f(x)| ≤ ε| = +� +Ω +χ[0,1] +�|f(x)| +ε +� +dx ≤ +� +Ω +ω +�f(x) +ε +� +dx. +Now we will use the Fourier inversion formula, and rewrite the last integral as +� +Ω +ω +�f(x) +ε +� +dx = 1 +2π +� +Ω +� ∞ +−∞ +ˇω(ξ)eiξ f(x) +ε dξdx. +As ˇω(ξ) is a Schwartz function, we can use Fubini theorem and change the order of +integration. So we have +� +Ω +� ∞ +−∞ +ˇω(ξ)eiξ f(x) +ε dξdx = +� ∞ +−∞ +ˇω(ξ) +� +Ω +eiξ f(x) +ε dxdξ. +We use inequality (3.2) for the inner integral and get +���� +� +Ω +eiξ f(x) +ε dx +���� ≤ C| ln(2 + ξ +ε)|m +(1 + | ξ +ε|)δ +. +As ˇω(ξ) is a Schwartz function, we also have +|ˇω(ξ)| ≤ +C +1 + |ξ|. +So +����� +� ∞ +−∞ +C ˇω(ξ)| ln(2 + ξ +ε)|m +(2 + | ξ +ε|)δ +dξ +����� ≲ +� ∞ +0 +2C| ln( ξ +ε)|m +(1 + |ξ|)(2 + | ξ +ε|)δ dξ. +Now we change the variable as ξ = ηε, and we get +� ∞ +0 +| ln( ξ +ε)|m +(1 + |ξ|)(2 + | ξ +ε|)δ dξ = +� ∞ +0 +ε| ln η|m +(1 + |εη|)(2 + |η|)δ dη. +Now we estimate the last integral for different values of δ. +If δ < 1 then we have +� ∞ +0 +ε| ln η|m +(1 + |εη|)(2 + |η|)δ dη ≤ Cε +� +1 +ε +0 +| ln η|mdη +(2 + η)δ + Cε +� ∞ +1 +ε +| ln η|mdη +εηδ+1 +. +We represent +1 +(2+η)δ = +1 +ηδ(1+ 2 +η )δ = +1 +ηδ + O( +1 +ηδ+1). So +Cε +� +1 +ε +0 +| ln η|mdη +(2 + η)δ = ε +� 2 +0 +| ln η|mdη +(2 + η)δ + ε +� +1 +ε +2 +| ln η|mdη +(2 + η)δ . +Integrating by parts we obtain +ε +� +1 +ε +2 +| ln η|mdη +(2 + η)δ ≤ ε +� +1 +ε +2 +| ln η|mdη +ηδ +≤ Cεδ| ln ε|m. + +6 +I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV +As δ < 1, the integrals +� 2 +0 +| ln η|mdη +(2+η)δ +and +� ∞ +1 +ε +| ln η|mdη +εηδ+1 +convergence. +If δ > 1 then we trivially obtain +���� +� ∞ +0 +Cε| ln η|m +(1 + |εη|)(2 + |η|)δ dη +���� ≤ Cε. +If δ = 1 then assuming 0 < ε < 1 +2 we get |εη| < 1 (for |η| < 2), then write the integral +as the sum of three integrals and obtain +���� +� ∞ +0 +Cε| ln η|m +(1 + |εη|)(1 + |η|)dη +���� ≤ +���� +� 2 +0 +Cε| ln η|mdη +���� + +����� +� +1 +ε +2 +Cε| ln η|m +η +dη +����� + +����� +� ∞ +1 +ε +Cε| lnη|m +η +dη +����� . +Then we have +���� +� 2 +0 +Cε| ln η|mdη +���� ≤ Cε, +and we get with simple calculating that +����� +� +1 +ε +2 +Cε| lnη|m +η +dη +����� ≤ Cε| ln ε|m+1. +We use the formula of integrating by parts several times, to get +����� +� ∞ +1 +ε +Cε| ln η|m +η +dη +����� ≤ Cε| ln ε|m, +completing the proof. +□ +From Proposition 3.2 we get the following corollaries. +Corollary 3.3. Let f(x1, x2) be a smooth function with f(0, 0) = 0, ∇f(0, 0) = 0, +and h be the height of the function f(x1, x2), and let m = 0, 1 be the multiplicity of +its Newton polyhedron. Let also a(x) = +� +1, +when |x| ≤ σ, +0, +when |x| ≥ 2σ, +σ > 0, and a(x) ≥ 0 +with a ∈ C∞ +0 (R2). If for all real λ ≫ 1 and for any positive δ ̸= 1, the following +inequality holds +���� +� +R2 eiλf(x)a(x)dx +���� ≤ C|λ|−δ| ln λ|m, +(3.3) +then we have +||x| ≤ σ : |f(x)| ≤ ε| ≤ Cεδ| ln ε|m, +where m ≥ 0. See [8, 12, 14, 22]. +Corollary 3.4. Let f(x1, x2) be a smooth function with f(0, 0) = 0, ∇f(0, 0) = 0, +and let Ω be a sufficiently small compact set around the origin. Let also h be the +height of the function f(x1, x2), and let m = 0, 1 be the multiplicity of its Newton +polyhedron. Then for all 0 < ε ≪ 1 we have +|x ∈ Ω : |f(x)| ≤ ε| ≤ Cε +1 +h| ln ε|m, +where h is the height of f and m is its multiplicity [8]. + +OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS +7 +4. Proof of the main result +Proof of Theorem 2.2. As for λ < 2 the integral (2.3) is just bounded, we +consider the case λ ≥ 2. Without loss of generality, we can consider the integral over +U. Using inequality (3.1), we have +|Eα,β(iλf(x))| ≤ +C +1 + λ|f(x)|. +(4.1) +We then use (4.1) for the integral (2.3), and get that +|Iα,β| ≤ +���� +� +U +Eα,β(iλf(x))ψ(x)dx +���� ≤ C +� +U +|ψ(x)|dx +1 + λ|f(x)|. +(4.2) +Now we represent the integral Iα,β over the union of sets Ω1 := Ω ∩ {λ|f(x1, x2)| < +M} and Ω2 := Ω ∩ {λ|f(x1, x2)| ≥ M} respectively, where M is a positive real +number. +We estimate the integral Iα,β over the sets Ω1 and Ω2, respectively, +|Iα,β| ≤ C +� +U +|ψ(x)|dx +1 + λ|f(x)| = J1 + J2 := C +� +Ω1 +|ψ(x)|dx +1 + λ|f(x)| + C +� +Ω2 +|ψ(x)|dx +1 + λ|f(x)|. +First we estimate the integral over the set Ω1. Using the results of the paper ([17] +page 31) (see also Corollary 3.4) we obtain +|J1| = C +� +Ω1 +|ψ(x)|dx +1 + λ|f(x)| ≤ +C| ln λ|m∥ψ∥L∞(Ω1) +λ +1 +h +. +Lemma 4.1. Let f ∈ C∞ and h be the height of the function f, and let m = 0, 1 be +the multiplicity of its Newton polyhedron. For any smooth function a = a(x, y) with +sufficiently small support and for h > 1 the following inequality holds +I := +� +{|f(x,y)|≥ M +λ } +a(x, y) +1 + λ|f(x, y)|dxdy ≤ +C| ln λ|m∥a∥L∞(U) +λ +1 +h +, +(4.3) +where supp{a(x, y)} = U. +Proof. Let h > 1. Consider the sets +Ak = +� +x ∈ U : 2k +λ ≤ |f(x)| ≤ 2k+1 +λ +� +. +For the measure of a set of smaller values we use Lemma 1 +′ in the paper [16] (see also +Corollary 3.4), and we have +µ +� +|f(x)| ≤ 2k+1 +λ , x ∈ U +� +≤ C +�2k+1 +λ +� 1 +h � +ln +���� +λ +2k+1 +���� +�m +. +Let +Ik := +� +Ak +a(x, y) +1 + λ|f(x, y)|dxdy. +For the integral +� +2k≤λ|f(x)|≤2k+1 +Ik = +� +Ω2 +a(x, y) +1 + λ|f(x, y)|dxdy, + +8 +I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV +we find the following estimate: +|Ik| = +���� +� +Ak +a(x, y) +1 + λ|f(x, y)|dxdy +���� ≤ C∥a∥L∞(U) +�2k+1 +λ +� 1 +h ����ln 2k+1 +λ +���� +m +2−k. +From here we find the sum of Ik and, by estimating the integral I, we get +I ≤ ∥a∥L∞(U) +∞ +� +k=1 +Ik ≤ ∥a∥L∞(U) +∞ +� +k=1 +�2k+1 +λ +� 1 +h ����ln 2k+1 +λ +���� +m +2−k +≤ ∥a∥L∞(U) +| ln λ|m +λ +1 +h +∞ +� +k=1 +2 +k+1 +h −kkm. +As h > 1, the last series is convergent, proving the lemma. +□ +Remark 4.2. Consider the case h = 1. +The smooth function has non-degenerate +critical point at the origin if and only if h = 1. As f(x, y) is a smooth function with +∇f(0, 0) = 0, using Morse lemma we have f ∼ x2 ± y2. So in this case we estimate +two sets ∆ = ∆1 ∪ ∆2, where ∆1 := {(x, y) : λ|x2 ± y2| ≤ M, |x| ≤ 1, |y| ≤ 1} and +∆2 := {(x, y) : λ|x2 ± y2| > M, |x| ≤ 1, |y| ≤ 1}. First we consider the integral over +the set ∆1. Then we have +���� +� +∆1 +a(x, y) +1 + λ|x2 ± y2|dxdy +���� ≤ C∥a∥L∞(∆1) +���� +� +∆1 +dxdy +����. +Now we estimate the last integral as +���� +� +λ|x2+y2|≤M +dxdy +���� ≤ C +λ . +Then we estimate the measure of the set {|x2 − y2| ≤ εM}, where ε = 1 +λ. We have, +for simplicity putting M = 1, +���� +� +|x2−y2|≤εM +dxdy +���� ≤ C +����� +� √1−ε +√ε +dy +� √ +y2+ε +√ +y2−ε +dx +����� = +����� +� √1−ε +√ε +�� +y2 + ε − +� +y2 − ε +� +dy +����� = += +�y +2 +� +y2 + ε + ε +2 ln |y + +� +y2 + ε| +� ��� +√1−ε +√ε +− +�y +2 +� +y2 − ε − ε +2 ln |y + +� +y2 − ε| +� ��� +√1−ε +√ε += += +����� +√1 − ε +2 ++ ε +2 ln +√1 − ε + 1 +√ε +− +√ +2 +2 ε − ε +2 ln |√ε(1 + +√ +2)|− +− +�� +(1 − ε)(1 − 2ε) +2 +− ε +2 ln | +√ +1 − ε + +√ +1 − 2ε| + ε +2 ln √ε| +������ ≤ Cε ln ε. +Now we consider the integral over the set ∆2. In this case we change the variables +to polar coordinate system and with easy calculating we get +���� +� +{λ|x2+y2|≥M} +a(x, y) +1 + λ|x2 + y2|dxdy +���� ≤ C| ln λ|∥a∥L∞(∆2) +λ +(4.4) +and +���� +� +{λ|x2−y2|≥M} +a(x, y) +1 + λ|x2 − y2|dxdy +���� ≤ C| ln λ|2∥a∥L∞(∆2) +λ +. +(4.5) + +OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS +9 +Now we continue the proof of Theorem 2.2. Let h > 1. We use Proposition 3.2 for +the integral J1, to get +|J1| ≤ +C| ln λ|m∥a∥L∞(U) +λ +1 +h +. +Let consider the integral J2. If h > 1, then using Lemma 4.1 we get +|J2| ≤ +C| ln λ|∥a∥L∞(U) +λ +1 +h +. +If h = 1, using the Remark 4.2 we get the inequality (2.5). The proof is complete. +The proof of Theorem 2.2 shows that if h = 1, we can get a more precise result. +Proposition 4.3. If h = 1 and f has an extremal point at the point (0,0) (then f is +diffeomorhic equivalent to x2 +1 + x2 +2 or −x2 +1 − x2 +2), then we have +|Iα,β| ≤ +C| ln λ|∥ψ∥L∞(U) +λ +, +for all λ ≥ 2. +Declaration of competing interest +This work does not have any conflicts of interest. +Acknowledgements +The second author was supported in parts by the FWO Odysseus 1 grant G.0H94.18N: +Analysis and Partial Differential Equations and by the Methusalem programme of the +Ghent University Special Research Fund (BOF) (Grant number 01M01021) and also +supported by EPSRC grant EP/R003025/2. +Data availability. 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Mittag-Leffler, Une g´en´eralization de l’int´egrale de Laplace-Abel, Comp. Rend. Acad. +Sci. Paris 136, 537-539 (1903). +[20] M. G. Mittag-Leffler, Sur la nouvelle fonction Eα(x), Comp. Rend. Acad. Sci. Paris 137, 554- +558 (1903). +[21] M. G. Mittag-Leffler, Sopra la funzione Eα(x), Rend.R.Acc.Lincei, (Ser.5)13, 3-5 (1904). +[22] D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operator, Acta +Math. 179(1), 1997, 105-152. +[23] I. Podlubny, Fractional Differensial Equations, Academic Press, New York, 1999. +[24] M. Ruzhansky, Pointwise van der Corput Lemma for Functions of Several Variables, Functional +Analysis and Its Applications, 43 (2009), no.1, 75–77. +[25] M. Ruzhansky, Multidimensional decay in the van der Corput Lemma, Studia Mathematica, +208 (2012), no.1, 1–9. +[26] M. Ruzhansky, B. Torebek, Van der Corput lemmas for Mittag-Leffler functions, Fractional +Calculus and Applied Analysis, 23 (6), (2021), 1663–1677. +[27] M. Ruzhansky, +B. Torebek, +Van der Corput lemmas for Mittag-Leffler functions. II. +α−directions , Bull. Sci. Math., 171 (2021), 103016, 23 pp. +[28] M. Ruzhansky, A. R. Safarov, G. A. Khasanov, Uniform estimates for oscillatory integrals with +homogeneous polynomial phases of degree 4, Analysis and Mathematical Physics, 12(130), +(2022). +[29] A. Safarov, Invariant estimates of two-dimensional oscillatory integrals // Math. Notes. 104, +2018. P.293–302. +[30] A. Safarov, On invariant estimates for oscillatory integrals with polynomial phase, // J. Sib. +Fed. Univ. Math. Phys. 9 (2016), P.102–107. +[31] A. Safarov, On a problem of restriction of Fourier transform on a hypersurface // Russian +Mathematics, 63 (4), P.57-63. +[32] A. R. Safarov, Estimates for Mittag-–Leffler Functions with Smooth Phase Depending on Two +Variables, J. Sib. Fed. Univ. Math. Phys., 15(4) (2022), P.459-–466. +[33] A. N. Varchenko, Newton polyhedra and estimation of oscillating integrals //Functional Analysis +and Its Applications, vol. 10, pages 175-–196 (1976). +[34] Van der Korput, Zur Methode der stationaren phase// Compositio Math. V.1. 1934. P. 15-38. + +OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS +11 +Isroil A. Ikromov +V.I. Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan +Olmazor district, University 46, Tashkent, Uzbekistan +Samarkand State University +Department of Mathematics, 15 University Boulevard +Samarkand, 140104, Uzbekistan +Email address: ikromov1@rambler.ru +Michael Ruzhansky +Department of Mathematics: Analysis, Logic and Discrete Mathematics +Ghent University, +Krijgslaan 281, Ghent, Belgium, +School of Mathematical Sciences, Queen Mary University of London, +United Kingdom +Email address: michael.ruzhansky@ugent.be +Akbar R.Safarov +Uzbek-Finnish Pedagogical Institute +Spitamenshox 166, Samarkand, Uzbekistan +Samarkand State University +Department of Mathematics, 15 University Boulevard +Samarkand, 140104, Uzbekistan +Email address: safarov-akbar@mail.ru + diff --git a/-9E3T4oBgHgl3EQfSwmi/content/tmp_files/load_file.txt b/-9E3T4oBgHgl3EQfSwmi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..77a30e542a3898e74f8e6ab5ac253b23bd2cf60b --- /dev/null +++ b/-9E3T4oBgHgl3EQfSwmi/content/tmp_files/load_file.txt @@ -0,0 +1,458 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf,len=457 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='04436v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='CA] 11 Jan 2023 OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS WITH TWO VARIABLES ISROIL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' IKROMOV, MICHAEL RUZHANSKY, AKBAR R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' SAFAROV∗ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' In this paper we consider the problem of estimation of oscillatory in- tegrals with Mittag-Leffler functions in two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory type integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Preliminaries 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Auxiliary statements 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Proof of the main result 7 Acknowledgements 9 Data availability 9 References 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Introduction The function Eα(z) is named after the Swedish mathematican G¨osta Magnus Mittag-Leffler (1846-1927) who defined it by a power series Eα(z) = ∞ � k=0 zk Γ(αk + 1), α ∈ C, Re(α) > 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1) and studied its properties in 1902-1905 in several subsequent notes [18, 19, 20, 21] in connection with his summation method for divergent series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' A classical generalization of the Mittag-Leffler function, namely the two-parametric Mittag-Leffler function is Eα,β(z) = ∞ � k=0 zk Γ(αk + β), α, β ∈ C, Re(α) > 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2) which was deeply investigated independently by Humbert and Agarval in 1953 ([1, 10, 11]) and by Dzherbashyan in 1954 ([4, 5, 6]) as well as in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' ∗Corresponding author 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' 35D10, 42B20, 26D10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Mittag-Leffler functions, phase function, amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' All authors contributed equally to the writing of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' All authors read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' 1 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='IKROMOV, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='RUZHANSKY, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='SAFAROV It has the property that E1,1(x) = ex, and we can refer to [23] for other properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='3) In harmonic analysis one of the most important estimates for oscillatory integral is van der Corput lemma [24, 25, 26, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Estimates for oscillatory integrals with poly- nomial phases can be found, for instance, in papers [2, 15, 29, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' In the current paper we replace the exponential function with the Mittag-Leffler-type function and study oscillatory type integrals (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' In the papers [26] and [27] analogues of the van der Corput lemmas involving Mittag-Leffler functions for one dimensional integrals have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We extend results of [26] and [27] for two-dimensional inte- grals with phase having some simple singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Analogous problem on estimates for Mittag-Leffler functions with the smooth phase functions of two variables having simple singularities was considered in [28] and [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Preliminaries Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' An oscillatory integral with phase f and amplitude a is an integral of the form J(λ, f, a) = � Rn a(x)eiλf(x)dx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1) where a ∈ C∞ 0 (Rn) and λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If the support of a lies in a sufficiently small neighborhood of the origin and f is an analytic function at x = 0, then for λ → ∞ the following asymptotic expansion holds ([17]): J(λ, f, a) ≈ eiλf(0) � s n−1 � k=0 bs,k(a)λs(ln λ)k, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2) where s belongs to a finite number of arithmetic progressions, independent of a, composed of negative rational numbers, bs,k is a distribution with support in the critical set {x : ∇f(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Inspired by the terminology from [3], we refer to the maximal value of s, denoting it by α in this case, as the growth index of f, or the oscillation index at the origin, and the corresponding value of k is referred to as the multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' More precisely, the multiplicity of the oscillation index of an analytic phase at a critical point is the maximal number k possessing the property: for any neighbour- hood of the critical point there is an amplitude with support in this neighbourhood for which in the asymptotic series (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2) the coefficient bs,k(a) is not equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The multiplicity of the oscillation index will be denoted by m (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let f be a smooth real-valued function defined on a neighborhood of the origin in R2 with f(0, 0) = 0, ∇f(0, 0) = 0, and consider the associated Taylor series f(x1, x2) ∼ ∞ � j,k=0 cjkxj 1xk 2 of f centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The set ℑ(f) := {(j, k) ∈ N2 : cjk = 1 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='∂j x1∂k x2f(0, 0) ̸= 0} OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS 3 is called the Taylor support of f at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We shall always assume that ℑ(f) ̸= ∅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=', that the function f is of finite type at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If f is real analytic, so that the Taylor series converges to f near the origin, this just means that f ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The Newton polyhedron ℵ(f) of f at the origin is defined to be the convex hull of the union of all the quadrants (j, k) + R2 +, with (j, k) ∈ ℑ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The associated Newton diagram ℵd(f) in the sense of Varchenko [33] is the union of all compact faces of the Newton polyhedron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' here, by a face, we mean an edge or a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We shall use coordinates (t1, t2) for points in the plane containing the Newton polyhedron, in order to distinguish this plane from the (x1, x2) - plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The distance d = d(f) between the Newton polyhedron and the origin in the sense of Varchenko is given by the coordinate d of the point (d, d) at which the bisectrix t1 = t2 intersects the boundary of the Newton polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The principal face π(f) of the Newton polyhedron of f is the face of minimal dimension containing the point (d, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Deviating from the notation in [33], we shall call the series fp(x1, x2) := � j,k∈π(f) cjkxj 1xk 2 the principal part of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' In the case that π(f) is compact, fπ is a mixed homogeneous polynomial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' otherwise, we shall consider fπ as a formal power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Note that the distance between the Newton polyhedron and the origin depends on the chosen local coordinate system in which f is expressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' By a local analytic (respectively smooth) coordinate system at the origin we shall mean an analytic (re- spectively smooth) coordinate system defined near the origin which preserves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If we work in the category of smooth functions f, we shall always consider smooth co- ordinate systems, and if f is analytic, then one usually restricts oneself to analytic coordinate systems (even though this will not really be necessary for the questions we are going to study, as we will see).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The height of the analytic (respectively smooth) function f is defined by h := h(f) := sup{dx}, where the supremum is taken over all local analytic (respectively smooth) coordinate systems x at the origin, and where dx is the distance between the Newton polyhedron and the origin in the coordinates x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' A given coordinate system x is said to be adapted to f if h(f) = dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let π be the principal face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We assume that π is a point or a compact edge, then fπ is a weighted homogeneous polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Denote by ν the maximal order of roots of fπ on the unit circle at the origin, so ν := max S1 ord(fπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If there exists a coordinate system x such that ν = dx then we set m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' It can be shown that in this case x is adapted to f (see [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Otherwise we take m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Following A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Varchenko we call m the multiplicity of the Newton polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' In the classical paper by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Varchenko [33], he obtained the sharp estimates for oscillatory integrals in terms of the height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Also in the paper [13] the height was used to get the sharp bound for maximal operators associated to smooth surfaces in 4 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='IKROMOV, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='RUZHANSKY, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='SAFAROV R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' It turns out that analogous quantities can be used for oscillatory integrals with the Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We consider the following integral with phase f and amplitude ψ, of the form Iα,β = � U Eα,β(iλf(x))ψ(x)dx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='3) where 0 < α < 1, β > 0, U is a sufficiently small neighborhood of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We are interested in particular in the behavior of Iα,β when λ is large, as for small λ the integral is just bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' In particular if α = 1 and β = 1 we have oscillatory integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The main result of the work is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let f be a smooth finite type function of two variables defined in a sufficiently small neighborhood of the origin and let ψ ∈ C∞ 0 (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let h be the height of the function f, and let m = 0, 1 be the multiplicity of its Newton polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If 0 < α < 1, β > 0, h > 1, and λ ≫ 1 then we have the estimate ���� � U Eα,β(iλf(x1, x2))ψ(x)dx ���� ≤ C| ln λ|m∥ψ∥L∞(U) λ 1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='4) If 0 < α < 1, β > 0, h = 1 and λ ≫ 1, then we have following estimate ���� � U Eα,β(iλf(x1, x2))ψ(x)dx ���� ≤ C| ln λ|2∥ψ∥L∞(U) λ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='5) where the constants C are independent of the phase, amplitude and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Auxiliary statements We first recall some useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If 0 < α < 2, β is an arbitrary real number, µ is such that πα/2 < µ < min{π, πα}, then there is C > 0, such that we have |Eα,β(z)| ≤ C 1 + |z|, z ∈ C, µ ≤ | arg(z)| ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1) See [4], [9], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let Ω be an open, bounded subset of R2, and let f : Ω → R be a measurable function such that for all λ ≫ 1 and for some positive δ ̸= 1, we have ���� � Ω eiλf(x)dx ���� ≤ C|λ|−δ| ln λ|m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2) with m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Then, we have ��x ∈ Ω : |f(x)| ≤ ε| ≤ Cδεδ| ln ε|m, for δ < 1, for 0 < ε ≪ 1, and for δ > 1, |x ∈ Ω : |f(x)| ≤ ε| ≤ Cδε , for δ = 1, ��x ∈ Ω : |f(x)| ≤ ε| ≤ Cδε| ln ε|m+1, where Cδ depends only on δ, |A| means the Lebesgue measure of a set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' See [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' For the convenience of the reader we give an independent proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We consider an even non-negative smooth function ω(x) = � 1, when |x| ≤ 1, 0, when |x| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' For the characteristic function of Ω with Ω ⊂ U, the following inequality holds true |x ∈ Ω : |f(x)| ≤ ε| = � Ω χ[0,1] �|f(x)| ε � dx ≤ � Ω ω �f(x) ε � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Now we will use the Fourier inversion formula, and rewrite the last integral as � Ω ω �f(x) ε � dx = 1 2π � Ω � ∞ −∞ ˇω(ξ)eiξ f(x) ε dξdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' As ˇω(ξ) is a Schwartz function, we can use Fubini theorem and change the order of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' So we have � Ω � ∞ −∞ ˇω(ξ)eiξ f(x) ε dξdx = � ∞ −∞ ˇω(ξ) � Ω eiξ f(x) ε dxdξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We use inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2) for the inner integral and get ���� � Ω eiξ f(x) ε dx ���� ≤ C| ln(2 + ξ ε)|m (1 + | ξ ε|)δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' As ˇω(ξ) is a Schwartz function, we also have |ˇω(ξ)| ≤ C 1 + |ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' So ����� � ∞ −∞ C ˇω(ξ)| ln(2 + ξ ε)|m (2 + | ξ ε|)δ dξ ����� ≲ � ∞ 0 2C| ln( ξ ε)|m (1 + |ξ|)(2 + | ξ ε|)δ dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Now we change the variable as ξ = ηε, and we get � ∞ 0 | ln( ξ ε)|m (1 + |ξ|)(2 + | ξ ε|)δ dξ = � ∞ 0 ε| ln η|m (1 + |εη|)(2 + |η|)δ dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Now we estimate the last integral for different values of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If δ < 1 then we have � ∞ 0 ε| ln η|m (1 + |εη|)(2 + |η|)δ dη ≤ Cε � 1 ε 0 | ln η|mdη (2 + η)δ + Cε � ∞ 1 ε | ln η|mdη εηδ+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We represent 1 (2+η)δ = 1 ηδ(1+ 2 η )δ = 1 ηδ + O( 1 ηδ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' So Cε � 1 ε 0 | ln η|mdη (2 + η)δ = ε � 2 0 | ln η|mdη (2 + η)δ + ε � 1 ε 2 | ln η|mdη (2 + η)δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Integrating by parts we obtain ε � 1 ε 2 | ln η|mdη (2 + η)δ ≤ ε � 1 ε 2 | ln η|mdη ηδ ≤ Cεδ| ln ε|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' 6 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='IKROMOV, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='RUZHANSKY, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='SAFAROV As δ < 1, the integrals � 2 0 | ln η|mdη (2+η)δ and � ∞ 1 ε | ln η|mdη εηδ+1 convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If δ > 1 then we trivially obtain ���� � ∞ 0 Cε| ln η|m (1 + |εη|)(2 + |η|)δ dη ���� ≤ Cε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If δ = 1 then assuming 0 < ε < 1 2 we get |εη| < 1 (for |η| < 2), then write the integral as the sum of three integrals and obtain ���� � ∞ 0 Cε| ln η|m (1 + |εη|)(1 + |η|)dη ���� ≤ ���� � 2 0 Cε| ln η|mdη ���� + ����� � 1 ε 2 Cε| ln η|m η dη ����� + ����� � ∞ 1 ε Cε| lnη|m η dη ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Then we have ���� � 2 0 Cε| ln η|mdη ���� ≤ Cε, and we get with simple calculating that ����� � 1 ε 2 Cε| lnη|m η dη ����� ≤ Cε| ln ε|m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We use the formula of integrating by parts several times, to get ����� � ∞ 1 ε Cε| ln η|m η dη ����� ≤ Cε| ln ε|m, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' □ From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2 we get the following corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let f(x1, x2) be a smooth function with f(0, 0) = 0, ∇f(0, 0) = 0, and h be the height of the function f(x1, x2), and let m = 0, 1 be the multiplicity of its Newton polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let also a(x) = � 1, when |x| ≤ σ, 0, when |x| ≥ 2σ, σ > 0, and a(x) ≥ 0 with a ∈ C∞ 0 (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If for all real λ ≫ 1 and for any positive δ ̸= 1, the following inequality holds ���� � R2 eiλf(x)a(x)dx ���� ≤ C|λ|−δ| ln λ|m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='3) then we have ||x| ≤ σ : |f(x)| ≤ ε| ≤ Cεδ| ln ε|m, where m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' See [8, 12, 14, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let f(x1, x2) be a smooth function with f(0, 0) = 0, ∇f(0, 0) = 0, and let Ω be a sufficiently small compact set around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let also h be the height of the function f(x1, x2), and let m = 0, 1 be the multiplicity of its Newton polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Then for all 0 < ε ≪ 1 we have |x ∈ Ω : |f(x)| ≤ ε| ≤ Cε 1 h| ln ε|m, where h is the height of f and m is its multiplicity [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Proof of the main result Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' As for λ < 2 the integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='3) is just bounded, we consider the case λ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Without loss of generality, we can consider the integral over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Using inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1), we have |Eα,β(iλf(x))| ≤ C 1 + λ|f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1) We then use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1) for the integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='3), and get that |Iα,β| ≤ ���� � U Eα,β(iλf(x))ψ(x)dx ���� ≤ C � U |ψ(x)|dx 1 + λ|f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2) Now we represent the integral Iα,β over the union of sets Ω1 := Ω ∩ {λ|f(x1, x2)| < M} and Ω2 := Ω ∩ {λ|f(x1, x2)| ≥ M} respectively, where M is a positive real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We estimate the integral Iα,β over the sets Ω1 and Ω2, respectively, |Iα,β| ≤ C � U |ψ(x)|dx 1 + λ|f(x)| = J1 + J2 := C � Ω1 |ψ(x)|dx 1 + λ|f(x)| + C � Ω2 |ψ(x)|dx 1 + λ|f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' First we estimate the integral over the set Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Using the results of the paper ([17] page 31) (see also Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='4) we obtain |J1| = C � Ω1 |ψ(x)|dx 1 + λ|f(x)| ≤ C| ln λ|m∥ψ∥L∞(Ω1) λ 1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let f ∈ C∞ and h be the height of the function f, and let m = 0, 1 be the multiplicity of its Newton polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' For any smooth function a = a(x, y) with sufficiently small support and for h > 1 the following inequality holds I := � {|f(x,y)|≥ M λ } a(x, y) 1 + λ|f(x, y)|dxdy ≤ C| ln λ|m∥a∥L∞(U) λ 1 h , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='3) where supp{a(x, y)} = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let h > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Consider the sets Ak = � x ∈ U : 2k λ ≤ |f(x)| ≤ 2k+1 λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' For the measure of a set of smaller values we use Lemma 1 ′ in the paper [16] (see also Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='4), and we have µ � |f(x)| ≤ 2k+1 λ , x ∈ U � ≤ C �2k+1 λ � 1 h � ln ���� λ 2k+1 ���� �m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let Ik := � Ak a(x, y) 1 + λ|f(x, y)|dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' For the integral � 2k≤λ|f(x)|≤2k+1 Ik = � Ω2 a(x, y) 1 + λ|f(x, y)|dxdy, 8 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='IKROMOV, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='RUZHANSKY, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='SAFAROV we find the following estimate: |Ik| = ���� � Ak a(x, y) 1 + λ|f(x, y)|dxdy ���� ≤ C∥a∥L∞(U) �2k+1 λ � 1 h ����ln 2k+1 λ ���� m 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' From here we find the sum of Ik and, by estimating the integral I, we get I ≤ ∥a∥L∞(U) ∞ � k=1 Ik ≤ ∥a∥L∞(U) ∞ � k=1 �2k+1 λ � 1 h ����ln 2k+1 λ ���� m 2−k ≤ ∥a∥L∞(U) | ln λ|m λ 1 h ∞ � k=1 2 k+1 h −kkm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' As h > 1, the last series is convergent, proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Consider the case h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The smooth function has non-degenerate critical point at the origin if and only if h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' As f(x, y) is a smooth function with ∇f(0, 0) = 0, using Morse lemma we have f ∼ x2 ± y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' So in this case we estimate two sets ∆ = ∆1 ∪ ∆2, where ∆1 := {(x, y) : λ|x2 ± y2| ≤ M, |x| ≤ 1, |y| ≤ 1} and ∆2 := {(x, y) : λ|x2 ± y2| > M, |x| ≤ 1, |y| ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' First we consider the integral over the set ∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Then we have ���� � ∆1 a(x, y) 1 + λ|x2 ± y2|dxdy ���� ≤ C∥a∥L∞(∆1) ���� � ∆1 dxdy ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Now we estimate the last integral as ���� � λ|x2+y2|≤M dxdy ���� ≤ C λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Then we estimate the measure of the set {|x2 − y2| ≤ εM}, where ε = 1 λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' for simplicity putting M = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' ���� � |x2−y2|≤εM dxdy ���� ≤ C ����� � √1−ε √ε dy � √ y2+ε √ y2−ε dx ����� = ����� � √1−ε √ε �� y2 + ε − � y2 − ε � dy ����� = = �y 2 � y2 + ε + ε 2 ln |y + � y2 + ε| � ��� √1−ε √ε − �y 2 � y2 − ε − ε 2 ln |y + � y2 − ε| � ��� √1−ε √ε = = ����� √1 − ε 2 + ε 2 ln √1 − ε + 1 √ε − √ 2 2 ε − ε 2 ln |√ε(1 + √ 2)|− − �� (1 − ε)(1 − 2ε) 2 − ε 2 ln | √ 1 − ε + √ 1 − 2ε| + ε 2 ln √ε| ������ ≤ Cε ln ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Now we consider the integral over the set ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' In this case we change the variables to polar coordinate system and with easy calculating we get ���� � {λ|x2+y2|≥M} a(x, y) 1 + λ|x2 + y2|dxdy ���� ≤ C| ln λ|∥a∥L∞(∆2) λ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='4) and ���� � {λ|x2−y2|≥M} a(x, y) 1 + λ|x2 − y2|dxdy ���� ≤ C| ln λ|2∥a∥L∞(∆2) λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='5) OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS 9 Now we continue the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let h > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' We use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2 for the integral J1, to get |J1| ≤ C| ln λ|m∥a∥L∞(U) λ 1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Let consider the integral J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If h > 1, then using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1 we get |J2| ≤ C| ln λ|∥a∥L∞(U) λ 1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If h = 1, using the Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2 we get the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='2 shows that if h = 1, we can get a more precise result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' If h = 1 and f has an extremal point at the point (0,0) (then f is diffeomorhic equivalent to x2 1 + x2 2 or −x2 1 − x2 2), then we have |Iα,β| ≤ C| ln λ|∥ψ∥L∞(U) λ , for all λ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Declaration of competing interest This work does not have any conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Acknowledgements The second author was supported in parts by the FWO Odysseus 1 grant G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='0H94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021) and also supported by EPSRC grant EP/R003025/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' The manuscript has no associated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Agarwal, A propos d’une note de 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Torebek, Van der Corput lemmas for Mittag-Leffler functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' α−directions , Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=', 171 (2021), 103016, 23 pp.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=', 15(4) (2022), P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='459-–466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Varchenko, Newton polyhedra and estimation of oscillating integrals //Functional Analysis and Its Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' 10, pages 175-–196 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' [34] Van der Korput, Zur Methode der stationaren phase// Compositio Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' 1934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' 15-38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS 11 Isroil A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Ikromov V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content=' Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan Olmazor district, University 46, Tashkent, Uzbekistan Samarkand State University Department of Mathematics, 15 University Boulevard Samarkand, 140104, Uzbekistan Email address: ikromov1@rambler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='ru Michael Ruzhansky Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University, Krijgslaan 281, Ghent, Belgium, School of Mathematical Sciences, Queen Mary University of London, United Kingdom Email address: michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='ruzhansky@ugent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='be Akbar R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='Safarov Uzbek-Finnish Pedagogical Institute Spitamenshox 166, Samarkand, Uzbekistan Samarkand State University Department of Mathematics, 15 University Boulevard Samarkand, 140104, Uzbekistan Email address: safarov-akbar@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} +page_content='ru' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'} diff --git a/.gitattributes b/.gitattributes index 58b36d4b7dbc3b05a5c4f92e3240ef9a1510c237..ea05126d8697cac3b28b17746e8d3c69c539f3db 100644 --- a/.gitattributes +++ b/.gitattributes @@ -3731,3 +3731,67 @@ stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf filter=lfs diff=lfs merge=lfs -tex sdE0T4oBgHgl3EQfbAAV/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text B9E5T4oBgHgl3EQfTg8R/content/2301.05536v1.pdf filter=lfs diff=lfs merge=lfs -text stE2T4oBgHgl3EQffQeV/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf filter=lfs diff=lfs merge=lfs -text +zNFRT4oBgHgl3EQfjjf7/vector_store/index.faiss filter=lfs 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+through the mask +Madeline Rachow∗, Thomas Karnowski† and Alice J. O’Toole‡ +∗University of Arkansas +† Oak Ridge National Laboratory +∗ The University of Texas at Dallas +Email: ∗mrachow@uark.edu, †karnowskitp@ornl.gov, ‡otoole@utdallas.edu +Abstract—Face identity masking algorithms developed in re- +cent years aim to protect the privacy of people in video +recordings. These algorithms are designed to interfere with +identification, while preserving information about facial actions. +An important challenge is to preserve subtle actions in the eye +region, while obscuring the salient identity cues from the eyes. We +evaluated the effectiveness of identity-masking algorithms based +on Canny filters, applied with and without eye enhancement, for +interfering with identification and preserving facial actions. In +Experiments 1 and 2, we tested human participants’ ability to +match the facial identity of a driver in a low resolution video to +a high resolution facial image. Results showed that both masking +methods impaired identification, and that eye enhancement did +not alter the effectiveness of the Canny filter mask. In Experiment +3, we tested action preservation and found that neither method in- +terfered significantly with driver action perception. We conclude +that relatively simple, filter-based masking algorithms, which are +suitable for application to low quality video, can be used in +privacy protection without compromising action perception. +Index Terms—identity-masking, face recognition, privacy, hu- +man visual perception, driver behavior, de-identification, action +preservation. +I. INTRODUCTION +Video recordings for security and surveillance are now +ubiquitous in public and private spaces. This has lead to a +pressing need to develop face identity masking algorithms +aimed at protecting the privacy of people in the recordings. +Facial identity masking technology also needs to preserve +the facial actions (gestures and expressions) of those being +photographed for applications that require action classification +without identification. Understanding and measuring the extent +to which identity-masking algorithms effectively accomplish +both goals is a challenging problem. Because identification +and action classification are tasks that can be done accurately +by humans, the success of masking algorithms cannot be eval- +uated comprehensively without examining human perception. +Human identification and gesture categorization of identity- +masked faces have been examined previously [1]. The effec- +tiveness of eight different identity masking algorithms was +evaluated using human perception and a deep convolutional +neural network (DCNN) trained for face identification. Human +participants and the DCNN were tested with videos taken of +drivers actively operating a motor vehicle. For the human ex- +periment, people studied high-resolution images of the drivers +to learn their identities and were tested on their recognition +of those drivers in low-resolution videos. Test videos were +low resolution and showed drivers actively operating a motor +vehicle. Videos were either unmasked or masked by one +of eight algorithms, including methods that rely on Facial +Action Transfer (FAT) (cf., [2], [3]), a DMask [4], Canny +filtering [5], and Scharr filtering [6]. The results showed that +all of the algorithms reduced human face recognition accuracy. +Moreover, people made their recognition decisions with a +conservative response bias (i.e., a tendency to indicate that +they did not recognize drivers, when they were uncertain). +This bias indicates that the participants had low confidence in +their identification decisions—supporting the effectiveness of +the masking methods. +In the machine evaluation of that test [1], the DCNN +matched identities between the high-resolution images and +masked videos, and between the unmasked and masked +videos. DCNN performance matching high-resolution images +to masked and unmasked videos showed a pattern of poor +performance approximately comparable to human behavior— +echoing the effectiveness of the masking algorithms for both +humans and the CNN. The results showed that even simple +methods, such as edge-detection, can impair identification +performance. +It is worth noting that more sophisticated methods than +filtering have been developed for identity masking, including +generative adversarial networks, GANS (e.g., [7]). However, +these techniques can only be applied to high quality (frontal) +images and are computationally intense, which limits their util- +ity for high volume throughput (e.g., videos). Many important +applications of face identity masking must deal with large +quantities of low resolution, poor quality video. Therefore, +there is a need to consider the effectiveness of simpler methods +that can be applied in these less controlled circumstances. +The present work builds on previous work [1], with the +goal of looking more carefully at the role the eyes play in +facilitating face recognition in the context of identity mask- +ing. Simple filtering operations can preserve eye information, +which is both valuable for gesture recognition, but may also +inadvertently boost face recognition by people. Specifically, +in human perception experiments, the eye region of the face +is known to support particularly good face recognition (e.g., +arXiv:2301.08408v1 [cs.CV] 20 Jan 2023 + +IDENTITY MASKING WITH EYE ENHANCEMENT +2 +Fig. 1: Example stimuli from the mask conditions. a. Canny+Eyezoom; b. (left) Unmasked, (right) Canny +[8]). In this study, we tested whether eye enhancement of an +identity masked face would increase human face identification +performance. To that end, we created a set of stimuli in which +the eye region was localized, expanded in size, and enhanced +with a Scharr filter [6]. We compared face identification in +three masking conditions: 1.) unmasked driver videos, 2.) +driver videos masked with the Canny method [5], and 3.) +a combination method that showed the Canny-masked video +with an inset of the Scharr-enhanced eye region. See Figure 1 +for an example of the stimulus conditions. Note that we chose +the Canny method filter for our masking algorithm, because +it is relatively simple, easy to implement, and is effective for +both identity-masking and action preservation [1]. +In the first and second experiments, we focused on the +effectiveness of identity masking. Videos were either shown +unmasked (unaltered), masked solely with Canny, or masked +with Canny and Canny+EyeZoom (see details, section II-B). +The third experiment examined action preservation in these +conditions. +A. Study contributions +• Masking the face of a driver in a video using a Canny +filter effectively impairs face identification by comparison +to an unmasked video. +• Enhancing and enlarging the eye region (Eyezoom of the +face) and masking it with a Schaar filter does not alter +the effectiveness of the Canny filter mask. +• Facial actions are preserved, in large part, when drivers’ +faces are masked with both the Canny and Canny + +Eyezoom manipulations. +II. METHODS +A. Dataset +Stimuli for the present experiment came from a set of +driver videos in the Head Pose Validation (HPV) database. +The HPV dataset was created to emulate data from the SHRP2- +Naturalistic Driving Study (SHRP2-NDS) database [9], which +is not easily available for research applications. The SHRP2- +NDS database is nearly unique in the range of imaging con- +ditions encompassed in the data. It includes approximately 2 +petabytes of video from approximately 3, 400 drivers obtained +over 1 to 2 years of observation. However, the dynamic video +nature of the dataset provides for highly salient, personally +identifiable, information about the drivers. The dataset is +characterized by extreme illumination conditions (e.g., night- +time shadowing, day-time bright spots, or illumination via +transient headlights as a car turns). There is also the problem +of quick driver movements (e.g., head turns and other actions +which are very common in real-world driving). +The HPV dataset used in the present study includes low- +resolution videos of people actively driving a car or performing +staged actions typical while driving, such as using a cellphone +and putting on headwear or glasses. The video resolution is +356 × 240 pixels, with a frame rate of 14.98 frames per second. +Each video segment was edited to 4s and masks were applied +to the segments for direct comparison of mask effectiveness +across conditions. Video length ranged from 1-4s depending on +the type of action (looking left, looking right, looking down). +The video segment lengths were identical for each identity +across conditions. +B. Conditions +The three masking conditions tested were implemented, as +follows: + +a +b.IDENTITY MASKING WITH EYE ENHANCEMENT +3 +• unmasked - drivers’ faces were unaltered. +• Canny mask - drivers’ faces were altered by applying +a series of processes aimed at producing optimal edge +detection, including the use of a Gaussian smoothing +filter, a set of gradient-based edge detectors to enhance +edges in the image, and then non-maximum suppression, +threshold, and tracking to produce thin, refined edges. +• Eyezoom condition– drivers’ faces were first masked +with the Canny process. Then the eyes were detected in +the original image using the retinaface algorithm [10]. +The original image was then expanded and masked with a +Schaar filter, and the region around the eye detection was +cropped. Finally the Canny-masked face was presented +in an inset showing the Schaar-filtered, zoomed eyes (see +Fig. II-B). +III. EXPERIMENT I: EFFECT OF EYEZOOM MASKING +METHOD +In Experiment 1, we investigated the effectiveness of the +Canny and Canny+Eyezoom filters at masking the identities +of drivers in low-resolution videos. +A. Participants +A total of 30 (11 male, 18 female, 1 other) undergraduate +student volunteers (ages 18-34) from the University of Texas +at Dallas (UTD) participated in the study in exchange for +research credit. All human experimental procedures were +approved by UTD’s Institutional Review Board. +B. Procedure +The experiment was composed of 72 trials in which a video +stimulus was displayed in the top center of the screen. The +response options were presented below the video and showed +two faces and silhouette (see Figure 2). Participants were asked +to select the face image that matched the driver in the video or +to select the silhouette if neither of the two images matched the +driver. In target-present trials (n = 36), one of the two faces +matched the driver. In target-absent trials (n = 36), neither of +the two faces matched the driver. In all cases, the two face +images presented as options showed similar-looking identities +from the dataset. Each of the dataset’s 36 identities was shown +twice, once with the correct response being one of the target- +present choices and once with the correct response being the +target-absent choice. +The video segments were shown in random order and looped +until the subjects responded. Subjects were assigned randomly +to one of the three masking conditions, with the unmasked +condition serving as a control for general recognition success. +Subjects were asked to determine whether the identity in the +video matched one of the two identity images shown or if the +identity was absent from the identity images shown. +C. Outcome Measures +1) Accuracy: Accuracy was assessed in two ways using a +signal detection-type calculation based on d’. This measure +depends on the proportion of hits p(hit) and false alarms +p(false alarms), as follows: +d′ = z(p(hit)) − z(p(false alarms), +where the z refers to the z-score. +In this experiment, hits were defined as target-present trials +in which participants correctly recognized a driver as the +matched-identity response choice. The design of the response +options in the experiment offered two ways to compute false +alarms. Specifically, false alarms can be defined as: a.) target- +present trials in which the participant choose the incorrect +identity; and/or b.) incorrect target-absent trials in which +neither image showed the identity (i.e., participants chose one +of the face images, when neither was an identity match to the +video). Because both options are consistent with the concept +of a false alarm, in what follows, we included both types of +false alarms (a and b) in the accuracy computation. +D. Results +1) Accuracy: Figure 3 shows the average d’ for each mask +condition. These values indicate that faces in the unmasked +condition were identified moderately well, but face recognition +in both masked conditions was significantly impaired. The +negative d’ values for the masked conditions are unusual +and suggest that participants used a systematically incorrect +decision strategy, which we will investigate further in Section +III-D2. +A one-factor Analysis of Variance (ANOVA) was performed +on accuracy (d’), with mask condition as the independent +variable. The resulting model yielded a main effect of mask +condition on d’, F(2, 27) = 11.03, p < .001. When comparing +the conditions, d’ accuracy was significantly higher in the +unmasked condition than in the masked conditions, with no +significant difference between the two masked conditions. This +suggests that Canny and ORNL masking are not significantly +less effective when used together than Canny masking alone. +As is clear from the Fig. 3, participant performance was more +variable in the Eyezoom condition. +2) Response Distribution: To further investigate the finding +of negative d’s, we examined the proportion of responses +allocated to each response type (face images chosen versus +no identity chosen). The pattern of responses is shown for +each mask type in Figure 4, with separate graphs for target- +present (correct identity was available as a choice) and target- +absent (correct identity was not available as a choice) trials. +For the unmasked condition, the graphs show a standard +(relatively accurate) pattern of responses as a function of +whether the target was present or absent. The graphs for +the masked conditions show inaccurate performance, but also +suggest that participants did not systematically choose the no- +identity match when a match was present, but instead often +chose the wrong face as the identity match. +We conclude tentatively that performance in the masked +conditions was very poor indicating the effectiveness of the +masks for preventing identification. However, given the un- +usual performance in the masked condition (i.e., negative d’s), + +IDENTITY MASKING WITH EYE ENHANCEMENT +4 +Fig. 2: Example trial in Experiment 1. +Fig. 3: Experiment 1 accuracy, measured as d’ across +conditions. Results show that both masking algorithms were +equally effective. +we retested the conditions with a design that eliminates the +possibility of response bias. +IV. EXPERIMENT II: EFFECT OF EYEZOOM MASKING +METHOD WITH A FORCED-CHOICE TASK +In this experiment, we used a two-alternative forced choice +(2AFC) task to test masking effectiveness. In the 2AFC, two +faces are presented as response options. In all cases, one of +the two images will be the same identity as the person in the +video. +A. Participants +A total of 30 (7 male, 22 female, 1 other) undergraduate +student volunteers (ages 18-26) from UTD participated in the +study in exchange for research credit. +B. Procedure +The experiment consisted of 72 trials. The video stimulus +was displayed in the top center of the screen with the two +face images beneath it. Participants were asked to determine +which of the two face images matched the identity shown +in the video. To make the task challenging, the two faces +presented had a similar appearance and were of the same race +and gender. An example trial is shown in Fig. 5. +Each of the dataset’s 36 identities was shown twice, once +with the correct response as the left-located option and once +with the correct response as the right-located option. The +video segments were shown in random order and looped until +the subjects responded. Participants were assigned randomly +to one of the three masking conditions, with the unmasked +condition serving as a baseline condition for identification +accuracy. +C. Results +Accuracy was assessed as the proportion of correct re- +sponses. Fig. 6 shows the proportion of correct responses +for each mask condition. These values indicate that face +recognition in the unmasked condition was more accurate +than face recognition in the masked conditions. A one-factor +ANOVA was performed on the accuracy data (proportion of +correct responses), with condition as the independent variable. +The model yielded a main effect of mask condition on + +? +Press "" if the person in the +Press "2" if the person in the +Press "3" if the person in the video +video is the person on the left. +video is the person in the middle. +is NOT either of the two people picturedcondition +unmasked +canny +eyezoom +-1 +-2 +conditionIDENTITY MASKING WITH EYE ENHANCEMENT +5 +Fig. 4: Proportion of responses by trial type in Experiment I. +proportion of correct responses, F(2, 27) = 9.68, p < .001. +As in the first experiment, participants were more accurate in +the unmasked condition than in the masked conditions, and +performed comparably for the two masked conditions. +The results replicate the pattern of performance across +conditions found for Experiment 1. As expected with a 2AFC +task, performance was more accurate in all three conditions +than it was in Experiment 1. Notably, average identification +was above chance in both masked conditions. Performance in +the Eyezoom condition was more variable than performance +in the Canny mask condition—replicating a similar finding in +Experiment I. +We conclude that the masks strongly inhibit identification, +but that when forced to guess between two images (with the +assurance that one was an identity match), participants fared +better than chance. Notwithstanding, applications of identity +masking would rarely if ever be able to assure a human or ma- +chine system that one of two candidates was an identity match. +Our goal in applying this method here was to test examine the +role of response bias in the unusual pattern of results found in +Experiment 1. The present results suggest that these masking +algorithms leave behind some residual identity information in +the face that humans can exploit when the response decision +is highly constrained. As noted, it is unlikely that that would + +Unmasked Target Present Trials +Unmasked Target Absent Trials +1.00 +response +1.00 +response +responses +chose either identity +chose either identity +0.75 +chose no identity +chose no identity +09'0 9. +prop +prop +0.00 +0.00 +response +response +Canny Target Present Trials +Canny Target Absent Trials +1.00 +response +1.00 +response +responses +chose either identity +chose either identity +0.75 + chose no identity +chose no identity +090 9 +pro +pro +0.00 +0.00 +response +response +Eyezoom Target Present Trials +Eyezoom Target Absent Trials +1.00 +response +1.00 +response +chose eitheridentity +chose either identity +chose noidentit +chose no identity +res +pro +pro +0.00 +0.00 +response +responseIDENTITY MASKING WITH EYE ENHANCEMENT +6 +Fig. 5: Example trial from Experiment II. +Fig. 6: Experiment 2 - identification accuracy across +conditions. +be the case in any applied scenario, and so we conclude that +these simple simple filtering procedures provide a reasonably +high degree of identity protection. Additionally, we conclude, +albeit more tentatively, that the eyezoom procedure does not +improve identification significantly over the Canny procedure. +V. EXPERIMENT III: EFFECT OF EYEZOOM MASKING +METHOD ON ACTION PRESERVATION +The effectiveness of the identity protection provided by +these masks opens the question of whether this protection +comes at the cost of preserving information about facial +actions. In this experiment, we examined whether the Canny +and Canny+Eyezoom mask conditions impaired driver facial +action perception. +A. Participants +A total of 30 (6 male, 23 female, 1 nonbinary) undergradu- +ate student volunteers (ages 18-30) from UTD participated in +the study in exchange for research credit. +B. Procedure +The experiment consisted of 100 trials, each with three +response options: a.) driver looking to the left, 2.) driver +looking to the right, and 3.) driver looking down. Each of the +36 identities in the dataset appeared between two and three +times, each with a different action (looking right, left, down). +Prior to the start of the main experiment, a pilot test with +only the unmasked condition was conducted to ensure that the +actions were identifiable in all videos. This test resulted in the +elimination of eight (of 108) videos segments in which actions +were not recognizable at sufficiently high levels of accuracy +for inclusion in the action preservation study. +The participants were assigned randomly to one of three +masking conditions with the unmasked condition providing a +baseline action recognition accuracy and were asked to identify +whether the driver was looking to the left, right, or down. The +video stimuli were shown in the upper center of the screen +with three written options below. See Fig. 7 for an example +trial. The clips were played in a random order and looped until +the participant responded. +C. Results +The proportion of correct responses was used to assess accu- +racy. Fig. 8 shows the proportion of correct responses for each +mask condition. These values indicate that action preservation + +Press"1" if the person in the +Press "2" if the person in the +videoisthepersonontheleft +video is the person in the rightcondition +0.9 +unmasked +canny +eyezoom +I of correct response +0.7 +ortion +propor +0.6 +0.5 +conditionIDENTITY MASKING WITH EYE ENHANCEMENT +7 +Fig. 7: Example trial from Experiment III. +was generally high, but also suggest a small advantage for +action perception in the unmasked condition. A one-factor +ANOVA, performed on the accuracy (proportion of correct +responses) data, with the independent variable of condition, +did not show a significant effect, but was generally consistent +with this conclusion. The model yielded a marginal main +effect of mask condition on proportion of correct responses, +F(2, 27) = 2.69, p = 0.086. This suggests a very slight +advantage for action perception without stimulus masking. +In conclusion, although the results did not reach statistical +significance, there is some indication that masking impaired +action perception. +VI. DISCUSSION +Our goal was to examine the effectiveness of simple +Canny-filtering based masking methods, with and without eye +enhancement, for interfering with face identification while +preserving facial actions. In Experiment I, face recognition +accuracy was diminished for both mask conditions, relative +to the unmasked condition. There was no difference between +the Canny mask alone and the mask with eye enhancement. In +Experiment II, we replicated this result with a 2AFC procedure +that controlled for response option bias, which may have been +a factor in the findings of negative ’. values for both masking +conditions. In combination, both studies point to the relative +effectiveness of the masks for interfering with identification. +They also point to the conclusion that eye enhancement did +not alter this effectiveness. Experiment III showed that facial +actions were preserved to a similar degree with both masks, +Fig. 8: ANOVA of proportion of correct responses. +though there was a marginal advantage for action perception +in the unmasked condition. +In summary, these results indicate that Eyezoom masking +does not significantly increase identification or alter facial +action preservation. +ACKNOWLEDGMENT +This work was supported through collaboration with Oak +Ridge National Laboratory and the Federal Highway Admin- + +Press "1" if the person looks toward the driver's side. +Press "2" if the person looks toward the passenger's side. +Press "3" if the person looks down.condition +unmasked +canny +eyezoom +f correct responses +0.96 +0.92 +of +proportion +0.88 +un +ma +ez +conditionIDENTITY MASKING WITH EYE ENHANCEMENT +8 +istration under the Exploratory Advanced Research Program. +The human experiment and analysis was subcontracted to +the University of Texas at Dallas from Oak Ridge National +Laboratory. +This manuscript has been authored in part by UT-Battelle, +LLC, under contract DE-AC05-00OR22725 with the US De- +partment of Energy (DOE). The US government retains and +the publisher, by accepting the article for publication, acknowl- +edges that the US government retains a nonexclusive, paid-up, +irrevocable, worldwide license to publish or reproduce the pub- +lished form of this manuscript, or allow others to do so, for US +government purposes. DOE will provide public access to these +results of federally sponsored research in accordance with the +DOE Public Access Plan (http://energy.gov/downloads/doe- +public-access-plan). +REFERENCES +[1] K. D. O. Hooge, A. Baragchizadeh, T. P. Karnowski, D. S. Bolme, +R. Ferrell, P. R. Jesudasen, C. D. Castillo, and A. J. O’toole, “Evaluating +automated face identity-masking methods with human perception and +a deep convolutional neural network,” ACM Transactions on Applied +Perception (TAP), vol. 18, no. 1, pp. 1–20, 2020. +[2] D. Huang and F. De La Torre, “Facial action transfer with personalized +bilinear regression,” in Computer Vision–ECCV 2012. +Springer, 2012, +pp. 144–158. +[3] X. Xiong and F. De la Torre, “Supervised descent method and its +applications to face alignment,” in Proceedings of the IEEE conference +on computer vision and pattern recognition, 2013, pp. 532–539. +[4] Federal +Highway +Administration +Active +Project: +Exploratory +Advanced +Research +Program, +“DMask: +A +reliable +identity +masking +system +for +driver +safety +video +data.” +FHWA-PROJ- +14-0054, 2014-2016. [Online]. Available: https://highways.dot.gov/ +dmask-reliable-identity-masking-system-driver-safety-video-data +[5] J. Canny, “A computational approach to edge detection,” IEEE Transac- +tions on pattern analysis and machine intelligence, no. 6, pp. 679–698, +1986. +[6] B. J¨ahne, H. Scharr, and S. K¨orkel, “Principles of filter design,” +Handbook of computer vision and applications, vol. 2, pp. 125–151, +1999. +[7] M. H. Khojaste, N. M. Farid, and A. Nickabadi, “Gmfim: A generative +mask-guided facial image manipulation model for privacy preservation,” +2022. +[8] J. Royer, C. Blais, I. Charbonneau, K. D´ery, J. Tardif, B. Duchaine, +F. Gosselin, and D. Fiset, “Greater reliance on the eye region predicts +better face recognition ability,” Cognition, vol. 181, pp. 12–20, 2018. +[9] M. Perez, S. Mclaughlin, T. Kondo, J. Antin, J. Mcclafferty, S. Lee, +J. Hankey, and T. Dingus, “Transportation safety meets big data: the +shrp 2 naturalistic driving database,” Journal of the Society of Instrument +and Control Engineers, no. 55.5, pp. 415–421, 2016. +[10] J. Deng, J. Guo, E. Ververas, I. Kotsia, S. Zafeiriou, and I. FaceSoft, +“Retinaface: Single-shot multi-level face localization in the wild,” Pro- +ceedings of the IEEE/CVF conference on computer vision and pattern +recognition, 2020. + diff --git a/0dFAT4oBgHgl3EQfCRwI/content/tmp_files/load_file.txt b/0dFAT4oBgHgl3EQfCRwI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e1d1bdd1d90e6dfe15788aea2e4b2c3c0129536 --- /dev/null +++ b/0dFAT4oBgHgl3EQfCRwI/content/tmp_files/load_file.txt @@ -0,0 +1,348 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf,len=347 +page_content='IDENTITY MASKING WITH EYE ENHANCEMENT 1 Identity masking effectiveness and gesture recognition: Effects of eye enhancement in seeing through the mask Madeline Rachow∗, Thomas Karnowski† and Alice J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' O’Toole‡ ∗University of Arkansas † Oak Ridge National Laboratory ∗ The University of Texas at Dallas Email: ∗mrachow@uark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='edu, †karnowskitp@ornl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='gov, ‡otoole@utdallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='edu Abstract—Face identity masking algorithms developed in re- cent years aim to protect the privacy of people in video recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' These algorithms are designed to interfere with identification, while preserving information about facial actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' An important challenge is to preserve subtle actions in the eye region, while obscuring the salient identity cues from the eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' We evaluated the effectiveness of identity-masking algorithms based on Canny filters, applied with and without eye enhancement, for interfering with identification and preserving facial actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In Experiments 1 and 2, we tested human participants’ ability to match the facial identity of a driver in a low resolution video to a high resolution facial image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Results showed that both masking methods impaired identification, and that eye enhancement did not alter the effectiveness of the Canny filter mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In Experiment 3, we tested action preservation and found that neither method in- terfered significantly with driver action perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' We conclude that relatively simple, filter-based masking algorithms, which are suitable for application to low quality video, can be used in privacy protection without compromising action perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Index Terms—identity-masking, face recognition, privacy, hu- man visual perception, driver behavior, de-identification, action preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' INTRODUCTION Video recordings for security and surveillance are now ubiquitous in public and private spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' This has lead to a pressing need to develop face identity masking algorithms aimed at protecting the privacy of people in the recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Facial identity masking technology also needs to preserve the facial actions (gestures and expressions) of those being photographed for applications that require action classification without identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Understanding and measuring the extent to which identity-masking algorithms effectively accomplish both goals is a challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Because identification and action classification are tasks that can be done accurately by humans, the success of masking algorithms cannot be eval- uated comprehensively without examining human perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Human identification and gesture categorization of identity- masked faces have been examined previously [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The effec- tiveness of eight different identity masking algorithms was evaluated using human perception and a deep convolutional neural network (DCNN) trained for face identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Human participants and the DCNN were tested with videos taken of drivers actively operating a motor vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' For the human ex- periment, people studied high-resolution images of the drivers to learn their identities and were tested on their recognition of those drivers in low-resolution videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Test videos were low resolution and showed drivers actively operating a motor vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Videos were either unmasked or masked by one of eight algorithms, including methods that rely on Facial Action Transfer (FAT) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=', [2], [3]), a DMask [4], Canny filtering [5], and Scharr filtering [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The results showed that all of the algorithms reduced human face recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Moreover, people made their recognition decisions with a conservative response bias (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=', a tendency to indicate that they did not recognize drivers, when they were uncertain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' This bias indicates that the participants had low confidence in their identification decisions—supporting the effectiveness of the masking methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In the machine evaluation of that test [1], the DCNN matched identities between the high-resolution images and masked videos, and between the unmasked and masked videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' DCNN performance matching high-resolution images to masked and unmasked videos showed a pattern of poor performance approximately comparable to human behavior— echoing the effectiveness of the masking algorithms for both humans and the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The results showed that even simple methods, such as edge-detection, can impair identification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' It is worth noting that more sophisticated methods than filtering have been developed for identity masking, including generative adversarial networks, GANS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=', [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' However, these techniques can only be applied to high quality (frontal) images and are computationally intense, which limits their util- ity for high volume throughput (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=', videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Many important applications of face identity masking must deal with large quantities of low resolution, poor quality video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Therefore, there is a need to consider the effectiveness of simpler methods that can be applied in these less controlled circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The present work builds on previous work [1], with the goal of looking more carefully at the role the eyes play in facilitating face recognition in the context of identity mask- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Simple filtering operations can preserve eye information, which is both valuable for gesture recognition, but may also inadvertently boost face recognition by people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Specifically, in human perception experiments, the eye region of the face is known to support particularly good face recognition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=', arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='08408v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='CV] 20 Jan 2023 IDENTITY MASKING WITH EYE ENHANCEMENT 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 1: Example stimuli from the mask conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Canny+Eyezoom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' (left) Unmasked, (right) Canny [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In this study, we tested whether eye enhancement of an identity masked face would increase human face identification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' To that end, we created a set of stimuli in which the eye region was localized, expanded in size, and enhanced with a Scharr filter [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' We compared face identification in three masking conditions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=') unmasked driver videos, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=') driver videos masked with the Canny method [5], and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=') a combination method that showed the Canny-masked video with an inset of the Scharr-enhanced eye region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' See Figure 1 for an example of the stimulus conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Note that we chose the Canny method filter for our masking algorithm, because it is relatively simple, easy to implement, and is effective for both identity-masking and action preservation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In the first and second experiments, we focused on the effectiveness of identity masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Videos were either shown unmasked (unaltered), masked solely with Canny, or masked with Canny and Canny+EyeZoom (see details, section II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The third experiment examined action preservation in these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Study contributions Masking the face of a driver in a video using a Canny filter effectively impairs face identification by comparison to an unmasked video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Enhancing and enlarging the eye region (Eyezoom of the face) and masking it with a Schaar filter does not alter the effectiveness of the Canny filter mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Facial actions are preserved, in large part, when drivers’ faces are masked with both the Canny and Canny + Eyezoom manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Dataset Stimuli for the present experiment came from a set of driver videos in the Head Pose Validation (HPV) database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The HPV dataset was created to emulate data from the SHRP2- Naturalistic Driving Study (SHRP2-NDS) database [9], which is not easily available for research applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The SHRP2- NDS database is nearly unique in the range of imaging con- ditions encompassed in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' It includes approximately 2 petabytes of video from approximately 3, 400 drivers obtained over 1 to 2 years of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' However, the dynamic video nature of the dataset provides for highly salient, personally identifiable, information about the drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The dataset is characterized by extreme illumination conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=', night- time shadowing, day-time bright spots, or illumination via transient headlights as a car turns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' There is also the problem of quick driver movements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=', head turns and other actions which are very common in real-world driving).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The HPV dataset used in the present study includes low- resolution videos of people actively driving a car or performing staged actions typical while driving, such as using a cellphone and putting on headwear or glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The video resolution is 356 × 240 pixels, with a frame rate of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='98 frames per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Each video segment was edited to 4s and masks were applied to the segments for direct comparison of mask effectiveness across conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Video length ranged from 1-4s depending on the type of action (looking left, looking right, looking down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The video segment lengths were identical for each identity across conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Conditions The three masking conditions tested were implemented, as follows: a b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='IDENTITY MASKING WITH EYE ENHANCEMENT 3 unmasked - drivers’ faces were unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Canny mask - drivers’ faces were altered by applying a series of processes aimed at producing optimal edge detection, including the use of a Gaussian smoothing filter, a set of gradient-based edge detectors to enhance edges in the image, and then non-maximum suppression, threshold, and tracking to produce thin, refined edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Eyezoom condition– drivers’ faces were first masked with the Canny process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Then the eyes were detected in the original image using the retinaface algorithm [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The original image was then expanded and masked with a Schaar filter, and the region around the eye detection was cropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Finally the Canny-masked face was presented in an inset showing the Schaar-filtered, zoomed eyes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' EXPERIMENT I: EFFECT OF EYEZOOM MASKING METHOD In Experiment 1, we investigated the effectiveness of the Canny and Canny+Eyezoom filters at masking the identities of drivers in low-resolution videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Participants A total of 30 (11 male, 18 female, 1 other) undergraduate student volunteers (ages 18-34) from the University of Texas at Dallas (UTD) participated in the study in exchange for research credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' All human experimental procedures were approved by UTD’s Institutional Review Board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Procedure The experiment was composed of 72 trials in which a video stimulus was displayed in the top center of the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The response options were presented below the video and showed two faces and silhouette (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Participants were asked to select the face image that matched the driver in the video or to select the silhouette if neither of the two images matched the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In target-present trials (n = 36), one of the two faces matched the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In target-absent trials (n = 36), neither of the two faces matched the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In all cases, the two face images presented as options showed similar-looking identities from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Each of the dataset’s 36 identities was shown twice, once with the correct response being one of the target- present choices and once with the correct response being the target-absent choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The video segments were shown in random order and looped until the subjects responded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Subjects were assigned randomly to one of the three masking conditions, with the unmasked condition serving as a control for general recognition success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Subjects were asked to determine whether the identity in the video matched one of the two identity images shown or if the identity was absent from the identity images shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Outcome Measures 1) Accuracy: Accuracy was assessed in two ways using a signal detection-type calculation based on d’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' This measure depends on the proportion of hits p(hit) and false alarms p(false alarms), as follows: d′ = z(p(hit)) − z(p(false alarms), where the z refers to the z-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In this experiment, hits were defined as target-present trials in which participants correctly recognized a driver as the matched-identity response choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The design of the response options in the experiment offered two ways to compute false alarms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Specifically, false alarms can be defined as: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=') target- present trials in which the participant choose the incorrect identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' and/or b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=') incorrect target-absent trials in which neither image showed the identity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=', participants chose one of the face images, when neither was an identity match to the video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Because both options are consistent with the concept of a false alarm, in what follows, we included both types of false alarms (a and b) in the accuracy computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Results 1) Accuracy: Figure 3 shows the average d’ for each mask condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' These values indicate that faces in the unmasked condition were identified moderately well, but face recognition in both masked conditions was significantly impaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The negative d’ values for the masked conditions are unusual and suggest that participants used a systematically incorrect decision strategy, which we will investigate further in Section III-D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' A one-factor Analysis of Variance (ANOVA) was performed on accuracy (d’), with mask condition as the independent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The resulting model yielded a main effect of mask condition on d’, F(2, 27) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='03, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' When comparing the conditions, d’ accuracy was significantly higher in the unmasked condition than in the masked conditions, with no significant difference between the two masked conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' This suggests that Canny and ORNL masking are not significantly less effective when used together than Canny masking alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' As is clear from the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 3, participant performance was more variable in the Eyezoom condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 2) Response Distribution: To further investigate the finding of negative d’s, we examined the proportion of responses allocated to each response type (face images chosen versus no identity chosen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The pattern of responses is shown for each mask type in Figure 4, with separate graphs for target- present (correct identity was available as a choice) and target- absent (correct identity was not available as a choice) trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' For the unmasked condition, the graphs show a standard (relatively accurate) pattern of responses as a function of whether the target was present or absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The graphs for the masked conditions show inaccurate performance, but also suggest that participants did not systematically choose the no- identity match when a match was present, but instead often chose the wrong face as the identity match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' We conclude tentatively that performance in the masked conditions was very poor indicating the effectiveness of the masks for preventing identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' However, given the un- usual performance in the masked condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=', negative d’s), IDENTITY MASKING WITH EYE ENHANCEMENT 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 2: Example trial in Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 3: Experiment 1 accuracy, measured as d’ across conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Results show that both masking algorithms were equally effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' we retested the conditions with a design that eliminates the possibility of response bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' EXPERIMENT II: EFFECT OF EYEZOOM MASKING METHOD WITH A FORCED-CHOICE TASK In this experiment, we used a two-alternative forced choice (2AFC) task to test masking effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In the 2AFC, two faces are presented as response options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In all cases, one of the two images will be the same identity as the person in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Participants A total of 30 (7 male, 22 female, 1 other) undergraduate student volunteers (ages 18-26) from UTD participated in the study in exchange for research credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Procedure The experiment consisted of 72 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The video stimulus was displayed in the top center of the screen with the two face images beneath it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Participants were asked to determine which of the two face images matched the identity shown in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' To make the task challenging, the two faces presented had a similar appearance and were of the same race and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' An example trial is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Each of the dataset’s 36 identities was shown twice, once with the correct response as the left-located option and once with the correct response as the right-located option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The video segments were shown in random order and looped until the subjects responded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Participants were assigned randomly to one of the three masking conditions, with the unmasked condition serving as a baseline condition for identification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Results Accuracy was assessed as the proportion of correct re- sponses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 6 shows the proportion of correct responses for each mask condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' These values indicate that face recognition in the unmasked condition was more accurate than face recognition in the masked conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' A one-factor ANOVA was performed on the accuracy data (proportion of correct responses), with condition as the independent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The model yielded a main effect of mask condition on ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Press "" if the person in the Press "2" if the person in the Press "3" if the person in the video video is the person on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' video is the person in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' is NOT either of the two people picturedcondition unmasked canny eyezoom 1 2 conditionIDENTITY MASKING WITH EYE ENHANCEMENT 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 4: Proportion of responses by trial type in Experiment I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' proportion of correct responses, F(2, 27) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='68, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' As in the first experiment, participants were more accurate in the unmasked condition than in the masked conditions, and performed comparably for the two masked conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The results replicate the pattern of performance across conditions found for Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' As expected with a 2AFC task, performance was more accurate in all three conditions than it was in Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Notably, average identification was above chance in both masked conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Performance in the Eyezoom condition was more variable than performance in the Canny mask condition—replicating a similar finding in Experiment I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' We conclude that the masks strongly inhibit identification, but that when forced to guess between two images (with the assurance that one was an identity match), participants fared better than chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Notwithstanding, applications of identity masking would rarely if ever be able to assure a human or ma- chine system that one of two candidates was an identity match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Our goal in applying this method here was to test examine the role of response bias in the unusual pattern of results found in Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The present results suggest that these masking algorithms leave behind some residual identity information in the face that humans can exploit when the response decision is highly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' As noted, it is unlikely that that would Unmasked Target Present Trials Unmasked Target Absent Trials 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 response 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 response responses chose either identity chose either identity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content="75 chose no identity chose no identity 09'0 9." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' prop prop 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 response response Canny Target Present Trials Canny Target Absent Trials 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 response 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 response responses chose either identity chose either identity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='75 chose no identity chose no identity 090 9 pro pro 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 response response Eyezoom Target Present Trials Eyezoom Target Absent Trials 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 response 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 response chose eitheridentity chose either identity chose noidentit chose no identity res pro pro 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='00 response responseIDENTITY MASKING WITH EYE ENHANCEMENT 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 5: Example trial from Experiment II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 6: Experiment 2 - identification accuracy across conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' be the case in any applied scenario, and so we conclude that these simple simple filtering procedures provide a reasonably high degree of identity protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Additionally, we conclude, albeit more tentatively, that the eyezoom procedure does not improve identification significantly over the Canny procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' EXPERIMENT III: EFFECT OF EYEZOOM MASKING METHOD ON ACTION PRESERVATION The effectiveness of the identity protection provided by these masks opens the question of whether this protection comes at the cost of preserving information about facial actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In this experiment, we examined whether the Canny and Canny+Eyezoom mask conditions impaired driver facial action perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Participants A total of 30 (6 male, 23 female, 1 nonbinary) undergradu- ate student volunteers (ages 18-30) from UTD participated in the study in exchange for research credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Procedure The experiment consisted of 100 trials, each with three response options: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=') driver looking to the left, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=') driver looking to the right, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=') driver looking down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Each of the 36 identities in the dataset appeared between two and three times, each with a different action (looking right, left, down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Prior to the start of the main experiment, a pilot test with only the unmasked condition was conducted to ensure that the actions were identifiable in all videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' This test resulted in the elimination of eight (of 108) videos segments in which actions were not recognizable at sufficiently high levels of accuracy for inclusion in the action preservation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The participants were assigned randomly to one of three masking conditions with the unmasked condition providing a baseline action recognition accuracy and were asked to identify whether the driver was looking to the left, right, or down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The video stimuli were shown in the upper center of the screen with three written options below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 7 for an example trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The clips were played in a random order and looped until the participant responded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Results The proportion of correct responses was used to assess accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 8 shows the proportion of correct responses for each mask condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' These values indicate that action preservation Press"1" if the person in the Press "2" if the person in the videoisthepersonontheleft video is the person in the rightcondition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='9 unmasked canny eyezoom I of correct response 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='7 ortion propor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='5 conditionIDENTITY MASKING WITH EYE ENHANCEMENT 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 7: Example trial from Experiment III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' was generally high, but also suggest a small advantage for action perception in the unmasked condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' A one-factor ANOVA, performed on the accuracy (proportion of correct responses) data, with the independent variable of condition, did not show a significant effect, but was generally consistent with this conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The model yielded a marginal main effect of mask condition on proportion of correct responses, F(2, 27) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='69, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='086.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' This suggests a very slight advantage for action perception without stimulus masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In conclusion, although the results did not reach statistical significance, there is some indication that masking impaired action perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' DISCUSSION Our goal was to examine the effectiveness of simple Canny-filtering based masking methods, with and without eye enhancement, for interfering with face identification while preserving facial actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In Experiment I, face recognition accuracy was diminished for both mask conditions, relative to the unmasked condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' There was no difference between the Canny mask alone and the mask with eye enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In Experiment II, we replicated this result with a 2AFC procedure that controlled for response option bias, which may have been a factor in the findings of negative ’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' values for both masking conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In combination, both studies point to the relative effectiveness of the masks for interfering with identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' They also point to the conclusion that eye enhancement did not alter this effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Experiment III showed that facial actions were preserved to a similar degree with both masks, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' 8: ANOVA of proportion of correct responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' though there was a marginal advantage for action perception in the unmasked condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' In summary, these results indicate that Eyezoom masking does not significantly increase identification or alter facial action preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was supported through collaboration with Oak Ridge National Laboratory and the Federal Highway Admin- Press "1" if the person looks toward the driver\'s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Press "2" if the person looks toward the passenger\'s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' Press "3" if the person looks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='condition unmasked canny eyezoom f correct responses 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='92 of proportion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='88 un ma ez conditionIDENTITY MASKING WITH EYE ENHANCEMENT 8 istration under the Exploratory Advanced Research Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The human experiment and analysis was subcontracted to the University of Texas at Dallas from Oak Ridge National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' This manuscript has been authored in part by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US De- partment of Energy (DOE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' The US government retains and the publisher, by accepting the article for publication, acknowl- edges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the pub- lished form of this manuscript, or allow others to do so, for US government purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content=' DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} +page_content='gov/downloads/doe- public-access-plan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'} diff --git a/19FQT4oBgHgl3EQf1zZe/content/tmp_files/2301.13421v1.pdf.txt b/19FQT4oBgHgl3EQf1zZe/content/tmp_files/2301.13421v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a770c848631503eef3864c095f50741b1451641 --- /dev/null +++ b/19FQT4oBgHgl3EQf1zZe/content/tmp_files/2301.13421v1.pdf.txt @@ -0,0 +1,2036 @@ +MOAT: Towards Safe BPF Kernel Extension +Hongyi Lu1,2, Shuai Wang2,∗, Yechang Wu1, Wanning He1, Fengwei Zhang1,∗ +1Southern University of Science and Technology +2Hong Kong University of Science and Technology +Abstract +The Linux kernel makes considerable use of Berkeley Packet +Filter (BPF) to allow user-written BPF applications to execute +in the kernel space. BPF employs a verifier to statically check +the security of user-supplied BPF code. Recent attacks show +that BPF programs can evade security checks and gain unau- +thorized access to kernel memory, indicating that the verifica- +tion process is not flawless. In this paper, we present MOAT, +a system that isolates potentially malicious BPF programs +using Intel Memory Protection Keys (MPK). Enforcing BPF +program isolation with MPK is not straightforward; MOAT +is carefully designed to alleviate technical obstacles, such +as limited hardware keys and supporting a wide variety of +kernel BPF helper functions. We have implemented MOAT +in a prototype kernel module, and our evaluation shows that +MOAT delivers low-cost isolation of BPF programs under +various real-world usage scenarios, such as the isolation of a +packet-forwarding BPF program for the memcached database +with an average throughput loss of 6%. +1 +Introduction +It is common to extend kernel functionality by allowing user +applications to download code into the kernel space. In 1993, +the well-known Berkeley Packet Filter (BPF) was introduced +for this purpose [4]. The classic BPF is an infrastructure +that inspects network packets and decides whether or not +to forward or discard them. With the introduction of its ex- +tended version (referred to as eBPF) in the Linux kernel, BPF +soon became more powerful and is now utilized in numerous +real-life scenarios, such as load balancing, scheduling, and +auditing [18, 22, 28, 52, 62, 63]. +To ensure security, BPF is equipped with a verifier [6]. +The verifier performs a variety of static analyses to ensure +the user-supplied code is secure. For instance, the verifier +tracks the bounds of all pointers to prevent an out-of-bound +access. Given that BPF code runs directly within the kernel, +∗Shuai Wang and Fengwei Zhang are the corresponding authors. +the verifier becomes crucial for the BPF security. Neverthe- +less, as pointed out by recent studies [25, 31, 32, 50, 60], the +currently available verifier has various limitations, and is in- +sufficient for the overall security of BPF. First, the current +BPF ecosystem supports a variety of kernel functionalities +with over 200 dedicated APIs [2], resulting in a complicated +verification process. Even though the verifier’s correctness has +been formally proved [59], the gap between abstraction and +implementation may still result in vulnerabilities [35–41, 43]. +Second, BPF Just-In-Time (JIT) is currently supported on +multiple platforms, including x86, ARM, and RISC-V, whose +differences frequently result in subtle vulnerabilities [44, 45]; +note that the verifier cannot detect vulnerabilities in the JIT +stage. Third, due to the rapid expansion of BPF capabilities, +the verifier has to be frequently updated. Nonetheless, it is +inherently difficult to frequently update a complex static veri- +fication tool without introducing new vulnerabilities [42]. To +date, the BPF subsystem has been constantly exploited. For +instance, two privileged-escalation vulnerabilities have been +discovered in the implementation of bpf_ringbuf, a rather +new BPF feature introduced in 2020 [4]. Further, the veri- +fier’s register-value tracking is quite complex and has been +bypassed by several severe vulnerabilities [35–38]. +Given the increasing security threats in BPF and the chal- +lenge of enforcing safe BPF programs with merely static +verification, we seek to employ hardware extensions to sand- +box untrusted BPF programs. In particular, we leverage Intel +Memory Protection Keys (MPK) [9], an emerging hardware +extension which partitions memory into distinct permission +groups by assigning up to 16 keys to their Page Table En- +trys (PTEs). With the aid of MPK and the BPF verifier’s +analysis results, we present MOAT, which isolates untrusted +BPF programs in a low-cost and principled manner. For in- +stance, two MPK protection keys K and E may be assigned to +the kernel and a BPF program, respectively. When the kernel +transfers control to the BPF program, it can set K as access- +disabled to prevent the potentially malicious BPF program +from tampering with kernel memory regions. +Despite its promising potential, we observe that using MPK +1 +arXiv:2301.13421v1 [cs.CR] 31 Jan 2023 + +tracepoint +packet filter +schduler +tracepoint +packet filter +schduler +User +Application +Kernel +packet filter +schduler +tracepoint +BPF Programs +BPF Bytecode +Verifier +Maps +Helpers +call bpf_pid +... +log next_sched +ret next_sched +Kernel +BPF (Runtime) Utilities +BPF Bytecode +BPF Compiler +Figure 1: BPF overview. We illustrate the BPF compilation procedure, and the execution context of a sample BPF program attached to the +kernel scheduler. Note that BPF verification is conducted at the BPF bytecode loading time. +to enforce BPF isolation is not straightforward. MOAT is de- +liberately designed to overcome two major technical hurdles. +First, Intel MPK provides a maximum of 16 keys. Thus, it +becomes challenging to support many BPF programs running +concurrently with this limited number of hardware keys. Exist- +ing workarounds like key virtualization [51] are incompatible +with the BPF scenario and challenging to be implemented in +kernel. This is because the key virtualization heavily relies on +scheduling and notification facilities that are only available +to userspace; directly reusing them in the kernel space may +largely block kernel threads. To address this hurdle, we pro- +pose a novel dynamic/fixed key allocation scheme that can +support multiple BPF programs with a small overhead. +Second, while MPK-based hardware isolation mitigates ma- +licious BPF programs, helper functions provided by the BPF +subsystem may be exploited by attackers. Overall, the growth +of the BPF ecosystem is accompanied by the expansion of its +dedicated helper functions; helper functions facilitate various +tasks commonly conducted by a BPF program. On one hand, +MOAT should allow benign BPF programs to freely use these +helpers. On the other hand, MOAT must be cautious enough +with these APIs to ensure they are not exploited by attackers. +Given that there are over 200 helpers [2] provided in the latest +Linux kernel, designing individual security policy for each +of them is impractical and less extensible. To this end, we +analyze all existing helpers with static dependency-analysis, +and propose several general defense schemes, each of which +is applicable to a group of helpers. We envision that when +a new helper is added, MOAT can be applied easily without +introducing new schemes. +To evaluate the security impact of MOAT, we systemati- +cally examined how MOAT mitigates attack surfaces due to +untrusted BPF programs. We also empirically analyze all +recent CVEs within MOAT’s application scope. The result +shows that MOAT successfully mitigates each CVE. More- +over, we evaluate the performance overhead of MOAT under +representative and edge-case scenarios. First, we examine the +performance of our dynamic/fixed key allocation policy by +assessing a use case where multiple programs are executed +concurrently to use all MPK keys. Then, we build a real- +life port-forwarding BPF program for the memcached [24] +database, and secure it with MOAT to measure how MOAT +influences memcached’s throughput. Furthermore, we apply +MOAT to several real-world BPF applications [52] to illus- +trate that MOAT can be directly applied to the current BPF +ecosystem with minimal engineering effort. MOAT’s worst +case performance overhead in all these experiments is less +than 30%, which is acceptable given the security benefits +MOAT provides. Moreover, MOAT imposes only 6% over- +head on average to the memcache’s throughput. In sum, we +make the following contributions. +• Instead of merely relying on BPF verifiers to statically +validate untrusted BPF programs, this paper for the first +time advocates to isolate user-supplied BPF programs with +an emerging hardware extension, Intel MPK. +• Technically, MOAT is specially designed to address domain- +specific challenges including limited hardware keys and +protecting over 200 helper functions in the BPF ecosystem. +• We implement a prototype of MOAT as a Loadable Kernel +Module (LKM) of the latest Linux (v5.19) and conduct a +thorough evaluation of its security and performance. The +evaluation shows that MOAT delivers a principled security +guarantee with moderate performance penalty. +2 +Background +2.1 +Berkeley Packet Filter (BPF) +BPF Overview. BPF [4] was originally introduced to facili- +tate flexible network package filtering. Instead of inspecting +packages in the userspace, users can provide BPF instructions +specifying package filter rules, which are directly executed in +the kernel. This allows configurable package filtering without +costly context switching and data copying. Modern Linux +kernel features extended BPF (eBPF), a Linux subsystem, +which supports a wide range of use cases such as kernel pro- +filing, load balancing, and firewalls.1 Popular applications like +Docker [34], web browsers [30, 48], and kernel debugging +utilities like Kprobes [8] are built on top of BPF. +Fig. 1 depicts an overview of how BPF programs are com- +piled and then deployed. The eBPF subsystem offers ten +general-purpose 64-bit registers, memory stack, eBPF cus- +tomized data structures (often referred to as eBPF maps), and +a set of eBPF helper functions. To use eBPF (e.g., for ker- +nel profiling), users can first write their own BPF programs +(in C code) to specify the functionality. The BPF programs +will then be compiled into BPF bytecode and downloaded +into the kernel. Given that eBPF code is written by untrusted +1While there are indeed two variants of BPF: classic BPF (cBPF) and +eBPF, cBPF is internally converted into the latter variant by the kernel. +2 + +users, the kernel employs a verifier to conduct several checks +during the bytecode loading time (see below). By default, +the bytecode is executed by the BPF interpreter (omitted in +Fig. 1). Additionally, depending on the kernel configuration +and architectural support, an optional JIT compilation may +be applied to the bytecode for better performance. The BPF +bytecode is then attached to certain kernel components, based +on its specific end goal. For instance, as shown in Fig. 1, a +BPF program attached to kernel scheduler collects relevant +statistics and decides which thread should be running next to +improve overall performance. +- len < INSN_MAX +- no loop +- no dead code +- no OOB jmp +Unverified +CFG +Check Phase +Data-Flow +Check Phase +- register tracking +- access check +- helper check +- misc fixups +Verified +Figure 2: BPF verification process. +BPF Verifier. BPF programs are written in C, and compiled +into a RISC-like instruction set. As aforementioned, the kernel +strictly verifies the BPF programs upon loading to ensure +that they are safe to execute. Fig. 2 illustrates the verifying +process in a holistic manner. First, a BPF program is parsed +into a control flow graph (CFG) by the verifier, which first +performs a CFG check phase to ensure four key properties: +1) the program size is within a limit; 2) there exists no back +edges (loops) on its CFG; 3) there exists no unreachable +codes; and 4) all jumps are direct jump and they refer to a +valid destination. Overall, given that BPF programs must be +terminated, the CFG check phase ensures that all jumps are +direct jumps and there are no back edges. Given that said, +loops are still feasible via unrolling at the cost of binary size. +The verifier further performs finer-grained data flow anal- +ysis. It first tracks the value flow of every register to deduce +its value ranges conservatively. Based on these ranges, the +verifier can decide if a pointer accesses safe memory regions, +and if a parameter is valid. Since this analysis is performed +statically prior to execution, there exists possibility that a ma- +licious BPF program uses certain operations to bypass this +analysis [35–41, 43]. Last, the verifier also performs some +miscellaneous fixups, like rewriting certain instructions to +simplify the follow-up JIT compilation. +BPF Maps. Out of security concern, the kernel also sets a +rather strict space limit on BPF programs. Each program +by default can only use up to 512 bytes stack space and 10 +registers, which is far from enough for certain BPF programs. +To address this problem, BPF maps can be allocated and +provide additional space for BPF programs. Up to now, there +are over 30 types of maps supported by kernel, such as array +map and hash map [5]. Moreover, as demonstrated in Fig. 1, +maps may act as a communication channel between BPF +programs and user applications, since some of these maps can +be accessed by both the BPF program and the user application. +BPF Helpers. Kernel also limits the kernel functions a BPF +program may call. Those functions are dubbed BPF helper +functions, as shown in Fig. 1. Up to now, there are over 200 +helpers scattered across subsystems of the kernel [2]. Note +that depending on the specific task, a BPF program can usually +call a group of relevant helpers. For example, a BPF program +attached to the scheduler is not allowed to call any helper +related to kernel probing, but it can call bpf_pid2 to obtain +the PID of the current process and chooses which process to +be scheduled next. +00 +01 +... +10 +00 +32 +0 +PKR +PTE[62:59] = 0xF +PTE[62:59] = 0xE +PTE[62:59] = 0x1 +PKR Entry Options +00 +Access Enable (EN) +01 +Access Disabled (AD) +10 +Write Disabled (WD) +11 +Access Disabled (AD) +Page Table Entry +Figure 3: Intel MPK overview. +Intel MPK. Intel introduces MPK [9] to provide efficient +page table permissions control. By assigning a MPK protec- +tion key to each page table entry (PTE) of one process, users +can enable intra-process isolation and confidential data access +control [17, 27, 33, 51, 58]. As illustrated in Fig. 3, MPK uses +four reserved bits [62:59]in each PTE to indicate which pro- +tection key is attached with this page. Those three PTEs in +Fig. 3 are assigned with keys 0x1, 0xE and 0xF, respectively. +Since there are only 4 bits involved, the maximum number +of keys is 16. Then, a new 32-bit register named Protection +Key Register (PKR) is introduced to specify the access per- +mission of these protection keys. Each key occupies two bits +in PKR, whose values denote either access-disabled (AD) or +write-disabled (WD), respectively. By writing to certain bits +in PKR, the access permission of corresponding pages can be +configured accordingly. It is worth noting that one key may +be assigned to arbitrary number of pages by modifying their +PTEs. This facilitates changing the access permission of a +large number of pages without severe performance penalty. +Clarification and Notations. As a side note, there are +actually two versions of Intel MPK. One applies to the +user-mode while the other applies to the supervisor-mode. +For brevity, we refer these two versions in their conven- +tional abbreviations as Protection Key Supervisor (PKS) +and Protection Key User (PKU), respectively. Most existing +works [17, 26, 27, 33, 51, 58] are based on PKU. In MOAT, +we use PKS instead since our goal is to isolate BPF programs, +which execute in the supervisor-mode. The logistics behind +these two versions are mostly identical with slight variations. +For instance, the permission configuration register in PKS is +a Model Specific Register (MSR) named IA32_PKRS, which +is inaccessible from userspace, whereas in PKU, this role is +assigned to a dedicated register PKRU. In addition to PKR, +there also exists a bit in the control register CR4 that can dis- +able/enable MPK entirely; for PKU, this bit is CR4.PKE; for +2Here, bpf_pid refers to bpf_get_current_pid_tgid. +3 + +PKS, this bit is CR4.PKS. To avoid potential misleading, the +rest of the paper directly refers MPK leveraged by MOAT as +PKS. +3 +Motivation and Assumptions +3.1 +Typical Threats to BPF Verifier +Fast Feature Evolving. As a fast developing technology, +threats may come from the inconsistency between the con- +stantly expanding BPF capabilities and the rigorous static +verification process imposed on them [39, 42]. It is a common +practice to add corresponding verification procedures simul- +taneously when introducing new features to BPF programs. +However, it is very hard to make changes correctly to the BPF +verifier, a critical security kernel component, which has over +10K LoC and a variety of functionalities [6]. +Challenging Pointer Tracking. Second type of threats origi- +nates from the complexity of pointer tracking mechanism. Al- +though the correctness of the verifier is formally proved [59], +there still exist gaps between the implementation and the +abstraction, especially in some corner cases, such as sign ex- +tension, truncation, and bit operators [35–41, 43]. +The fact that the contemporary BPF verifier only performs +static analysis is a severe deficiency, as evidenced by the +threats noted above. Performing sound and complete static +analysis toward BPF programs to uncover potential threats is +fundamentally challenging, and from the disclosed BPF vul- +nerabilities, we find that there frequently exists a gap between +verifier’s static analysis results and BPF programs’ runtime +behavior. For instance, the verifier, based on its static analysis +results, may conclude that a program is benign because it +only accesses a memory region ranging from [0x0,0x1000]. +However, by leveraging vulnerabilities like noted above, the +software may behave differently during execution. Therefore, +a hardware feature, Intel MPK, is utilized to enforce further +isolation, such that a BPF program is constrained in its own +memory regions, and any runtime accesses that violate this +constraint are effectively flagged and terminated by MOAT. +3.2 +Threat Model +Assumption. Our threat model considers a practical setting +which is aligned with existing BPF vulnerabilities [35–41, 43]. +In particular, we assume attackers are non-privileged users +with BPF access, since a root user already has the control over +almost the entire kernel. Attackers can download their pre- +pared BPF code into the kernel space to launch exploitation. +Attacker Capability and Application Scope. MOAT iso- +lates user-submitted BPF programs and prevent them from +accessing privileged kernel memory regions. As will be intro- +duced in Sec. 4, a BPF program is given the minimum neces- +sary resources and privileges to complete its task. To clarify, +there are also other more subtle vulnerabilities (not relevant +to memory exploitations) such as speculation, race condition, +and DoS occurred to exploit the BPF subsystem [46, 47]; a +well-isolated BPF program can still launch these attacks to +jeopardize the BPF subsystem and the kernel. This research +views them as common security defects shared by many other +applications such as SGX enclaves [15, 21]. To date, coun- +termeasures have been deployed by the BPF verifier [11], +and we assume the standard BPF verifier can handle them +properly. In contrast, correctly detecting memory-related BPF +exploitation requires systematic and rigorous static analysis +over BPF programs and is fundamentally hard for BPF ver- +ifiers; MOAT enhances mitigating memory-related exploita- +tions with hardware-assisted isolation. Next, we present anal- +ysis of three major components in our research context as +follows. +BPF Programs. This includes the BPF bytecodes or the JIT- +emitted native instructions. Our threat model takes the as- +sumption that malicious BPF programs are able to bypass +checks statically performed by the verifier; they may thus +behave maliciously during runtime. Our threat model deems +BPF programs as untrusted, and MOAT is designed to isolate +them from the rest of the kernel. More specifically, every BPF +program, during its runtime, is only allowed to access its own +stack, allocated maps, and certain helper functions. +BPF Helper Functions. These helpers act as the interme- +diate layer between the BPF subsystem and kernel. Certain +malicious BPF programs can abuse these helpers to perform +attacks, and therefore, we assume they are also untrusted. +MOAT mitigates risks raised by adversarial-manipulated +helper functions with practical defenses. +Kernel. Kernel is the target to protect. We assume the kernel +is functioning normally, and attackers aim to leverage mali- +cious BPF programs to gain unauthorized access to kernel +data or executing arbitrary privileged kernel code. +4 +Design +MOAT Overview. As motivated in Sec. 3, current security +design against malicious BPF programs solely relies on the +static analysis performed by the BPF verifier, which is seen as +a weak point and exploitable by non-privileged users. MOAT +instead delivers a principled isolation of BPF programs using +MPK from the rest part of the kernel and prevent bypasses. +bpf_lookup_elem +call bpf_run +... +bpf_delete_elem +mov %rax, $0x1 +... +call bpf_helper +st %(rax), $0x10 +Helper +Auditor +BPF Memory +... +bpf_get_time +mov %rax, %rbx +MOAT +BPF Payload +Access +Rules +Stack +... +Maps +MPK +Verifier +Kernel Memory +1 +2 +3 +4 +Figure 4: MOAT overview. +4 + +Fig. 4 illustrates the lifecycle of a BPF program with the +presence of MOAT. 1 Given a user-submitted BPF program +P, MOAT statically derives the minimum necessary memory +regions the program needs, such as stack, used maps and +context by reading metadata from P. 2 These regions (“BPF +Memory” in Fig. 4) are assigned to P using PKS, forming its +runtime environment. 3 When the kernel invokes P, MOAT +configures PKS to constrain P to its own regions and forbids +its access to other memory regions. 4 On the occasions that P +requires helper calls to interact with the kernel, depending on +the helper types, MOAT may adjust involved kernel memory +region permissions and also validate the helper parameter +values to prevent helpers from being abused. +Security Guarantees. Overall, MOAT provides the following +two key secure design guarantees. +(i) A BPF program is given the minimum necessary ker- +nel resources and privileges for completing its task, +preventing any malicious behavior. +(ii) The interactions (e.g. helper calls) between the BPF +program and the kernel are audited thus not abused. +Extensibility. MOAT leverages MPK, a de facto hardware ex- +tension available on mainstream Intel architectures to isolate +BPF programs. We view this design choice is consistent with +recent hardware-assisted security enforcement works [17, 58]. +Nevertheless, we clarify that the design of MOAT is not lim- +ited to leveraging MPK. There exist similar hardware security +mechanisms on other platforms and architectures such as the +Memory Domains [3] on ARM and the Domain Keys [53] +on RISC-V. These mechanisms can be used to replace MPK +on these platforms with a small amount of engineering effort; +see Sec. 8 for our discussion on migration and extension. +4.1 +General BPF Isolation +In accordance with the BPF program lifecycle depicted in +Fig. 4, this section elaborates on the general isolation ap- +proach offered by MOAT. We further discuss two key techni- +cal challenges in Sec. 4.2. +0x0 +... +59 +62 +Kernel +BPF +BPF +0x3 +... +... +Shared by + & +Kernel Data +Kernel Code +Stack +Maps +Context +Code +Stack +Context +Code +Shared Maps +Page Table Entries +Data Regions +Runtime PKR Value +Enable +01 +00 +01 +00 +AD +EN +EN +N/A +AD +01 +01 +00 +00 +AD +EN EN AD +8 +.. +.. +.. +.. +32 +... +0x2 +... +... +0x1 +... +... +AD Access-Disabled +EN Access-Enabled +Figure 5: BPF memory regions. +4.1.1 +BPF Memory Regions +Fig. 5 depicts the memory regions of BPF programs and +the kernel. By default, all pages should belong to the kernel +memory region, and each page is initialized with a default +MPK key value 0. Then, when a BPF program P is newly +loaded into the kernel, MOAT decides the minimum pages +it needs, and assigns these amount of pages to the memory +region of P. Note that the necessary memory sections of a +BPF program includes its code, stack, and the context; many +non-trivial BPF programs also require BPF maps (e.g., array +and hash maps) to use. After assigning these sections to the +memory region of P, MOAT restricts P to its own memory +regions by configuring the PKR register. Take the BPF P1 in +Fig. 5 as an example, most of its sections (including a number +of BPF maps) solely belong to itself. Furthermore, P1 and +P2 share several extra BPF maps. Thus, at its runtime, MOAT +configures the PKR register of P1 to enable its access (EN; +denoted as 00 in the runtime PKR value column of Fig. 5) +to its own region 0x1 and the shared region 0x3. Moreover, +MOAT disables any accesses from P1 to the kernel region 0x0 +and the P2 memory region 0x2 by setting corresponding bits +in P1’s PKR register as 01 (denoting AD). +To clarify, the code and map sections of a BPF program +requires are trivially known (by reading the metadata in the +BPF program) once it is loaded. Thus, MOAT can assign these +pages to its designated region by modifying their PTEs during +the program loading phase without any runtime overhead. The +assignment for stack, context and some special types of maps +will be discussed in the next section. +4.1.2 +BPF Runtime Environment +Apart from the program itself and the maps it uses, a BPF +program requires additional kernel structures to function prop- +erly. These structures include descriptor tables, stacks, and the +program’s runtime context. Furthermore, certain maps (such +as the hash map) are not stored continuously in the kernel +and cannot be assigned trivially during initialization. MOAT +assigns entries of this kind of maps on the fly. +Descriptor Tables. On x86 platforms, descriptor tables such +as Global Descriptor Table (GDT) and Interrupt Descriptor +Table (IDT) are essential for basic operations like interrupt. +These kernel data structures are assigned to a shared region +that all BPF programs can access. To prevent tampering those +critical structures, they are made read-only when shared. +Dedicated Stack. BPF programs require a 512-byte stack +space to store local variables and function frames. The ver- +ifier is in charge of determining if a program makes Out of +Bound (OOB) accesses toward this stack. Thus, when the +BPF program passes the static checks, its required stack is di- +rectly allocated from the kernel stack. However, as discussed +in Sec. 3, certain vulnerabilities may allow BPF programs +to bypass this check at runtime. Given that this stack is uti- +lized so frequently, we note that executing dynamic auditing +5 + +Table 1: BPF context of common program types. +Program Type +Context Type +Note +Socket Filter +__sk_buff * +Metadata of sk_buff +Socket Ops +bpf_sock_ops * +Socket events (timeout, retransmission, ...) +XDP +xdp_md * +Metadata of xdp_buff +Kprobe +pt_regs * +Register status +Tracepoints +Depending on Tracepoint Types +Relevant Tracepoint information +Perf Event +bpf_perf_event_data * +Perf. event (register status, sample period) +Cgroup Device +bpf_cgroup_dev_ctx * +Device ID, access type (read, write, ...) +on it, as MOAT does for helper calls (see Sec. 4.2.2), would +incur an unreasonable level of overhead. Thus, to prevent +malicious BPF programs from tampering the kernel stack, +MOAT allocates per-CPU stacks for BPF programs to use. To +do so, similar to the descriptor tables, these per-CPU stacks +are shared by all BPF programs running on the same CPU +core. Consequently, they are also assigned to the shared re- +gion. To prevent a malicious BPF program from tampering +stacks of other CPU cores, the stack beginning addresses are +randomized for each CPU core. +Runtime Context. The context refers to BPF program param- +eters, which vary depending on the BPF program types. For +instance, if the BPF program serves as the filter attached to a +particular socket, its runtime context is a pointer to the socket +buffer, which stores packets for the attached socket. Since +these contexts are not available until runtime, MOAT assigns +these contexts upon the entry point of each BPF program. +Table 1 lists common BPF contexts: These contexts are rather +simple and only a few of them are nested data structures (i.e., +containing pointers to other structures). Thus, this assignment +can be performed efficiently upon each entry point. +Incontiguous Maps. Despite the fact that there are over 30 +distinct types of maps, their implementations can be roughly +divided into only two types: Array maps and hash maps. +The array maps are easy for MOAT to isolate since they +are stored in a continuous form and of a fixed size. For +these maps, MOAT determines its isolation when loading +the BPF programs. The hash maps, however, are stored non- +contiguously in the memory and can be dynamically expanded +upon map insertion. This prevents MOAT from determining +the addresses and sizes of the maps before executing the +BPF programs. To overcome this issue, MOAT attaches to +the bpf_map_lookup_elem, which is used to lookup a map +entry and return its pointer. If the pointer is retrieved from +an non-contiguous map, the memory to which it points is +dynamically assigned to the BPF program. These entries are +returned to the kernel once the program exits. +4.1.3 +Lifecycle of a BPF Program +This section has described how MOAT uses PKS to grant a +BPF program accesses to its minimum necessary memory +regions required to complete its task. This protects the ker- +nel from being attacked by malicious BPF programs while +allowing benign BPF programs to operate smoothly. We sum- +marize all these details and depict the lifecycle of an isolated +BPF program in Fig. 6. +BPF Program +Used Map +BPF Program +Used Map +Ctx +Stack +Entry +BPF Program +Used Map +Ctx +Stack +Dynamic +Map Entry +Run +Exit +BPF Program +Load +1 +4 +2 +3 +1 Load: The program itself and its maps are assigned to its region. +2 Entry: Context is assigned and stack is swapped. +3 Runtime: Entries of incontiguous maps are assigned on the fly. +4 Exit: Memory assigned during runtime is returned. +Figure 6: BPF program lifecycle under isolation of MOAT. +4.2 +Challenges for MOAT +The preceding section illustrates the overall procedure of +MOAT. However, to effectively isolate a BPF program using +PKS, MOAT needs to overcome the following obstacles. +C1: Limited Hardware Regions. In PKS, only 16 hardware +keys are available. This means there can be no more than +16 memory regions concurrently, but there may be signifi- +cantly more than 16 BPF programs running in the kernel. To +overcome this limitation, we propose a novel dynamical key +allocation policy in Sec. 4.2.1. +C2: Helpers. BPF is a complex ecosystem containing over +200 helper functions [2]. Unlike BPF programs, these helper +functions must have access to certain kernel memory to func- +tion properly. Thus, MOAT must ensure that these helper func- +tions are secure and not being abused. However, designing +specific isolation policy for every one of these helpers requires +massive human effort. Even worse, designing individualized +isolation strategy for each helper may impede the applica- +bility to helpers added in the future. To this end, we analyze +these BPF helper functions with static analysis techniques and +propose three general security isolation schemes in Sec. 4.2.2. +4.2.1 +Dynamic Key Allocation +Currently, PKS supports up to 16 memory regions, whose +permissions are decided by a 32-bit PKR. Although works +like libmpk [51] propose key virtualization to enable key +sharing, these works typically focus on isolating userspace +applications. Therefore, they rely on scheduling and notifica- +tion mechanisms that are exclusive to userspace. However, +6 + +after examining their methods, we conclude that porting these +userspace mechanisms to kernel is difficult, if at all possible. +Intuitively, we may explore making key a shared resource; +each BPF program will dynamically fetch and return a key +upon its entry point and exit. Our tentative study shows that +this approach works well with small BPF programs consum- +ing few pages. Nevertheless, this approach may incur signifi- +cant runtime overhead, as assigning these pages to a specific +region upon each entry and exit can be time-consuming, partic- +ularly if the program is attached with large maps. For instance, +a 512KB map consists of over 100 pages. If a BPF tracepoint +program employs this map to log kernel events, there will +be over 200 page assignments every time this BPF program +is invoked. These frequent assignments bring unacceptable +overhead. Overall, given that frequent key retrieval and return +is too expensive due to the presence of large BPF programs +with many pages, we propose an adaptive dynamic key allo- +cation scheme that shares keys across relatively small BPF +programs and assigns fixed keys to large BPF programs. +Dynamic keys +K1 +K2 +Run +Fixed +keys +Exit +K1 +Wait +Wait +... +Entry +Large BPF +Programs +1 +2 +3 +4 +Figure 7: Adaptive key allocation. +As illustrated in Fig. 7, we divide PKS keys into two cate- +gories — dynamic keys and fixed keys. We allocate dynamic +keys to small BPF programs, whose allocation procedure are +specified as follows. 1 Upon a BPF program P’s entry point, +MOAT fetches a dynamic key and assigns this key to all pages +of P. 2 During the runtime, MOAT can detect if P accesses +pages not assigned to it via PKS. 3 When P exits, all of its +pages are returned to the kernel, and the key is deallocated. +4 If currently no key available when the kernel launches a +BPF program, then this program is placed in a queue to wait. +In contrast, fixed key allocation is straightforward. Once +a large BPF program is loaded by the kernel, MOAT assigns +a fixed key to it. In extreme cases where multiple large BPF +programs are loaded into the kernel, and fixed keys are insuffi- +cient, the smallest and least frequently invoked BPF program +running will be evicted to use dynamic keys. +We need to decide a threshold to determine whether a BPF +program is “small” or “large.” Note that the current BPF sub- +system only accepts programs that with fewer than 4,096 +instructions, which occupy about eight pages. Considering +that the majority of BPF programs use a small map to com- +municate with userspace, we select ten pages as the threshold +for dynamic key allocation. That is, a BPF program using up +to ten pages is configured to use dynamic keys, whereas BPF +programs with more than ten pages uses fixed keys. +4.2.2 +Helper Security Mechanism +As interfaces between kernel and BPF programs, a set of +BPF helper functions has been provided for kernel interac- +tion. Since these helper functions serve as interfaces, most of +them have to access certain kernel memory to function prop- +erly. Therefore, these helpers may be leveraged by malicious +BPF programs to launch attacks. Thus, MOAT has to prevent +these helpers from being abused. However, there are over 200 +helpers provided by the BPF subsystem; it is impractical to +design individual protection policy for each one. To overcome +this obstacle, we analyze these helper functions and propose +three defenses based on their interaction with the kernel. Each +of these defenses applies to a large number of helpers and can +be combined to enhance the offered protection guarantee. +Analyzing all these helpers manually requires a significant +amount of human effort. We leverage a de facto static pointer +analysis library, SVF[55, 56], to perform dependency analysis. +SVF performs sparse value flow analysis to establish value +flow and pointer analysis results. SVF has been widely used +to analyze large-size production software [54]. We use the +default flow-sensitive pointer analysis [55] provided by SVF. +Specifically, we use it to track the value flow of the parameters +of these helper functions. Based on the value flow, we can +scope the usage (read or write) of parameters and decide +which category (see below) a helper function belongs to. With +the help of SVF, this categorization process can be conducted +in a principled way and scalable to analyze all helpers. +Attackers might manipulate the parameters of these helpers +to launch attacks. Therefore, based on the above analysis +results, we divide 260 BPF helper functions into five types. +As shown in Table 2, the first type (No Arg.) has no argu- +ments, which does not need any extra protection. The second +type (Pure Arg.) operates solely on its own arguments and +does not access kernel memory, which is also safe. The third +type (Read Only) accesses kernel in a read-only manner, and +the forth type (Write) may use its argument to modify the +kernel memory. The fifth type (Other) includes helpers that +are hard to categorize. For example, bpf_loopis the auxiliary +function that simplifies the verification process of loops. Note +that the last three types may interact with the kernel space +and potentially cause unauthorized access or even kernel ex- +ploitations by being abused by malicious BPF programs. +Table 2: BPF helper analysis result. CRP denotes critical region +protection, ROK denotes read-only kernel space, and DPA denotes +dynamic parameter auditing. +Type +# +Example +Applicable Defense +No Arg. +30 +bpf_get_retval() +No Need +Pure Arg. +16 +bpf_strncmp() +No Need +Read Only +75 +bpf_get_stackid_tp() +ROK/CRP/DPA +Write +129 +bpf_skb_set_tstamp() +CRP/DPA +Other +10 +bpf_loop() +CRP/DPA +With this categorization, we now present three mechanisms +7 + +in MOAT that ensure helper security as follows. +Read-Only Kernel Space (ROK). Our analysis reveals that +the majority of helpers only access the kernel in a read-only +manner. These read-only helpers account for near one third +of all helpers. Even though in most cases, read-only helpers +do not alter the kernel state and are considered safe, MOAT +still sets the kernel space as read-only when executing these +helpers.3 This nullifies possibility of potentially tempering +kernel spaces, and it does not impose extra runtime overhead. +Normal Regions +AD +Critical Regions +AD +PKR +... +AD +Critical Regions +BPF Region +EN +PKR +... +Helper Call +Kernel Address Space +Kernel Address Space +AD Access-Disabled +EN Access-Enabled +Normal Regions +BPF Region +EN +EN +Figure 8: Critical region protection (CRP). +Critical Region Protection (CRP) in Kernel. Further to the +discussion in ROK, though many helpers only access kernel +in a read-only manner, they may still be abused to probe sen- +sitive data of the kernel, such as task_struct. Moreover, a +considerable number of helpers, as illustrated in Table 2, may +modify kernel memory. To prevent such abuse, we protect +these critical kernel regions with PKS. As shown in Fig. 8, +instead of treating the entire kernel memory as a whole, we +divide it into normal regions and critical regions. When enter- +ing helper functions, instead of setting the entire kernel space +as access-enabled (EN), those critical memory regions remain +access-disabled (AD), preventing any access to these regions. +Once the helper finishes, these normal region will be set back +to access-disabled (AD) to avoid potential security risk. It is +worth noting these critical memory regions do not vary with +helpers. That is, only helpers manipulated by attackers (e.g., +via deliberately crafted helper parameters) may attempt to +access these critical regions. These critical regions can be +specified in the configurations of MOAT. +r0 = 0x10 +r1 = r0 + 0x1 +call BPF_HELPER +BPF Instructions +Static Register Value +Inferred by Verifier +0x10 +0x11 +Runtime Register Values +for Each Instruction +... +0x10 +0xbe +0x10 +0x11 +r0 +r1 +r0 = 0x10 +r0 = 0x10 r1 = 0x11 +r0 = 0x10 r1 = 0x11 +... +... +Figure 9: Register value tracking of the verifier. While the veri- +fier can indeed deduce a possible value range of each register, for +simplicity, we use a value point (e.g., r1 = 0x11) here. +Dynamic Parameter Auditing (DPA). To further regulate +helpers, we propose dynamic parameters auditing (DPA), +which leverages the information obtained from the BPF ver- +3There exist few functions in this category that rely on synchronization +facilities like Read-Copy Update (RCU), which cannot be applied with this +protection scheme. +ifier to dynamically check if the parameters are within their +legitimate ranges. As illustrated in Fig. 9, the verifier can +deduce the value range of each register via static analysis +(as a practical assumption, we allow the statically deduced +value ranges to be invalid; see below for clarification). MOAT +logs such value range information, and during runtime, MOAT +serves as a “gateway” when the BPF program enters a helper +function to check if the provided parameter values are within +the verifier-deduced value ranges. In our example, we can +check if r0==0x10;r1==0x11 when BPF_HELPER is called. +If the parameter runtime values do not match with the static +analysis results, the BPF program is terminated immediately. +Clarification. In the aforementioned DPA strategy, one may +question if the “legitimate value ranges” inferred by the veri- +fier are correct. Recall as discussed in our research motivation +in Sec. 3, there exist several vulnerabilities that can be lever- +aged to bypass verifier static checks. Overall, we clarify that +we do not need the verifier’s static analysis results as always +correct. Nevertheless, as long as the runtime input values are +inconsistent with the static analysis results, we terminate the +BPF program. For such cases, either the verifier is wrong or +the BPF program is behaving maliciously, both are highly +severe and we require manual inspection of the triage. We +assume the chance of both verifier and BPF program being +unsafe (but still appear to be consistent) is extremely low, +if at all possible. In fact, for today’s known BPF exploita- +tions, the verifier’s static analysis results (e.g., deciding the +value ranges of certain pointers) are safe, though incomplete +(omitting some data facts on subtle variables) and thus being +leveraged by malicious BPF programs. Also, even though it +may be technically feasible to perform dynamic auditing to +validate the data facts after executing every BPF instruction, it +is apparently too costly. MOAT thus leverages PKS to deliver +a low-cost and principled isolation. +Hybrid Usage. We summarize the applicability of these three +defense mechanisms in Table 2. On the one hand, DPA pro- +tects helpers from being abused by ensuring the validity of +their parameters. On the other hand, even if the helpers are +already compromised, ROK and CRP can still protect the +kernel from these compromised helpers. Thus, combining +these mechanisms together improves the overall security for +both BPF helpers and the kernel itself. Moreover, we want to +emphasize that these defenses are not dependent on a partic- +ular helper. Instead, they are applicable to helper groups, as +listed in Table 2. Although it can be argued all three defenses +may be evaded in extreme circumstances, we believe the at- +tack feasibility is very low (if it exists at all), given that the +BPF program has been isolated by MOAT and these restricted +helpers constitute a relatively minor attack surface. Our inves- +tigation on existing vulnerabilities supports this assumption. +8 + +5 +Implementation +MOAT is written in 2,075 lines of C code, as a loadable kernel +module.4 It includes three components: a BPF loader, a BPF +executor, and a key allocator. We explain key points below. +Portable Implementation. The major components of MOAT +are implemented as hooks to replace their corresponding ker- +nel functions. This is accomplished using an existing ker- +nel hook utility named ftrace [7]. This introduces a small +amount of overhead, but it allows these major components to +be kernel-agnostic and can be easily ported across different +kernel versions. Though the overhead of the current MOAT +prototype is reasonable (see details in Sec. 6.2), we anticipate +to further reduce the performance overhead of MOAT, if it is +implemented via directly modifying kernel. +Kernel Interrupt Handling. Though the major components +of MOAT are implemented as loadable modules, certain low- +level codes still require direct kernel modification. For in- +stance, during the execution of BPF programs, an interrupt +may occur and take over the control flow to its handler. Note +that most interrupt handlers require access to kernel memory +and as a result, the PKS would presumably raise spurious +alerts. Thus, we need to temporarily disable PKS inside these +handlers and re-enable it once the handlers are finished. The +modified code is shown in Fig. 10. Additionally, the exception +handler of the kernel is also modified to support terminating +and detaching malicious BPF programs upon violation. +1 +mov +%cr4,%rbx +2 +push %rbx +; save CR4 +3 +and +$0xfffffffffeffffff, %rbx ; clear CR4.PKS +4 +mov +%rbx,%cr4 +5 +call \cfunc +; invoke handler +6 +pop +%rbx +7 +mov +%rbx,%cr4 +; restore CR4 +Figure 10: The modified kernel interrupt handler in entry_64.S. +6 +Evaluation +To evaluate MOAT, we first analyze how MOAT mitigates +various attack interfaces, and then benchmark its CVEs de- +tectability in Sec. 6.1. We then assess the performance of +MOAT under different BPF program setups in Sec. 6.2. Lastly, +the functionality of MOAT is tested using various types of BPF +programs and under different scenarios in Sec. 6.3. +6.1 +Security Evaluation +6.1.1 +Analysis of Attack Surface Mitigation +We first systematically analyze how MOAT mitigates five rep- +resentative attack interfaces presented in the BPF ecosystem. +These potential attack interfaces are illustrated in Fig. 11. +4We will release the codebase of MOAT once this paper is published. We +will maintain MOAT to benefit the community and follow-up research. +PTEs +IDT/GDT +Memory +BPF +Program +Helper +Auditor +BPF +Helper +IA32_PKRS +CR4.PKS +3 +4 +1 +2 +5 +PKS Region +Write Disabled +Access Disabled +Figure 11: Analysis of mitigating potential attack surfaces. +1 Arbitrary Kernel Accesses. Currently, the most prevalent +threat to the BPF ecosystem is the ability of malicious BPF +programs to arbitrarily modify kernel memory. In order to +accomplish this, these BPF programs typically employ corner- +case operations to deceive the verifier during the loading +phase and to behave maliciously during runtime. This type +of attack is effectively mitigated due to the fact that MOAT +derives the minimum necessary memory regions of each BPF +program and uses PKS to prevent any runtime access beyond +this region (Sec. 4.1), mitigating such illegal accesses. +2 Helper Function Abuse. Apart from launching attack di- +rectly from BPF programs, a malicious BPF program may +carefully prepare parameter values by exploiting similar +corner-cases operations in 1 and pass them to abuse certain +helpers. To prevent such abuse, MOAT features three security +enforcement schemes (Sec. 4.2.2) to dynamically audit helper +parameters and also protect critical kernel memory regions +during the execution of these helpers. Thus, the attacker can +no longer take advantage of these helpers. +3 PTE Corruption. A page’s PKS region is configured via +its PTE. Consequently, a malicious BPF program may attempt +to tamper these PTEs to disable MOAT. However, this is im- +possible since MOAT sets these PTEs as access-disabled; they +are thus protected by PKS like other kernel resources. +4 Descriptor Table Tampering. Descriptor tables like GDT +and IDT are essential for segmentation and interrupt handling. +Since they are needed for these critical functions, blindly set- +ting them as access-disabled would cause system crashes. +However, since these descriptor tables are only accessed in +a read-only manner, MOAT sets them as write-disabled to +thwart any tampering made by malicious BPF programs. This +effectively prevents malicious BPF programs from compro- +mising the kernel using these tables. +5 Hardware Configuration Tampering. Besides memory- +based attacks discussed above, attackers may also directly +disable PKS through hardware configurations. As described +in Sec. 2, CR4.PKS and IA32_PKRS are two critical registers +for configuring PKS. One may disable PKS via modifying +these two registers. However, both registers can only be mod- +ified via special instructions, and BPF instruction sets do not +include any of these. Therefore, BPF bytecodes containing +these instructions are rejected immediately. Since the BPF +programs are set to W ⊕ X (meaning write and executable +permissions cannot be simultaneously enabled), adding these +instructions via self-modification is also impossible. +9 + +6.1.2 +Real-world CVE Evaluation +We analyzed all 37 CVEs relating to BPF since 2020 and +found that nine of them are related to runtime memory corrup- +tion caused by malicious BPF programs, which falls within +the application scope of MOAT. Even though these memory +corruption vulnerabilities only account for about one-forth +of all CVEs, they all result in privilege escalation and pose a +severe security threat to the kernel. As listed in Table 3, five +of these vulnerabilities have PoC exploits available and are +evaluated at this step. +We report that MOAT can successfully mitigate all of them. +We clarify that these five are not cherry-picked; the untested +four only have high-level text descriptions without further de- +tails or any PoC, making it extremely hard for us to build +a workable exploit based on these descriptions alone. In- +stead, we thoroughly analyze these four vulnerabilities. Due +to their conceptual similarity to the other five tested cases, +it should be accurate to conclude that these four can also be +mitigated by MOAT. For instance, although there is no exploit +for CVE-2021-3444, it shares the same logistics with CVE- +2021-31440, albeit with different BPF instructions. Note that +both originate from incorrect truncation. From the fact that +CVE-2021-31440 is mitigated by MOAT, we would believe +the same for CVE-2021-3444. +Table 3: BPF CVE detectability evaluation. +denotes experimented +and mitigated by MOAT. +denotes the CVEs share conceptually +identical patterns, though they lack available PoC exploit. +CVE ID +Description +Status +2022-2785 [43] +Incorrect Instruction Rewrite +2022-23222 [42] +Mischeck *_OR_NULL Pointer +2021-45402 [41] +Incorrect MOV32 Bound +2021-3490 [40] +Incorrect ALU32 Bound +2021-31440 [37] +Incorrect 32-bit Truncation +2021-3444 [39] +Incorrect MOD32 Truncation +2021-33200 [38] +Incorrect Pointer Arithmetic +2020-8835 [36] +Incorrect 32-bit Bound +2020-27194 [35] +Incorrect OR32 Bound +CVE Case Study. To better explain how MOAT mitigates +these CVEs, we elaborate on the exploit paths for two of +them, CVE-20222-23222 and CVE-2020-27194. +CVE-2022-23222 is a pointer mischeck vulnerability intro- +duced via a rather new feature of BPF named bpf_ringbuf. +This new feature was brought to BPF in 2020 along with +a new pointer type named PTR_TO_MEM_OR_NULL. However, +the verifier had not been updated to track the bounds of this +new type, resulting in this vulnerability. As illustrated in +Fig. 12, the malicious payload first retrieves a nullptr via +bpf_ringbuf_reserve (line 1), which returns this newly- +added pointer type named PTR_TO_MEM_OR_NULL. Since this +new type is not tracked by the verifier, the payload can bypass +pointer checks by convincing the verifier that r1 is 0x0 when +it is actually 0x1 (line 3). This pointer can then be multiplied +with any offset to perform arbitrary kernel accesses (line 9). +However, such arbitrary access violates PKS immediately and +is terminated by MOAT (line 10). +1 +r0 = bpf_ringbuf_reserve(fd, INT_MAX, 0) +2 +r1 = r0 +// R:r0=0;r1=0 V:r0=r1=? +3 +r1 = r0 + 1 +// R:r0=0;r1=1 V:r0=r1=? +4 +if (r0 != nullptr) { +// R:r0=0;r1=1 V:r0=r1=? +5 +ringbuf_discard(r0, 1) +6 +exit(2) +7 +} +8 +off = +// R:r0=0;r1=1 V:r0=r1=0 +9 +off = off * r1 +// R:off= V:off=0 +10 +*(ptr+off) = 0xbad +// PKS violation! +Figure 12: Code snippet of CVE-2022-23222. R denotes variable +runtime statuses. V denotes verifier-deduced values of variables. +CVE-2020-27194 is a vulnerability due to incorrect trunca- +tion. As in Fig. 13, the user first inputs an arbitrary value +in the range of [0,0x600000001] (line 1). Then, two con- +ditional clauses help the verifier to determine its lower and +upper bounds (line 3 and line 5). However, when tracking +the BPF_OR operator (line 7), the verifier performs a wrong +truncation on its upper bound. After the truncation, the user- +controlled r5is viewed by the verifier as a legitimate constant +scalar 0x1(line 7), which can later be used as the offset to per- +form arbitrary accesses to the kernel (line 8). Similarly, such +accesses can be detected by MOAT and terminated instantly. +1 +r5 = +2 +r6 = 0x600000002 +3 +if (r5 >= r6) +// R&V:r5<=0x600000001 +4 +exit(2) +5 +if (r5 <= 0) +// R&V:0x1<=r5<=0x600000001 +6 +exit(2) +7 +r5 = r5 | 0 +// R:r5= V: r5=0x1 +8 +*(ptr+r5)=0xbad +// PKS violation! +Figure 13: Code snippet of CVE-2020-27194. R denotes variable +runtime statuses. V denotes verifier-deduced values of variables. +6.2 +Performance Evaluation +We assess MOAT performance overhead on Linux v5.195 and +a 16-core Intel 12700H, whose efficiency cores are disabled +and performance cores are locked to 4 GHz to avoid random- +ness. As a common setup, the cycle and time statistics are +measured via the rdtscp instruction and the kernel utility +get_ktime_raw(), respectively. +6.2.1 +Micro Benchmark +For micro benchmark, we measure the CPU cycles of four +key operations in MOAT. We list the the four operations in +Table 4. switch_pks() enables/disables PKS by setting/- +clearing the corresponding control bit in CR4. set_pkrs() +changes region permissions by changing IA32_PKRS via +WRMSR. get_pkrs() returns current permission configuration +by reading IA32_PKRS via RDMSR. assign_page() changes +5The kernel is slightly modified as described in Sec. 5. +10 + +the permission region of one page by modifying its PTE. Each +operation is measured by averaging ten runs of one million +invocations to eliminate randomness. +Table 4: Micro benchmark results. As a reference [51], userspace +RDPKRU, WRPKRU, and pkey_assign() take 0.5, 23.3, and 1104.9 +cycles, respectively. +Operation +# Cycle +Note +switch_pks() +4.2 +Set/Clear CR4.PKS +set_pkrs()/WRMSR +71.7 +Set region permissions +get_pkrs()/RDMSR +25.8 +Get region permissions +assign_page() +1120.4 +Assign a page to region +As Table 4 shows, the most expensive operation is +assign_page() which modifies the region one page be- +longs to, including locating its PTE and changing specific +bits within. Notably, setting and getting the region permis- +sions (set_pkrs()/get_pkrs()) in PKS is much more ex- +pensive than its userspace variant in libmpk [51] (see the +caption of Table 4). We presume that this is because in PKU, +the region permission is controlled via a dedicated register +named PKRU with two special instructions RDPKRU/WRPKRU, +whereas in PKS employed by MOAT, its region permission +is stored in an MSR named IA32_PKRS without any special +instruction. To configure the permission in IA32_PKRS, one +has to use the general RDMSR/WRMSR instructions with the +MSR ID 0x6E1, which requires additional cycles to complete. +Similarly, directly enabling/disabling PKS via switch_pks() +also takes fewer cycle than set_pkrs(). +Since configuring permission via set_pkrs() is more +expensive than switch_pks(), on situations where MOAT +needs to temporarily switch back to kernel regions (e.g. inter- +rupt handling), it uses switch_pks() to disable PKS instead +of using set_pkrs(). Then, before returning to BPF pro- +grams, we reactive PKS to maintain isolation. +6.2.2 +Macro Benchmark +To prepare the macro benchmark suite, we consider the fol- +lowing properties. +(a) To test the performance of MOAT conducting fixed and +dynamic key allocation, it is necessary to include BPF +programs of varying sizes. +(b) The number of BPF programs should exceed the num- +ber of available keys to test MOAT in situations where +hardware keys are insufficient. +(c) The BPF programs should be highly parallel to evaluate +the waiting time when dynamic keys are insufficient. +(d) The execution order should reflect actual system behav- +ior with high enough frequency to stress MOAT. +To simultaneously fulfill these requirements, we prepare +macro benchmark as follows. We choose seven different +events frequently triggered in the kernel, which are sys_open, +sys_close, +sys_read, +sys_write, +sched_switch, +page_fault_user, and page_fault_kernel. These events +are of high frequency (e.g., sched_switch occurs on every +context switch) and can reflect actual BPF running behavior. +For each of these events, we attach three BPF tracepoints of +varying sizes to log this event. This ensures that these BPF +programs are highly parallel. +MOAT Configuration. In both regular and extreme cases (see +below), we choose the configuration as follows: the threshold +for dynamic key allocation is ten pages. The number of fixed +keys is ten, while the number of dynamic keys is four. Two +keys are reserved for the kernel memory region and the shared +region (i.e., for per-CPU stack, IDT, GDT), respectively. +Regular Case. In the regular case, we attach each one of +these events with three types of BPF tracepoints, i.e., small (1 +page), medium (10 pages) and large (200 pages). We run +each setup ten times, and each run consists of 1,000 invoca- +tions of each tracepoint. The average results are reported in +Fig. 14. We find that even in the worst case, MOAT imposes a +moderate overhead of less than 30%. This overhead occurs +when launching the medium-size BPF program attached to +the event page_fault_kernel. Since its size (10 pages) does +not exceed the threshold of dynamic key allocation, it has to +repetitively assign and return the dynamic key to its pages +upon every entry point and exit. As reflected on the micro +benchmark in Sec. 6.2.1, such key assignment is quite costly. +Overall, we interpret the performance penalty is aligned with +our expectation, and the overall overhead is reasonable. +All large-size BPF programs exceed the page number +threshold of dynamic key allocation. Therefore, MOAT as- +signs fixed keys to them during their loading phase without +incurring runtime overhead. The incurred overheads are gen- +erally moderate: for all cases, the overheads are less than 10%. +Moreover, the overheads for those small-size BPF programs +are all less than 22%, which lie between the large-size and +the medium-size ones. Apart from the total overhead reported +above, we also investigate the waiting overhead, which is the +amount of time a BPF program must wait if there is no dy- +namic key available. Note that in the regular cases above, 14 +programs are smaller than the page number threshold; they +are configured to use the dynamic key allocation scheme, +although there are only four dynamic keys available. Their +waiting statistics are shown in Table 5. It is seen that although +the average waiting time is near 1µs, less than 1% BPF exe- +cutions really experience this delay. Considering there are 14 +running processes and only four dynamic keys available, we +can conclude that the dynamic key allocation policy handles +parallelism reasonably well. Moreover, this also shows that +four dynamic keys are sufficient for most scenarios; adding +more dynamic keys brings marginal benefit. +Table 5: Waiting time statistics. +Avg. (ns) +Waited Avg. (ns) +Max. (ns) +# Waited +7.1 +915.2 +2559 +0.8% +Extreme Cases. The above regular cases only evaluate MOAT +under situations where dynamic keys are limited but fixed +keys are sufficient. Here, we further explore MOAT’s overhead +via extreme cases. Instead of attaching three BPF programs +11 + +1.00 +1.00 +1.00 +1.00 +1.00 +1.00 +1.00 +1.19 +1.22 +1.07 +1.04 +1.14 +1.06 +1.06 +1.25 +1.29 +1.25 +1.12 +1.18 +1.22 +1.24 +1.08 +1.09 +1.03 +1.04 +1.05 +1.03 +1.03 +0.00 +0.50 +1.00 +1.50 +pf_u +pf_k +sched +open +close +read +write +Relative Time +Base +Small +Medium +Large +Figure 14: Regular macro benchmark. +of varying sizes, as we did in the regular cases above, in the +extreme case evaluation we attach three large (200 pages) +BPF programs to each tracepoint. Under this setting, there are +only ten fixed keys available, although there are 21 large-size +BPF programs, requiring dynamic key allocation for over half +of these programs. Since each of these programs contains +over 200 pages, there are a large number of page assignments +occurring upon their program entry points and exits. +Table 6: Extreme overhead. +Static Keys (ns) +Dynamic Keys (ns) +Avg. +Max. +Avg. +Max. +Waited +# Waited +140.7 +202.8 +3630 +4401 +1968.1 +4% +We report the evaluation results of extreme cases in Ta- +ble 6. We find that MOAT imposes a negligible overhead to +BPF programs that use fixed keys even under such extreme +cases. And for those large BPF programs that use dynamic +keys, the average overhead is still reasonably low (around +3.6µs). Overall, we point out that real-life scenarios seldomly +require this many BPF programs with large maps running +concurrently. Moreover, the currently observed overhead can +be further reduced by sharing these large maps between BPF +programs, thereby reducing the need for fixed keys. We also +report that the waiting time due to the shortage of dynamic +keys shows a similar pattern to the regular cases. Although +the average waiting time is near 2µs, less than 5% of the +executions would experience this delay. +6.2.3 +Real-world Case Study +To evaluate the performance of MOAT under real-world sce- +narios, we setup a BPF port forwarding program which redi- +rects incoming requests to the memcached [24] memory +database. To prepare the benchmark, we choose YCSB [19] to +generate six distinct workloads and test the overall throughput +of the memcached service. The results are shown in Fig. 15. +From the figure, we can see that MOAT imposes on average +6% (up to 14%) slowdown to the overall performance of the +BPF-based port forwarding, which is acceptable considering +the security benefits MOAT provides. Note that this overhead +is far less than the worst overhead we observed from the +regular/extreme cases above, which further justifies our as- +sumption that BPF programs are invoked less frequently in +real-world applications than in extreme cases. +5586 +5649 +7407 +5649 +14084 +4975 +5464 +5681 +6493 +5050 +13889 +4366 +0 +5000 +10000 +15000 +YCSB_A +YCSB_B +YCSB_C +YCSB_D +YCSB_E +YCSB_F +Throughput +(ops/sec) +Base +MOAT +Figure 15: Overall throughput of the memcached case study. +6.3 +Functionality Evaluation +To show that MOAT is able to support various BPF features, +we select seven BPF applications with varying functionalities +from the famous bcc toolbox [52]. Among them, execsnoop +and opensnoopare used for kernel profiling, recording differ- +ent system events; tcptrace and net_monitor are used for +network monitoring, collecting packet statistics; xdp_drop, +xdp_cpu and xdp_interface can be used in firewalls and +various load balancing scenarios, redirecting or dropping +packages. These applications cover the majority of contem- +porary BPF ecosystem usage scenarios. After securing these +applications with MOAT, we examine the runtime status of +these applications and confirm that they are operating cor- +rectly and are not affected by MOAT. Furthermore, Fig. 16 +reports the performance evaluation results of these applica- +tions with MOAT enabled. The extra overhead incurred by +MOAT under different scenarios is reasonably low. Overall, +the evaluation shows that MOAT can be smoothly applied to +secure de facto BPF applications under various scenarios with +minimal engineering effort and moderate cost. +1.00 +1.00 +1.00 +1.00 +1.00 +1.00 +1.00 +1.07 +1.01 +1.25 +1.10 +1.21 +1.11 +1.07 +0.00 +0.50 +1.00 +1.50 +execsnoop +opensnoop +tcptrace +net_monitor +xdp_drop +xdp_cpu +xdp_interface +Relative Time +Base +MOAT +Figure 16: Application benchmark. +7 +Related Work +In-Kernel Isolation. Most existing works [10, 12–14, 16, 23, +26, 29, 49, 61, 64] on kernel isolation focuses kernel com- +ponents like device drivers and file systems, which are dis- +tinct from BPF programs and hence cannot be reused directly +in our scenario. Existing works can be roughly divided into +three categories: virtualization, Software Fault Isolation (SFI), +and formal methods. Narayanan et al. [49] propose LVD, +which isolates kernel components in a virtualized environ- +ment. Based on LVD, Huang et al. [29] split kernel modules +into individual components for finer-grained isolation. SFI +12 + +is employed to instrument programs at the source or binary +level [13, 14, 23]. These works ensure kernel security by in- +serting pointer checks prior to memory accesses. Furthermore, +formal methods enable principled isolation of kernel compo- +nents, e.g., separating kernel code from untrusted drivers [61], +or verifying file system correctness [10, 16]. +We believe none of these methods are readily re-usable +in our BPF scenario. Virtualization method [12, 29, 49, 64] +require placing the program in a separated address space, +making it hard for BPF programs to interact with kernel. +SFI [13, 14, 23] is based on program (compile-time) instru- +mentation, whose inserted software checks often lead to high +runtime overhead. Lastly, the BPF verifier itself performs +formal verification, which shares conceptually similar advan- +tage and drawbacks with existing formal method-based kernel +isolation methods [10, 16, 61]; MOAT employs hardware ex- +tensions to offer more principled BPF isolation. +MPK-Based Isolation. Prior to PKS, Intel first announced +its userspace variant PKU. Consequently, most existing +works [27, 51, 58] using MPK focus on userspace isolation. +To better utilize PKU as an isolation primitive, Park et al. [51] +proposed libmpk that resolves the semantic discrepancies +between PKU and conventional mprotect. There are also +works [27, 58] that leverage this hardware feature to protect +confidential data. Apart of using PKU to isolate normal user +applications, efforts are made to isolate trusted applications +in SGX via PKU [17, 33]. SGXLock [17] establishes mu- +tual distrust between kernel and the trusted SGX applications, +while EnclaveDom [33] enables intra-isolation within one +enclave. PKU has been used for kernel security [26, 57] as +well. IskiOS [26] applies PKU to kernel pages by marking +them as user-owned, while Sung et al. [57] employ PKU to +protect userspace unikernels. As a new feature introduced in +2021, research works using PKS are rather rare comparing +to PKU. Linux community attempted to use PKS to prevent +stray writes [1], which refers to kernel accidentally writing to +wrong addresses. +BPF Security. There also exist many works [25, 31, 32, 50, +60] on securing the the BPF ecosystem. However, most of +these works use formal methods to enhance the following +BPF components: the verifier, the JIT compiler and the BPF +program itself. To enhance the standard BPF verifier, Ger- +shuni et al. [25] built PREVAIL based on abstract interpre- +tation [20], which supports more program structures (e.g. +loops) and is more efficient comparing to the standard verifier. +PRSafe [32], on the other hand, designs a new domain-specific +language based on primitive recursive functions, whose prop- +erties ensure that all computations must terminate. The ul- +timate goal of PRSafe is to build a mathematically verifi- +able compiler for BPF programs. As for BPF JIT compiler, +Jitk [60] is a classic BPF JIT compiler whose correctness is +proven manually. Further, Nelson et al. [50] propose Jitterbug +to generate automated proof for real-world BPF JIT compilers. +Lastly, Luke Nelson [31] build proof-carrying BPF programs, +requiring developers to provide a correctness proof alongside +with the program before loading it into the kernel. +8 +Discussion +Platform Migration. The current prototype implementation +of MOAT is based on MPK, a hardware extension available on +Intel platforms. Below, we discuss migrating MOAT to other +platforms with similar hardware extensions. +ARM Memory Domains. “Domain” is a MPK-like feature +supported since ARMv7 [3]. It employs 4-bit domain keys in +PTEs and a Domain Access Control Register (DACR) in su- +pervisor mode. Following a similar rationale to MPK, DACR +allows accesses to be configured as denied, fully-allowed, or +the same as PTEs. Since this feature is only supported on first- +level and section-level PTEs, the domain boundaries must +be aligned to 1 megabyte. Due to the similarity between this +feature and MPK, we expect MOAT to be implemented on +ARM with a moderate effort using this feature. +RISC-V Domain Keys. As an open-source architecture, there +exists a hardware extension on the RISC-V platform that +supports similar features as MPK named Donky [53]. Donky +leverages ten unused bits in the PTEs as a protection key, +hence supporting 1,024 permission regions. Since Donky +supports 1,024 keys, it is no longer possible to control permis- +sions for all these regions using a single register, like MPK +does. Donky thus introduces a 64-bit DKRU register with four +key slots. Each slot can be loaded with a 10-bit protection key. +Only when a key is loaded in DKRU can its associated region +be written to or read from. From the description above, we +interpret that Donky is quite flexible, and therefore, MOAT +may be smoothly implemented on RISC-V using Donky. +BPF JIT Support. As described in Sec. 2, there are two ways +of executing a BPF program: directly interpreting the BPF +bytecode, or using a JIT compiler for improved performance. +Our prototype implementation of MOAT is based on the BPF +interpreter. However, we note that the design of MOAT is +compatible with the JIT compiler. First, the PKS is config- +ured at the entry and exit points of running a BPF program, +which is independent of the BPF program execution method. +Second, the operations that MOAT performs during the BPF +execution, such as helper auditing, are implemented as part +of BPF helpers and also decoupled from how BPF programs +are executed. Therefore, MOAT is essentially agnostic about +the BPF program execution method, and it is adaptive to the +native code produced by the BPF JIT compiler. Moreover, +unlike the JIT compiler in Java virtual machine (JVM), which +compiles only hotspot code chunks of Java bytecode each +time, the BPF JIT compiler compiles the entire BPF program +bytecode into native code once. This further reduces the effort +of adapting MOAT to BPF programs compiled by JIT. +13 + +9 +Conclusion +Despite the increasing popularity of using BPF to extend +kernel functionality, the security of BPF programs is still a +concern. Recent attacks reveal that BPF applications can by- +pass static security checks and conduct unauthorized kernel +memory accesses. This paper has presented MOAT, which iso- +lates potentially malicious BPF applications from the kernel +using Intel MPK. MOAT addresses technical challenges and +delivers a practical and extensible protection mechanism, in +compensation to the contemporary BPF verifiers. 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In 2014 IEEE Symposium on Security and Privacy, +pages 308–323, 2014. doi: 10.1109/SP.2014.27. +17 + diff --git a/19FQT4oBgHgl3EQf1zZe/content/tmp_files/load_file.txt b/19FQT4oBgHgl3EQf1zZe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f431557fb4bfa6f4b8ed2d7af43e5860ba3cdcb --- /dev/null +++ b/19FQT4oBgHgl3EQf1zZe/content/tmp_files/load_file.txt @@ -0,0 +1,1212 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf,len=1211 +page_content='MOAT: Towards Safe BPF Kernel Extension Hongyi Lu1,2, Shuai Wang2,∗, Yechang Wu1, Wanning He1, Fengwei Zhang1,∗ 1Southern University of Science and Technology 2Hong Kong University of Science and Technology Abstract The Linux kernel makes considerable use of Berkeley Packet Filter (BPF) to allow user-written BPF applications to execute in the kernel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF employs a verifier to statically check the security of user-supplied BPF code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Recent attacks show that BPF programs can evade security checks and gain unau- thorized access to kernel memory, indicating that the verifica- tion process is not flawless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In this paper, we present MOAT, a system that isolates potentially malicious BPF programs using Intel Memory Protection Keys (MPK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Enforcing BPF program isolation with MPK is not straightforward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT is carefully designed to alleviate technical obstacles, such as limited hardware keys and supporting a wide variety of kernel BPF helper functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We have implemented MOAT in a prototype kernel module, and our evaluation shows that MOAT delivers low-cost isolation of BPF programs under various real-world usage scenarios, such as the isolation of a packet-forwarding BPF program for the memcached database with an average throughput loss of 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1 Introduction It is common to extend kernel functionality by allowing user applications to download code into the kernel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In 1993, the well-known Berkeley Packet Filter (BPF) was introduced for this purpose [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The classic BPF is an infrastructure that inspects network packets and decides whether or not to forward or discard them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' With the introduction of its ex- tended version (referred to as eBPF) in the Linux kernel, BPF soon became more powerful and is now utilized in numerous real-life scenarios, such as load balancing, scheduling, and auditing [18, 22, 28, 52, 62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To ensure security, BPF is equipped with a verifier [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The verifier performs a variety of static analyses to ensure the user-supplied code is secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For instance, the verifier tracks the bounds of all pointers to prevent an out-of-bound access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Given that BPF code runs directly within the kernel, ∗Shuai Wang and Fengwei Zhang are the corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' the verifier becomes crucial for the BPF security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Neverthe- less, as pointed out by recent studies [25, 31, 32, 50, 60], the currently available verifier has various limitations, and is in- sufficient for the overall security of BPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' First, the current BPF ecosystem supports a variety of kernel functionalities with over 200 dedicated APIs [2], resulting in a complicated verification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Even though the verifier’s correctness has been formally proved [59], the gap between abstraction and implementation may still result in vulnerabilities [35–41, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Second, BPF Just-In-Time (JIT) is currently supported on multiple platforms, including x86, ARM, and RISC-V, whose differences frequently result in subtle vulnerabilities [44, 45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' note that the verifier cannot detect vulnerabilities in the JIT stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Third, due to the rapid expansion of BPF capabilities, the verifier has to be frequently updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Nonetheless, it is inherently difficult to frequently update a complex static veri- fication tool without introducing new vulnerabilities [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To date, the BPF subsystem has been constantly exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For instance, two privileged-escalation vulnerabilities have been discovered in the implementation of bpf_ringbuf, a rather new BPF feature introduced in 2020 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Further, the veri- fier’s register-value tracking is quite complex and has been bypassed by several severe vulnerabilities [35–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Given the increasing security threats in BPF and the chal- lenge of enforcing safe BPF programs with merely static verification, we seek to employ hardware extensions to sand- box untrusted BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In particular, we leverage Intel Memory Protection Keys (MPK) [9], an emerging hardware extension which partitions memory into distinct permission groups by assigning up to 16 keys to their Page Table En- trys (PTEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' With the aid of MPK and the BPF verifier’s analysis results, we present MOAT, which isolates untrusted BPF programs in a low-cost and principled manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For in- stance, two MPK protection keys K and E may be assigned to the kernel and a BPF program, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' When the kernel transfers control to the BPF program, it can set K as access- disabled to prevent the potentially malicious BPF program from tampering with kernel memory regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Despite its promising potential, we observe that using MPK 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='13421v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='CR] 31 Jan 2023 tracepoint packet filter schduler tracepoint packet filter schduler User Application Kernel packet filter schduler tracepoint BPF Programs BPF Bytecode Verifier Maps Helpers call bpf_pid .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' log next_sched ret next_sched Kernel BPF (Runtime) Utilities BPF Bytecode BPF Compiler Figure 1: BPF overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We illustrate the BPF compilation procedure, and the execution context of a sample BPF program attached to the kernel scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Note that BPF verification is conducted at the BPF bytecode loading time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' to enforce BPF isolation is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT is de- liberately designed to overcome two major technical hurdles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' First, Intel MPK provides a maximum of 16 keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Thus, it becomes challenging to support many BPF programs running concurrently with this limited number of hardware keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Exist- ing workarounds like key virtualization [51] are incompatible with the BPF scenario and challenging to be implemented in kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This is because the key virtualization heavily relies on scheduling and notification facilities that are only available to userspace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' directly reusing them in the kernel space may largely block kernel threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To address this hurdle, we pro- pose a novel dynamic/fixed key allocation scheme that can support multiple BPF programs with a small overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Second, while MPK-based hardware isolation mitigates ma- licious BPF programs, helper functions provided by the BPF subsystem may be exploited by attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Overall, the growth of the BPF ecosystem is accompanied by the expansion of its dedicated helper functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' helper functions facilitate various tasks commonly conducted by a BPF program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' On one hand, MOAT should allow benign BPF programs to freely use these helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' On the other hand, MOAT must be cautious enough with these APIs to ensure they are not exploited by attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Given that there are over 200 helpers [2] provided in the latest Linux kernel, designing individual security policy for each of them is impractical and less extensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To this end, we analyze all existing helpers with static dependency-analysis, and propose several general defense schemes, each of which is applicable to a group of helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We envision that when a new helper is added, MOAT can be applied easily without introducing new schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To evaluate the security impact of MOAT, we systemati- cally examined how MOAT mitigates attack surfaces due to untrusted BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We also empirically analyze all recent CVEs within MOAT’s application scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The result shows that MOAT successfully mitigates each CVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' More- over, we evaluate the performance overhead of MOAT under representative and edge-case scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' First, we examine the performance of our dynamic/fixed key allocation policy by assessing a use case where multiple programs are executed concurrently to use all MPK keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Then, we build a real- life port-forwarding BPF program for the memcached [24] database, and secure it with MOAT to measure how MOAT influences memcached’s throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Furthermore, we apply MOAT to several real-world BPF applications [52] to illus- trate that MOAT can be directly applied to the current BPF ecosystem with minimal engineering effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT’s worst case performance overhead in all these experiments is less than 30%, which is acceptable given the security benefits MOAT provides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Moreover, MOAT imposes only 6% over- head on average to the memcache’s throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In sum, we make the following contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Instead of merely relying on BPF verifiers to statically validate untrusted BPF programs, this paper for the first time advocates to isolate user-supplied BPF programs with an emerging hardware extension, Intel MPK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Technically, MOAT is specially designed to address domain- specific challenges including limited hardware keys and protecting over 200 helper functions in the BPF ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We implement a prototype of MOAT as a Loadable Kernel Module (LKM) of the latest Linux (v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='19) and conduct a thorough evaluation of its security and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The evaluation shows that MOAT delivers a principled security guarantee with moderate performance penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1 Berkeley Packet Filter (BPF) BPF Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF [4] was originally introduced to facili- tate flexible network package filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Instead of inspecting packages in the userspace, users can provide BPF instructions specifying package filter rules, which are directly executed in the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This allows configurable package filtering without costly context switching and data copying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Modern Linux kernel features extended BPF (eBPF), a Linux subsystem, which supports a wide range of use cases such as kernel pro- filing, load balancing, and firewalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1 Popular applications like Docker [34], web browsers [30, 48], and kernel debugging utilities like Kprobes [8] are built on top of BPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1 depicts an overview of how BPF programs are com- piled and then deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The eBPF subsystem offers ten general-purpose 64-bit registers, memory stack, eBPF cus- tomized data structures (often referred to as eBPF maps), and a set of eBPF helper functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To use eBPF (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=', for ker- nel profiling), users can first write their own BPF programs (in C code) to specify the functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The BPF programs will then be compiled into BPF bytecode and downloaded into the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Given that eBPF code is written by untrusted 1While there are indeed two variants of BPF: classic BPF (cBPF) and eBPF, cBPF is internally converted into the latter variant by the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 2 users, the kernel employs a verifier to conduct several checks during the bytecode loading time (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' By default, the bytecode is executed by the BPF interpreter (omitted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Additionally, depending on the kernel configuration and architectural support, an optional JIT compilation may be applied to the bytecode for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The BPF bytecode is then attached to certain kernel components, based on its specific end goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For instance, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1, a BPF program attached to kernel scheduler collects relevant statistics and decides which thread should be running next to improve overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' len < INSN_MAX no loop no dead code no OOB jmp Unverified CFG Check Phase Data-Flow Check Phase register tracking access check helper check misc fixups Verified Figure 2: BPF verification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF Verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF programs are written in C, and compiled into a RISC-like instruction set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As aforementioned, the kernel strictly verifies the BPF programs upon loading to ensure that they are safe to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 2 illustrates the verifying process in a holistic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' First, a BPF program is parsed into a control flow graph (CFG) by the verifier, which first performs a CFG check phase to ensure four key properties: 1) the program size is within a limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 2) there exists no back edges (loops) on its CFG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3) there exists no unreachable codes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' and 4) all jumps are direct jump and they refer to a valid destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Overall, given that BPF programs must be terminated, the CFG check phase ensures that all jumps are direct jumps and there are no back edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Given that said, loops are still feasible via unrolling at the cost of binary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The verifier further performs finer-grained data flow anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' It first tracks the value flow of every register to deduce its value ranges conservatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Based on these ranges, the verifier can decide if a pointer accesses safe memory regions, and if a parameter is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since this analysis is performed statically prior to execution, there exists possibility that a ma- licious BPF program uses certain operations to bypass this analysis [35–41, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Last, the verifier also performs some miscellaneous fixups, like rewriting certain instructions to simplify the follow-up JIT compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF Maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Out of security concern, the kernel also sets a rather strict space limit on BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Each program by default can only use up to 512 bytes stack space and 10 registers, which is far from enough for certain BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To address this problem, BPF maps can be allocated and provide additional space for BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Up to now, there are over 30 types of maps supported by kernel, such as array map and hash map [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Moreover, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1, maps may act as a communication channel between BPF programs and user applications, since some of these maps can be accessed by both the BPF program and the user application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF Helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Kernel also limits the kernel functions a BPF program may call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Those functions are dubbed BPF helper functions, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Up to now, there are over 200 helpers scattered across subsystems of the kernel [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Note that depending on the specific task, a BPF program can usually call a group of relevant helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For example, a BPF program attached to the scheduler is not allowed to call any helper related to kernel probing, but it can call bpf_pid2 to obtain the PID of the current process and chooses which process to be scheduled next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 00 01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 10 00 32 0 PKR PTE[62:59] = 0xF PTE[62:59] = 0xE PTE[62:59] = 0x1 PKR Entry Options 00 Access Enable (EN) 01 Access Disabled (AD) 10 Write Disabled (WD) 11 Access Disabled (AD) Page Table Entry Figure 3: Intel MPK overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Intel MPK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Intel introduces MPK [9] to provide efficient page table permissions control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' By assigning a MPK protec- tion key to each page table entry (PTE) of one process, users can enable intra-process isolation and confidential data access control [17, 27, 33, 51, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3, MPK uses four reserved bits [62:59]in each PTE to indicate which pro- tection key is attached with this page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Those three PTEs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3 are assigned with keys 0x1, 0xE and 0xF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since there are only 4 bits involved, the maximum number of keys is 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Then, a new 32-bit register named Protection Key Register (PKR) is introduced to specify the access per- mission of these protection keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Each key occupies two bits in PKR, whose values denote either access-disabled (AD) or write-disabled (WD), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' By writing to certain bits in PKR, the access permission of corresponding pages can be configured accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' It is worth noting that one key may be assigned to arbitrary number of pages by modifying their PTEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This facilitates changing the access permission of a large number of pages without severe performance penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Clarification and Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As a side note, there are actually two versions of Intel MPK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' One applies to the user-mode while the other applies to the supervisor-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For brevity, we refer these two versions in their conven- tional abbreviations as Protection Key Supervisor (PKS) and Protection Key User (PKU), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Most existing works [17, 26, 27, 33, 51, 58] are based on PKU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In MOAT, we use PKS instead since our goal is to isolate BPF programs, which execute in the supervisor-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The logistics behind these two versions are mostly identical with slight variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For instance, the permission configuration register in PKS is a Model Specific Register (MSR) named IA32_PKRS, which is inaccessible from userspace, whereas in PKU, this role is assigned to a dedicated register PKRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In addition to PKR, there also exists a bit in the control register CR4 that can dis- able/enable MPK entirely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' for PKU, this bit is CR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='PKE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' for 2Here, bpf_pid refers to bpf_get_current_pid_tgid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3 PKS, this bit is CR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='PKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To avoid potential misleading, the rest of the paper directly refers MPK leveraged by MOAT as PKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3 Motivation and Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1 Typical Threats to BPF Verifier Fast Feature Evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As a fast developing technology, threats may come from the inconsistency between the con- stantly expanding BPF capabilities and the rigorous static verification process imposed on them [39, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' It is a common practice to add corresponding verification procedures simul- taneously when introducing new features to BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, it is very hard to make changes correctly to the BPF verifier, a critical security kernel component, which has over 10K LoC and a variety of functionalities [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Challenging Pointer Tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Second type of threats origi- nates from the complexity of pointer tracking mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Al- though the correctness of the verifier is formally proved [59], there still exist gaps between the implementation and the abstraction, especially in some corner cases, such as sign ex- tension, truncation, and bit operators [35–41, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The fact that the contemporary BPF verifier only performs static analysis is a severe deficiency, as evidenced by the threats noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Performing sound and complete static analysis toward BPF programs to uncover potential threats is fundamentally challenging, and from the disclosed BPF vul- nerabilities, we find that there frequently exists a gap between verifier’s static analysis results and BPF programs’ runtime behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For instance, the verifier, based on its static analysis results, may conclude that a program is benign because it only accesses a memory region ranging from [0x0,0x1000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, by leveraging vulnerabilities like noted above, the software may behave differently during execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Therefore, a hardware feature, Intel MPK, is utilized to enforce further isolation, such that a BPF program is constrained in its own memory regions, and any runtime accesses that violate this constraint are effectively flagged and terminated by MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2 Threat Model Assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Our threat model considers a practical setting which is aligned with existing BPF vulnerabilities [35–41, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In particular, we assume attackers are non-privileged users with BPF access, since a root user already has the control over almost the entire kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Attackers can download their pre- pared BPF code into the kernel space to launch exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Attacker Capability and Application Scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT iso- lates user-submitted BPF programs and prevent them from accessing privileged kernel memory regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As will be intro- duced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4, a BPF program is given the minimum neces- sary resources and privileges to complete its task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To clarify, there are also other more subtle vulnerabilities (not relevant to memory exploitations) such as speculation, race condition, and DoS occurred to exploit the BPF subsystem [46, 47];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' a well-isolated BPF program can still launch these attacks to jeopardize the BPF subsystem and the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This research views them as common security defects shared by many other applications such as SGX enclaves [15, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To date, coun- termeasures have been deployed by the BPF verifier [11], and we assume the standard BPF verifier can handle them properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In contrast, correctly detecting memory-related BPF exploitation requires systematic and rigorous static analysis over BPF programs and is fundamentally hard for BPF ver- ifiers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT enhances mitigating memory-related exploita- tions with hardware-assisted isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Next, we present anal- ysis of three major components in our research context as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF Programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This includes the BPF bytecodes or the JIT- emitted native instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Our threat model takes the as- sumption that malicious BPF programs are able to bypass checks statically performed by the verifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' they may thus behave maliciously during runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Our threat model deems BPF programs as untrusted, and MOAT is designed to isolate them from the rest of the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' More specifically, every BPF program, during its runtime, is only allowed to access its own stack, allocated maps, and certain helper functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF Helper Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These helpers act as the interme- diate layer between the BPF subsystem and kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Certain malicious BPF programs can abuse these helpers to perform attacks, and therefore, we assume they are also untrusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT mitigates risks raised by adversarial-manipulated helper functions with practical defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Kernel is the target to protect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We assume the kernel is functioning normally, and attackers aim to leverage mali- cious BPF programs to gain unauthorized access to kernel data or executing arbitrary privileged kernel code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4 Design MOAT Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As motivated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3, current security design against malicious BPF programs solely relies on the static analysis performed by the BPF verifier, which is seen as a weak point and exploitable by non-privileged users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT instead delivers a principled isolation of BPF programs using MPK from the rest part of the kernel and prevent bypasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' bpf_lookup_elem call bpf_run .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' bpf_delete_elem mov %rax, $0x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' call bpf_helper st %(rax), $0x10 Helper Auditor BPF Memory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' bpf_get_time mov %rax, %rbx MOAT BPF Payload Access Rules Stack .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Maps MPK Verifier Kernel Memory 1 2 3 4 Figure 4: MOAT overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4 illustrates the lifecycle of a BPF program with the presence of MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1 Given a user-submitted BPF program P, MOAT statically derives the minimum necessary memory regions the program needs, such as stack, used maps and context by reading metadata from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 2 These regions (“BPF Memory” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4) are assigned to P using PKS, forming its runtime environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3 When the kernel invokes P, MOAT configures PKS to constrain P to its own regions and forbids its access to other memory regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4 On the occasions that P requires helper calls to interact with the kernel, depending on the helper types, MOAT may adjust involved kernel memory region permissions and also validate the helper parameter values to prevent helpers from being abused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Security Guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Overall, MOAT provides the following two key secure design guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' (i) A BPF program is given the minimum necessary ker- nel resources and privileges for completing its task, preventing any malicious behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' (ii) The interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' helper calls) between the BPF program and the kernel are audited thus not abused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Extensibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT leverages MPK, a de facto hardware ex- tension available on mainstream Intel architectures to isolate BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We view this design choice is consistent with recent hardware-assisted security enforcement works [17, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Nevertheless, we clarify that the design of MOAT is not lim- ited to leveraging MPK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' There exist similar hardware security mechanisms on other platforms and architectures such as the Memory Domains [3] on ARM and the Domain Keys [53] on RISC-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These mechanisms can be used to replace MPK on these platforms with a small amount of engineering effort;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 8 for our discussion on migration and extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1 General BPF Isolation In accordance with the BPF program lifecycle depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4, this section elaborates on the general isolation ap- proach offered by MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We further discuss two key techni- cal challenges in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 0x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 59 62 Kernel BPF BPF 0x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Shared by & Kernel Data Kernel Code Stack Maps Context Code Stack Context Code Shared Maps Page Table Entries Data Regions Runtime PKR Value Enable 01 00 01 00 AD EN EN N/A AD 01 01 00 00 AD EN EN AD 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='. 32 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 0x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 0x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' AD Access-Disabled EN Access-Enabled Figure 5: BPF memory regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1 BPF Memory Regions Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 5 depicts the memory regions of BPF programs and the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' By default, all pages should belong to the kernel memory region, and each page is initialized with a default MPK key value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Then, when a BPF program P is newly loaded into the kernel, MOAT decides the minimum pages it needs, and assigns these amount of pages to the memory region of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Note that the necessary memory sections of a BPF program includes its code, stack, and the context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' many non-trivial BPF programs also require BPF maps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=', array and hash maps) to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' After assigning these sections to the memory region of P, MOAT restricts P to its own memory regions by configuring the PKR register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Take the BPF P1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 5 as an example, most of its sections (including a number of BPF maps) solely belong to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Furthermore, P1 and P2 share several extra BPF maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Thus, at its runtime, MOAT configures the PKR register of P1 to enable its access (EN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' denoted as 00 in the runtime PKR value column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 5) to its own region 0x1 and the shared region 0x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Moreover, MOAT disables any accesses from P1 to the kernel region 0x0 and the P2 memory region 0x2 by setting corresponding bits in P1’s PKR register as 01 (denoting AD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To clarify, the code and map sections of a BPF program requires are trivially known (by reading the metadata in the BPF program) once it is loaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Thus, MOAT can assign these pages to its designated region by modifying their PTEs during the program loading phase without any runtime overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The assignment for stack, context and some special types of maps will be discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2 BPF Runtime Environment Apart from the program itself and the maps it uses, a BPF program requires additional kernel structures to function prop- erly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These structures include descriptor tables, stacks, and the program’s runtime context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Furthermore, certain maps (such as the hash map) are not stored continuously in the kernel and cannot be assigned trivially during initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT assigns entries of this kind of maps on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Descriptor Tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' On x86 platforms, descriptor tables such as Global Descriptor Table (GDT) and Interrupt Descriptor Table (IDT) are essential for basic operations like interrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These kernel data structures are assigned to a shared region that all BPF programs can access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To prevent tampering those critical structures, they are made read-only when shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Dedicated Stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF programs require a 512-byte stack space to store local variables and function frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The ver- ifier is in charge of determining if a program makes Out of Bound (OOB) accesses toward this stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Thus, when the BPF program passes the static checks, its required stack is di- rectly allocated from the kernel stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3, certain vulnerabilities may allow BPF programs to bypass this check at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Given that this stack is uti- lized so frequently, we note that executing dynamic auditing 5 Table 1: BPF context of common program types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Program Type Context Type Note Socket Filter __sk_buff * Metadata of sk_buff Socket Ops bpf_sock_ops * Socket events (timeout, retransmission, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=') XDP xdp_md * Metadata of xdp_buff Kprobe pt_regs * Register status Tracepoints Depending on Tracepoint Types Relevant Tracepoint information Perf Event bpf_perf_event_data * Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' event (register status, sample period) Cgroup Device bpf_cgroup_dev_ctx * Device ID, access type (read, write, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=') on it, as MOAT does for helper calls (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2), would incur an unreasonable level of overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Thus, to prevent malicious BPF programs from tampering the kernel stack, MOAT allocates per-CPU stacks for BPF programs to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To do so, similar to the descriptor tables, these per-CPU stacks are shared by all BPF programs running on the same CPU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Consequently, they are also assigned to the shared re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To prevent a malicious BPF program from tampering stacks of other CPU cores, the stack beginning addresses are randomized for each CPU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Runtime Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The context refers to BPF program param- eters, which vary depending on the BPF program types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For instance, if the BPF program serves as the filter attached to a particular socket, its runtime context is a pointer to the socket buffer, which stores packets for the attached socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since these contexts are not available until runtime, MOAT assigns these contexts upon the entry point of each BPF program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Table 1 lists common BPF contexts: These contexts are rather simple and only a few of them are nested data structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=', containing pointers to other structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Thus, this assignment can be performed efficiently upon each entry point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Incontiguous Maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Despite the fact that there are over 30 distinct types of maps, their implementations can be roughly divided into only two types: Array maps and hash maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The array maps are easy for MOAT to isolate since they are stored in a continuous form and of a fixed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For these maps, MOAT determines its isolation when loading the BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The hash maps, however, are stored non- contiguously in the memory and can be dynamically expanded upon map insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This prevents MOAT from determining the addresses and sizes of the maps before executing the BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To overcome this issue, MOAT attaches to the bpf_map_lookup_elem, which is used to lookup a map entry and return its pointer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' If the pointer is retrieved from an non-contiguous map, the memory to which it points is dynamically assigned to the BPF program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These entries are returned to the kernel once the program exits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='3 Lifecycle of a BPF Program This section has described how MOAT uses PKS to grant a BPF program accesses to its minimum necessary memory regions required to complete its task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This protects the ker- nel from being attacked by malicious BPF programs while allowing benign BPF programs to operate smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We sum- marize all these details and depict the lifecycle of an isolated BPF program in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF Program Used Map BPF Program Used Map Ctx Stack Entry BPF Program Used Map Ctx Stack Dynamic Map Entry Run Exit BPF Program Load 1 4 2 3 1 Load: The program itself and its maps are assigned to its region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 2 Entry: Context is assigned and stack is swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3 Runtime: Entries of incontiguous maps are assigned on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4 Exit: Memory assigned during runtime is returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Figure 6: BPF program lifecycle under isolation of MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2 Challenges for MOAT The preceding section illustrates the overall procedure of MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, to effectively isolate a BPF program using PKS, MOAT needs to overcome the following obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' C1: Limited Hardware Regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In PKS, only 16 hardware keys are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This means there can be no more than 16 memory regions concurrently, but there may be signifi- cantly more than 16 BPF programs running in the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To overcome this limitation, we propose a novel dynamical key allocation policy in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' C2: Helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF is a complex ecosystem containing over 200 helper functions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Unlike BPF programs, these helper functions must have access to certain kernel memory to func- tion properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Thus, MOAT must ensure that these helper func- tions are secure and not being abused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, designing specific isolation policy for every one of these helpers requires massive human effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Even worse, designing individualized isolation strategy for each helper may impede the applica- bility to helpers added in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To this end, we analyze these BPF helper functions with static analysis techniques and propose three general security isolation schemes in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1 Dynamic Key Allocation Currently, PKS supports up to 16 memory regions, whose permissions are decided by a 32-bit PKR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Although works like libmpk [51] propose key virtualization to enable key sharing, these works typically focus on isolating userspace applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Therefore, they rely on scheduling and notifica- tion mechanisms that are exclusive to userspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, 6 after examining their methods, we conclude that porting these userspace mechanisms to kernel is difficult, if at all possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Intuitively, we may explore making key a shared resource;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' each BPF program will dynamically fetch and return a key upon its entry point and exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Our tentative study shows that this approach works well with small BPF programs consum- ing few pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Nevertheless, this approach may incur signifi- cant runtime overhead, as assigning these pages to a specific region upon each entry and exit can be time-consuming, partic- ularly if the program is attached with large maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For instance, a 512KB map consists of over 100 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' If a BPF tracepoint program employs this map to log kernel events, there will be over 200 page assignments every time this BPF program is invoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These frequent assignments bring unacceptable overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Overall, given that frequent key retrieval and return is too expensive due to the presence of large BPF programs with many pages, we propose an adaptive dynamic key allo- cation scheme that shares keys across relatively small BPF programs and assigns fixed keys to large BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Dynamic keys K1 K2 Run Fixed keys Exit K1 Wait Wait .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Entry Large BPF Programs 1 2 3 4 Figure 7: Adaptive key allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 7, we divide PKS keys into two cate- gories — dynamic keys and fixed keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We allocate dynamic keys to small BPF programs, whose allocation procedure are specified as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1 Upon a BPF program P’s entry point, MOAT fetches a dynamic key and assigns this key to all pages of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 2 During the runtime, MOAT can detect if P accesses pages not assigned to it via PKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3 When P exits, all of its pages are returned to the kernel, and the key is deallocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4 If currently no key available when the kernel launches a BPF program, then this program is placed in a queue to wait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In contrast, fixed key allocation is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Once a large BPF program is loaded by the kernel, MOAT assigns a fixed key to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In extreme cases where multiple large BPF programs are loaded into the kernel, and fixed keys are insuffi- cient, the smallest and least frequently invoked BPF program running will be evicted to use dynamic keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We need to decide a threshold to determine whether a BPF program is “small” or “large.” Note that the current BPF sub- system only accepts programs that with fewer than 4,096 instructions, which occupy about eight pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Considering that the majority of BPF programs use a small map to com- municate with userspace, we select ten pages as the threshold for dynamic key allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' That is, a BPF program using up to ten pages is configured to use dynamic keys, whereas BPF programs with more than ten pages uses fixed keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2 Helper Security Mechanism As interfaces between kernel and BPF programs, a set of BPF helper functions has been provided for kernel interac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since these helper functions serve as interfaces, most of them have to access certain kernel memory to function prop- erly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Therefore, these helpers may be leveraged by malicious BPF programs to launch attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Thus, MOAT has to prevent these helpers from being abused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, there are over 200 helpers provided by the BPF subsystem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' it is impractical to design individual protection policy for each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To overcome this obstacle, we analyze these helper functions and propose three defenses based on their interaction with the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Each of these defenses applies to a large number of helpers and can be combined to enhance the offered protection guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Analyzing all these helpers manually requires a significant amount of human effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We leverage a de facto static pointer analysis library, SVF[55, 56], to perform dependency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' SVF performs sparse value flow analysis to establish value flow and pointer analysis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' SVF has been widely used to analyze large-size production software [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We use the default flow-sensitive pointer analysis [55] provided by SVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Specifically, we use it to track the value flow of the parameters of these helper functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Based on the value flow, we can scope the usage (read or write) of parameters and decide which category (see below) a helper function belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' With the help of SVF, this categorization process can be conducted in a principled way and scalable to analyze all helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Attackers might manipulate the parameters of these helpers to launch attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Therefore, based on the above analysis results, we divide 260 BPF helper functions into five types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As shown in Table 2, the first type (No Arg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=') has no argu- ments, which does not need any extra protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The second type (Pure Arg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=') operates solely on its own arguments and does not access kernel memory, which is also safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The third type (Read Only) accesses kernel in a read-only manner, and the forth type (Write) may use its argument to modify the kernel memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The fifth type (Other) includes helpers that are hard to categorize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For example, bpf_loopis the auxiliary function that simplifies the verification process of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Note that the last three types may interact with the kernel space and potentially cause unauthorized access or even kernel ex- ploitations by being abused by malicious BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Table 2: BPF helper analysis result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CRP denotes critical region protection, ROK denotes read-only kernel space, and DPA denotes dynamic parameter auditing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Type # Example Applicable Defense No Arg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 30 bpf_get_retval() No Need Pure Arg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 16 bpf_strncmp() No Need Read Only 75 bpf_get_stackid_tp() ROK/CRP/DPA Write 129 bpf_skb_set_tstamp() CRP/DPA Other 10 bpf_loop() CRP/DPA With this categorization, we now present three mechanisms 7 in MOAT that ensure helper security as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Read-Only Kernel Space (ROK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Our analysis reveals that the majority of helpers only access the kernel in a read-only manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These read-only helpers account for near one third of all helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Even though in most cases, read-only helpers do not alter the kernel state and are considered safe, MOAT still sets the kernel space as read-only when executing these helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='3 This nullifies possibility of potentially tempering kernel spaces, and it does not impose extra runtime overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Normal Regions AD Critical Regions AD PKR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' AD Critical Regions BPF Region EN PKR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Helper Call Kernel Address Space Kernel Address Space AD Access-Disabled EN Access-Enabled Normal Regions BPF Region EN EN Figure 8: Critical region protection (CRP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Critical Region Protection (CRP) in Kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Further to the discussion in ROK, though many helpers only access kernel in a read-only manner, they may still be abused to probe sen- sitive data of the kernel, such as task_struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Moreover, a considerable number of helpers, as illustrated in Table 2, may modify kernel memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To prevent such abuse, we protect these critical kernel regions with PKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 8, instead of treating the entire kernel memory as a whole, we divide it into normal regions and critical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' When enter- ing helper functions, instead of setting the entire kernel space as access-enabled (EN), those critical memory regions remain access-disabled (AD), preventing any access to these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Once the helper finishes, these normal region will be set back to access-disabled (AD) to avoid potential security risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' It is worth noting these critical memory regions do not vary with helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' That is, only helpers manipulated by attackers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=', via deliberately crafted helper parameters) may attempt to access these critical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These critical regions can be specified in the configurations of MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' r0 = 0x10 r1 = r0 + 0x1 call BPF_HELPER BPF Instructions Static Register Value Inferred by Verifier 0x10 0x11 Runtime Register Values for Each Instruction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 0x10 0xbe 0x10 0x11 r0 r1 r0 = 0x10 r0 = 0x10 r1 = 0x11 r0 = 0x10 r1 = 0x11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Figure 9: Register value tracking of the verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' While the veri- fier can indeed deduce a possible value range of each register, for simplicity, we use a value point (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=', r1 = 0x11) here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Dynamic Parameter Auditing (DPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To further regulate helpers, we propose dynamic parameters auditing (DPA), which leverages the information obtained from the BPF ver- 3There exist few functions in this category that rely on synchronization facilities like Read-Copy Update (RCU), which cannot be applied with this protection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ifier to dynamically check if the parameters are within their legitimate ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 9, the verifier can deduce the value range of each register via static analysis (as a practical assumption, we allow the statically deduced value ranges to be invalid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' see below for clarification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT logs such value range information, and during runtime, MOAT serves as a “gateway” when the BPF program enters a helper function to check if the provided parameter values are within the verifier-deduced value ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In our example, we can check if r0==0x10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='r1==0x11 when BPF_HELPER is called.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' If the parameter runtime values do not match with the static analysis results, the BPF program is terminated immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Clarification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In the aforementioned DPA strategy, one may question if the “legitimate value ranges” inferred by the veri- fier are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Recall as discussed in our research motivation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3, there exist several vulnerabilities that can be lever- aged to bypass verifier static checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Overall, we clarify that we do not need the verifier’s static analysis results as always correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Nevertheless, as long as the runtime input values are inconsistent with the static analysis results, we terminate the BPF program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For such cases, either the verifier is wrong or the BPF program is behaving maliciously, both are highly severe and we require manual inspection of the triage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We assume the chance of both verifier and BPF program being unsafe (but still appear to be consistent) is extremely low, if at all possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In fact, for today’s known BPF exploita- tions, the verifier’s static analysis results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=', deciding the value ranges of certain pointers) are safe, though incomplete (omitting some data facts on subtle variables) and thus being leveraged by malicious BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Also, even though it may be technically feasible to perform dynamic auditing to validate the data facts after executing every BPF instruction, it is apparently too costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT thus leverages PKS to deliver a low-cost and principled isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Hybrid Usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We summarize the applicability of these three defense mechanisms in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' On the one hand, DPA pro- tects helpers from being abused by ensuring the validity of their parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' On the other hand, even if the helpers are already compromised, ROK and CRP can still protect the kernel from these compromised helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Thus, combining these mechanisms together improves the overall security for both BPF helpers and the kernel itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Moreover, we want to emphasize that these defenses are not dependent on a partic- ular helper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Instead, they are applicable to helper groups, as listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Although it can be argued all three defenses may be evaded in extreme circumstances, we believe the at- tack feasibility is very low (if it exists at all), given that the BPF program has been isolated by MOAT and these restricted helpers constitute a relatively minor attack surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Our inves- tigation on existing vulnerabilities supports this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 8 5 Implementation MOAT is written in 2,075 lines of C code, as a loadable kernel module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='4 It includes three components: a BPF loader, a BPF executor, and a key allocator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We explain key points below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Portable Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The major components of MOAT are implemented as hooks to replace their corresponding ker- nel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This is accomplished using an existing ker- nel hook utility named ftrace [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This introduces a small amount of overhead, but it allows these major components to be kernel-agnostic and can be easily ported across different kernel versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Though the overhead of the current MOAT prototype is reasonable (see details in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2), we anticipate to further reduce the performance overhead of MOAT, if it is implemented via directly modifying kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Kernel Interrupt Handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Though the major components of MOAT are implemented as loadable modules, certain low- level codes still require direct kernel modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For in- stance, during the execution of BPF programs, an interrupt may occur and take over the control flow to its handler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Note that most interrupt handlers require access to kernel memory and as a result, the PKS would presumably raise spurious alerts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Thus, we need to temporarily disable PKS inside these handlers and re-enable it once the handlers are finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The modified code is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Additionally, the exception handler of the kernel is also modified to support terminating and detaching malicious BPF programs upon violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1 mov %cr4,%rbx 2 push %rbx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' save CR4 3 and $0xfffffffffeffffff, %rbx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' clear CR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='PKS 4 mov %rbx,%cr4 5 call \\cfunc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' invoke handler 6 pop %rbx 7 mov %rbx,%cr4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' restore CR4 Figure 10: The modified kernel interrupt handler in entry_64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6 Evaluation To evaluate MOAT, we first analyze how MOAT mitigates various attack interfaces, and then benchmark its CVEs de- tectability in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We then assess the performance of MOAT under different BPF program setups in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Lastly, the functionality of MOAT is tested using various types of BPF programs and under different scenarios in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1 Security Evaluation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1 Analysis of Attack Surface Mitigation We first systematically analyze how MOAT mitigates five rep- resentative attack interfaces presented in the BPF ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These potential attack interfaces are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4We will release the codebase of MOAT once this paper is published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We will maintain MOAT to benefit the community and follow-up research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' PTEs IDT/GDT Memory BPF Program Helper Auditor BPF Helper IA32_PKRS CR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='PKS 3 4 1 2 5 PKS Region Write Disabled Access Disabled Figure 11: Analysis of mitigating potential attack surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1 Arbitrary Kernel Accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Currently, the most prevalent threat to the BPF ecosystem is the ability of malicious BPF programs to arbitrarily modify kernel memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In order to accomplish this, these BPF programs typically employ corner- case operations to deceive the verifier during the loading phase and to behave maliciously during runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This type of attack is effectively mitigated due to the fact that MOAT derives the minimum necessary memory regions of each BPF program and uses PKS to prevent any runtime access beyond this region (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1), mitigating such illegal accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 2 Helper Function Abuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Apart from launching attack di- rectly from BPF programs, a malicious BPF program may carefully prepare parameter values by exploiting similar corner-cases operations in 1 and pass them to abuse certain helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To prevent such abuse, MOAT features three security enforcement schemes (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2) to dynamically audit helper parameters and also protect critical kernel memory regions during the execution of these helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Thus, the attacker can no longer take advantage of these helpers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3 PTE Corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' A page’s PKS region is configured via its PTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Consequently, a malicious BPF program may attempt to tamper these PTEs to disable MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, this is im- possible since MOAT sets these PTEs as access-disabled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' they are thus protected by PKS like other kernel resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4 Descriptor Table Tampering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Descriptor tables like GDT and IDT are essential for segmentation and interrupt handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since they are needed for these critical functions, blindly set- ting them as access-disabled would cause system crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, since these descriptor tables are only accessed in a read-only manner, MOAT sets them as write-disabled to thwart any tampering made by malicious BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This effectively prevents malicious BPF programs from compro- mising the kernel using these tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 5 Hardware Configuration Tampering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Besides memory- based attacks discussed above, attackers may also directly disable PKS through hardware configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 2, CR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='PKS and IA32_PKRS are two critical registers for configuring PKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' One may disable PKS via modifying these two registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, both registers can only be mod- ified via special instructions, and BPF instruction sets do not include any of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Therefore, BPF bytecodes containing these instructions are rejected immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since the BPF programs are set to W ⊕ X (meaning write and executable permissions cannot be simultaneously enabled), adding these instructions via self-modification is also impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2 Real-world CVE Evaluation We analyzed all 37 CVEs relating to BPF since 2020 and found that nine of them are related to runtime memory corrup- tion caused by malicious BPF programs, which falls within the application scope of MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Even though these memory corruption vulnerabilities only account for about one-forth of all CVEs, they all result in privilege escalation and pose a severe security threat to the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As listed in Table 3, five of these vulnerabilities have PoC exploits available and are evaluated at this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We report that MOAT can successfully mitigate all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We clarify that these five are not cherry-picked;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' the untested four only have high-level text descriptions without further de- tails or any PoC, making it extremely hard for us to build a workable exploit based on these descriptions alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In- stead, we thoroughly analyze these four vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Due to their conceptual similarity to the other five tested cases, it should be accurate to conclude that these four can also be mitigated by MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For instance, although there is no exploit for CVE-2021-3444, it shares the same logistics with CVE- 2021-31440, albeit with different BPF instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Note that both originate from incorrect truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' From the fact that CVE-2021-31440 is mitigated by MOAT, we would believe the same for CVE-2021-3444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Table 3: BPF CVE detectability evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' denotes experimented and mitigated by MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' denotes the CVEs share conceptually identical patterns, though they lack available PoC exploit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE ID Description Status 2022-2785 [43] Incorrect Instruction Rewrite 2022-23222 [42] Mischeck *_OR_NULL Pointer 2021-45402 [41] Incorrect MOV32 Bound 2021-3490 [40] Incorrect ALU32 Bound 2021-31440 [37] Incorrect 32-bit Truncation 2021-3444 [39] Incorrect MOD32 Truncation 2021-33200 [38] Incorrect Pointer Arithmetic 2020-8835 [36] Incorrect 32-bit Bound 2020-27194 [35] Incorrect OR32 Bound CVE Case Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To better explain how MOAT mitigates these CVEs, we elaborate on the exploit paths for two of them, CVE-20222-23222 and CVE-2020-27194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2022-23222 is a pointer mischeck vulnerability intro- duced via a rather new feature of BPF named bpf_ringbuf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This new feature was brought to BPF in 2020 along with a new pointer type named PTR_TO_MEM_OR_NULL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, the verifier had not been updated to track the bounds of this new type, resulting in this vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 12, the malicious payload first retrieves a nullptr via bpf_ringbuf_reserve (line 1), which returns this newly- added pointer type named PTR_TO_MEM_OR_NULL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since this new type is not tracked by the verifier, the payload can bypass pointer checks by convincing the verifier that r1 is 0x0 when it is actually 0x1 (line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This pointer can then be multiplied with any offset to perform arbitrary kernel accesses (line 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, such arbitrary access violates PKS immediately and is terminated by MOAT (line 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1 r0 = bpf_ringbuf_reserve(fd, INT_MAX, 0) 2 r1 = r0 // R:r0=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='r1=0 V:r0=r1=?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 3 r1 = r0 + 1 // R:r0=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='r1=1 V:r0=r1=?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 4 if (r0 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='= nullptr) { // R:r0=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='r1=1 V:r0=r1=?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 5 ringbuf_discard(r0, 1) 6 exit(2) 7 } 8 off = // R:r0=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='r1=1 V:r0=r1=0 9 off = off * r1 // R:off= V:off=0 10 (ptr+off) = 0xbad // PKS violation!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Figure 12: Code snippet of CVE-2022-23222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' R denotes variable runtime statuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' V denotes verifier-deduced values of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2020-27194 is a vulnerability due to incorrect trunca- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 13, the user first inputs an arbitrary value in the range of [0,0x600000001] (line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Then, two con- ditional clauses help the verifier to determine its lower and upper bounds (line 3 and line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, when tracking the BPF_OR operator (line 7), the verifier performs a wrong truncation on its upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' After the truncation, the user- controlled r5is viewed by the verifier as a legitimate constant scalar 0x1(line 7), which can later be used as the offset to per- form arbitrary accesses to the kernel (line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Similarly, such accesses can be detected by MOAT and terminated instantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1 r5 = 2 r6 = 0x600000002 3 if (r5 >= r6) // R&V:r5<=0x600000001 4 exit(2) 5 if (r5 <= 0) // R&V:0x1<=r5<=0x600000001 6 exit(2) 7 r5 = r5 | 0 // R:r5= V: r5=0x1 8 (ptr+r5)=0xbad // PKS violation!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Figure 13: Code snippet of CVE-2020-27194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' R denotes variable runtime statuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' V denotes verifier-deduced values of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2 Performance Evaluation We assess MOAT performance overhead on Linux v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='195 and a 16-core Intel 12700H, whose efficiency cores are disabled and performance cores are locked to 4 GHz to avoid random- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As a common setup, the cycle and time statistics are measured via the rdtscp instruction and the kernel utility get_ktime_raw(), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1 Micro Benchmark For micro benchmark, we measure the CPU cycles of four key operations in MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We list the the four operations in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' switch_pks() enables/disables PKS by setting/- clearing the corresponding control bit in CR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' set_pkrs() changes region permissions by changing IA32_PKRS via WRMSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' get_pkrs() returns current permission configuration by reading IA32_PKRS via RDMSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' assign_page() changes 5The kernel is slightly modified as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 10 the permission region of one page by modifying its PTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Each operation is measured by averaging ten runs of one million invocations to eliminate randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Table 4: Micro benchmark results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As a reference [51], userspace RDPKRU, WRPKRU, and pkey_assign() take 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='5, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='3, and 1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='9 cycles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Operation # Cycle Note switch_pks() 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2 Set/Clear CR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='PKS set_pkrs()/WRMSR 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='7 Set region permissions get_pkrs()/RDMSR 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='8 Get region permissions assign_page() 1120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='4 Assign a page to region As Table 4 shows, the most expensive operation is assign_page() which modifies the region one page be- longs to, including locating its PTE and changing specific bits within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Notably, setting and getting the region permis- sions (set_pkrs()/get_pkrs()) in PKS is much more ex- pensive than its userspace variant in libmpk [51] (see the caption of Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We presume that this is because in PKU, the region permission is controlled via a dedicated register named PKRU with two special instructions RDPKRU/WRPKRU, whereas in PKS employed by MOAT, its region permission is stored in an MSR named IA32_PKRS without any special instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To configure the permission in IA32_PKRS, one has to use the general RDMSR/WRMSR instructions with the MSR ID 0x6E1, which requires additional cycles to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Similarly, directly enabling/disabling PKS via switch_pks() also takes fewer cycle than set_pkrs().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since configuring permission via set_pkrs() is more expensive than switch_pks(), on situations where MOAT needs to temporarily switch back to kernel regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' inter- rupt handling), it uses switch_pks() to disable PKS instead of using set_pkrs().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Then, before returning to BPF pro- grams, we reactive PKS to maintain isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2 Macro Benchmark To prepare the macro benchmark suite, we consider the fol- lowing properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' (a) To test the performance of MOAT conducting fixed and dynamic key allocation, it is necessary to include BPF programs of varying sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' (b) The number of BPF programs should exceed the num- ber of available keys to test MOAT in situations where hardware keys are insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' (c) The BPF programs should be highly parallel to evaluate the waiting time when dynamic keys are insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' (d) The execution order should reflect actual system behav- ior with high enough frequency to stress MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To simultaneously fulfill these requirements, we prepare macro benchmark as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We choose seven different events frequently triggered in the kernel, which are sys_open, sys_close, sys_read, sys_write, sched_switch, page_fault_user, and page_fault_kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These events are of high frequency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=', sched_switch occurs on every context switch) and can reflect actual BPF running behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' For each of these events, we attach three BPF tracepoints of varying sizes to log this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This ensures that these BPF programs are highly parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT Configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In both regular and extreme cases (see below), we choose the configuration as follows: the threshold for dynamic key allocation is ten pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The number of fixed keys is ten, while the number of dynamic keys is four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Two keys are reserved for the kernel memory region and the shared region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=', for per-CPU stack, IDT, GDT), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Regular Case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In the regular case, we attach each one of these events with three types of BPF tracepoints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=', small (1 page), medium (10 pages) and large (200 pages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We run each setup ten times, and each run consists of 1,000 invoca- tions of each tracepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The average results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We find that even in the worst case, MOAT imposes a moderate overhead of less than 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This overhead occurs when launching the medium-size BPF program attached to the event page_fault_kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since its size (10 pages) does not exceed the threshold of dynamic key allocation, it has to repetitively assign and return the dynamic key to its pages upon every entry point and exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As reflected on the micro benchmark in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1, such key assignment is quite costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Overall, we interpret the performance penalty is aligned with our expectation, and the overall overhead is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' All large-size BPF programs exceed the page number threshold of dynamic key allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Therefore, MOAT as- signs fixed keys to them during their loading phase without incurring runtime overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The incurred overheads are gen- erally moderate: for all cases, the overheads are less than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Moreover, the overheads for those small-size BPF programs are all less than 22%, which lie between the large-size and the medium-size ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Apart from the total overhead reported above, we also investigate the waiting overhead, which is the amount of time a BPF program must wait if there is no dy- namic key available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Note that in the regular cases above, 14 programs are smaller than the page number threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' they are configured to use the dynamic key allocation scheme, although there are only four dynamic keys available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Their waiting statistics are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' It is seen that although the average waiting time is near 1µs, less than 1% BPF exe- cutions really experience this delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Considering there are 14 running processes and only four dynamic keys available, we can conclude that the dynamic key allocation policy handles parallelism reasonably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Moreover, this also shows that four dynamic keys are sufficient for most scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' adding more dynamic keys brings marginal benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Table 5: Waiting time statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' (ns) Waited Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' (ns) Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' (ns) # Waited 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1 915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2 2559 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='8% Extreme Cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The above regular cases only evaluate MOAT under situations where dynamic keys are limited but fixed keys are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Here, we further explore MOAT’s overhead via extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Instead of attaching three BPF programs 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='50 pf_u pf_k sched open close read write Relative Time Base Small Medium Large Figure 14: Regular macro benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' of varying sizes, as we did in the regular cases above, in the extreme case evaluation we attach three large (200 pages) BPF programs to each tracepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Under this setting, there are only ten fixed keys available, although there are 21 large-size BPF programs, requiring dynamic key allocation for over half of these programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since each of these programs contains over 200 pages, there are a large number of page assignments occurring upon their program entry points and exits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Table 6: Extreme overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Static Keys (ns) Dynamic Keys (ns) Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Waited # Waited 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='7 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='8 3630 4401 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1 4% We report the evaluation results of extreme cases in Ta- ble 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We find that MOAT imposes a negligible overhead to BPF programs that use fixed keys even under such extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' And for those large BPF programs that use dynamic keys, the average overhead is still reasonably low (around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='6µs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Overall, we point out that real-life scenarios seldomly require this many BPF programs with large maps running concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Moreover, the currently observed overhead can be further reduced by sharing these large maps between BPF programs, thereby reducing the need for fixed keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We also report that the waiting time due to the shortage of dynamic keys shows a similar pattern to the regular cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Although the average waiting time is near 2µs, less than 5% of the executions would experience this delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='3 Real-world Case Study To evaluate the performance of MOAT under real-world sce- narios, we setup a BPF port forwarding program which redi- rects incoming requests to the memcached [24] memory database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To prepare the benchmark, we choose YCSB [19] to generate six distinct workloads and test the overall throughput of the memcached service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' From the figure, we can see that MOAT imposes on average 6% (up to 14%) slowdown to the overall performance of the BPF-based port forwarding, which is acceptable considering the security benefits MOAT provides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Note that this overhead is far less than the worst overhead we observed from the regular/extreme cases above, which further justifies our as- sumption that BPF programs are invoked less frequently in real-world applications than in extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 5586 5649 7407 5649 14084 4975 5464 5681 6493 5050 13889 4366 0 5000 10000 15000 YCSB_A YCSB_B YCSB_C YCSB_D YCSB_E YCSB_F Throughput (ops/sec) Base MOAT Figure 15: Overall throughput of the memcached case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='3 Functionality Evaluation To show that MOAT is able to support various BPF features, we select seven BPF applications with varying functionalities from the famous bcc toolbox [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Among them, execsnoop and opensnoopare used for kernel profiling, recording differ- ent system events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' tcptrace and net_monitor are used for network monitoring, collecting packet statistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' xdp_drop, xdp_cpu and xdp_interface can be used in firewalls and various load balancing scenarios, redirecting or dropping packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These applications cover the majority of contem- porary BPF ecosystem usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' After securing these applications with MOAT, we examine the runtime status of these applications and confirm that they are operating cor- rectly and are not affected by MOAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 16 reports the performance evaluation results of these applica- tions with MOAT enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The extra overhead incurred by MOAT under different scenarios is reasonably low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Overall, the evaluation shows that MOAT can be smoothly applied to secure de facto BPF applications under various scenarios with minimal engineering effort and moderate cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='50 execsnoop opensnoop tcptrace net_monitor xdp_drop xdp_cpu xdp_interface Relative Time Base MOAT Figure 16: Application benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 7 Related Work In-Kernel Isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Most existing works [10, 12–14, 16, 23, 26, 29, 49, 61, 64] on kernel isolation focuses kernel com- ponents like device drivers and file systems, which are dis- tinct from BPF programs and hence cannot be reused directly in our scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Existing works can be roughly divided into three categories: virtualization, Software Fault Isolation (SFI), and formal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Narayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [49] propose LVD, which isolates kernel components in a virtualized environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Based on LVD, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [29] split kernel modules into individual components for finer-grained isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' SFI 12 is employed to instrument programs at the source or binary level [13, 14, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' These works ensure kernel security by in- serting pointer checks prior to memory accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Furthermore, formal methods enable principled isolation of kernel compo- nents, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=', separating kernel code from untrusted drivers [61], or verifying file system correctness [10, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' We believe none of these methods are readily re-usable in our BPF scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Virtualization method [12, 29, 49, 64] require placing the program in a separated address space, making it hard for BPF programs to interact with kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' SFI [13, 14, 23] is based on program (compile-time) instru- mentation, whose inserted software checks often lead to high runtime overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Lastly, the BPF verifier itself performs formal verification, which shares conceptually similar advan- tage and drawbacks with existing formal method-based kernel isolation methods [10, 16, 61];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT employs hardware ex- tensions to offer more principled BPF isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MPK-Based Isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Prior to PKS, Intel first announced its userspace variant PKU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Consequently, most existing works [27, 51, 58] using MPK focus on userspace isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To better utilize PKU as an isolation primitive, Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [51] proposed libmpk that resolves the semantic discrepancies between PKU and conventional mprotect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' There are also works [27, 58] that leverage this hardware feature to protect confidential data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Apart of using PKU to isolate normal user applications, efforts are made to isolate trusted applications in SGX via PKU [17, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' SGXLock [17] establishes mu- tual distrust between kernel and the trusted SGX applications, while EnclaveDom [33] enables intra-isolation within one enclave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' PKU has been used for kernel security [26, 57] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' IskiOS [26] applies PKU to kernel pages by marking them as user-owned, while Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [57] employ PKU to protect userspace unikernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As a new feature introduced in 2021, research works using PKS are rather rare comparing to PKU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Linux community attempted to use PKS to prevent stray writes [1], which refers to kernel accidentally writing to wrong addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' There also exist many works [25, 31, 32, 50, 60] on securing the the BPF ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, most of these works use formal methods to enhance the following BPF components: the verifier, the JIT compiler and the BPF program itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' To enhance the standard BPF verifier, Ger- shuni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [25] built PREVAIL based on abstract interpre- tation [20], which supports more program structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' loops) and is more efficient comparing to the standard verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' PRSafe [32], on the other hand, designs a new domain-specific language based on primitive recursive functions, whose prop- erties ensure that all computations must terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The ul- timate goal of PRSafe is to build a mathematically verifi- able compiler for BPF programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As for BPF JIT compiler, Jitk [60] is a classic BPF JIT compiler whose correctness is proven manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Further, Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [50] propose Jitterbug to generate automated proof for real-world BPF JIT compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Lastly, Luke Nelson [31] build proof-carrying BPF programs, requiring developers to provide a correctness proof alongside with the program before loading it into the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 8 Discussion Platform Migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The current prototype implementation of MOAT is based on MPK, a hardware extension available on Intel platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Below, we discuss migrating MOAT to other platforms with similar hardware extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ARM Memory Domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' “Domain” is a MPK-like feature supported since ARMv7 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' It employs 4-bit domain keys in PTEs and a Domain Access Control Register (DACR) in su- pervisor mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Following a similar rationale to MPK, DACR allows accesses to be configured as denied, fully-allowed, or the same as PTEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since this feature is only supported on first- level and section-level PTEs, the domain boundaries must be aligned to 1 megabyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Due to the similarity between this feature and MPK, we expect MOAT to be implemented on ARM with a moderate effort using this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' RISC-V Domain Keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As an open-source architecture, there exists a hardware extension on the RISC-V platform that supports similar features as MPK named Donky [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Donky leverages ten unused bits in the PTEs as a protection key, hence supporting 1,024 permission regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Since Donky supports 1,024 keys, it is no longer possible to control permis- sions for all these regions using a single register, like MPK does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Donky thus introduces a 64-bit DKRU register with four key slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Each slot can be loaded with a 10-bit protection key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Only when a key is loaded in DKRU can its associated region be written to or read from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' From the description above, we interpret that Donky is quite flexible, and therefore, MOAT may be smoothly implemented on RISC-V using Donky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF JIT Support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 2, there are two ways of executing a BPF program: directly interpreting the BPF bytecode, or using a JIT compiler for improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Our prototype implementation of MOAT is based on the BPF interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' However, we note that the design of MOAT is compatible with the JIT compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' First, the PKS is config- ured at the entry and exit points of running a BPF program, which is independent of the BPF program execution method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Second, the operations that MOAT performs during the BPF execution, such as helper auditing, are implemented as part of BPF helpers and also decoupled from how BPF programs are executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Therefore, MOAT is essentially agnostic about the BPF program execution method, and it is adaptive to the native code produced by the BPF JIT compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Moreover, unlike the JIT compiler in Java virtual machine (JVM), which compiles only hotspot code chunks of Java bytecode each time, the BPF JIT compiler compiles the entire BPF program bytecode into native code once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This further reduces the effort of adapting MOAT to BPF programs compiled by JIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 13 9 Conclusion Despite the increasing popularity of using BPF to extend kernel functionality, the security of BPF programs is still a concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Recent attacks reveal that BPF applications can by- pass static security checks and conduct unauthorized kernel memory accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' This paper has presented MOAT, which iso- lates potentially malicious BPF applications from the kernel using Intel MPK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' MOAT addresses technical challenges and delivers a practical and extensible protection mechanism, in compensation to the contemporary BPF verifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Our evalua- tion reveals that MOAT can isolate (malicious) BPF programs in various real-world circumstances at a low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' References [1] Memory protection keys for the kernel, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [2] BPF-Helpers(7) - Linux Manual Page, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [3] ARM Architecture Reference Manual, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [4] BPF Documentation — The Linux Kernel Documenta- tion, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [5] eBPF Maps — The Linux Kernel Documentation, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [6] eBPF Verifier — The Linux Kernel Documentation, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [7] ftrace - Function Tracer — The Linux Kernel Documen- tation, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [8] Kprobes Documentation — The Linux Kernel Documen- tation, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [9] Intel 64 and IA-32 Architectures Software Developer Manuals, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [10] Sidney Amani, Alex Hixon, Zilin Chen, Christine Rizkallah, Peter Chubb, Liam O’Connor, Joel Beeren, Yutaka Nagashima, Japheth Lim, Thomas Sewell, Joseph Tuong, Gabriele Keller, Toby Murray, Gerwin Klein, and Gernot Heiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Cogent: Verifying high- assurance file system implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In Proceedings of the Twenty-First International Conference on Archi- tectural Support for Programming Languages and Oper- ating Systems, ASPLOS ’16, page 175–188, New York, NY, USA, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ISBN 9781450340915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1145/2872362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2872404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [11] Daniel Borkmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' BPF and Spectre: Mitigating tran- sient execution attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' eBPF Summit, 2021.' metadata={'source': 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software fault isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In Proceedings of the ACM SIGOPS 22nd Symposium on Operating Systems Prin- ciples, SOSP ’09, page 45–58, New York, NY, USA, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ISBN 9781605587523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1145/1629575.' metadata={'source': 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untrusted linux kernel extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In Pro- ceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation, PLDI 2019, page 1069–1084, New York, NY, USA, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ISBN 9781450367127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1145/3314221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='3314590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [26] Spyridoula Gravani, Mohammad Hedayati, John Criswell, and Michael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Fast intra-kernel isolation and security with iskios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In 24th International Symposium on Research in Attacks, Intrusions and Defenses, RAID ’21, page 119–134, New York, NY, USA, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ISBN 9781450390583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1145/3471621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='3471849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [27] Mohammad Hedayati, Spyridoula Gravani, Ethan John- son, John Criswell, Michael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Scott, Kai Shen, and Mike Marty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Hodor: Intra-Process isolation for High- Throughput data plane libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In 2019 USENIX An- nual Technical Conference (USENIX ATC 19), pages 489–504, Renton, WA, July 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' USENIX Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ISBN 978-1-939133-03-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [28] Toke Høiland-Jørgensen, Jesper Dangaard Brouer, Daniel Borkmann, John Fastabend, Tom Herbert, David Ahern, and David Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The express data path: Fast programmable packet processing in the operating sys- tem kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In Proceedings of the 14th International Conference on Emerging Networking EXperiments and Technologies, CoNEXT ’18, page 54–66, New York, NY, USA, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ISBN 9781450360807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1145/3281411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='3281443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [29] Yongzhe Huang, Vikram Narayanan, David Detweiler, Kaiming Huang, Gang Tan, Trent Jaeger, and Anton Burtsev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' KSplit: Automating device driver isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In 16th USENIX Symposium on Operating Systems De- sign and Implementation (OSDI 22), pages 613–631, Carlsbad, CA, July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' USENIX Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ISBN 978-1-939133-28-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [30] Google Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The Chromium Projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' chromium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/chromium-projects/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [31] Emina Torlak Luke Nelson, Xi Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' A proof-carrying approach to building correct and flexible in-kernel verifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' https://homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='edu/ ~lukenels/slides/2021-09-23-lpc21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='pdf, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [32] Sai Veerya Mahadevan, Yuuki Takano, and Atsuko Miyaji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Prsafe: Primitive recursive function based domain specific language using llvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In 2021 In- ternational Conference on Electronics, Information, and Communication (ICEIC), pages 1–4, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1109/ICEIC51217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='9369763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [33] Marcela S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Melara, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Freedman, and Mic Bow- man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Enclavedom: Privilege separation for large-tcb applications in trusted execution environments, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [34] Dirk Merkel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Docker: lightweight linux containers for consistent development and deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Linux journal, 2014(239):2, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [35] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2020-27194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' http://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-2020-27194, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [36] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2020-8835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' http://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-2020-8835, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [37] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2021-31440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' http://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-2021-31440, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [38] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2021-33200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' http://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-2021-33200, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [39] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2021-3444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' http://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-2021-3444, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [40] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2021-3490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' http://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-2021-3490, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [41] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2021-45402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' http://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-2021-45402, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [42] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2022-23222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' http://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-2022-23222, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [43] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2022-2785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' https://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-CVE-2022-2785, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 15 [44] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2021-38300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' http://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-2021-38300, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [45] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2021-29154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' http://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-2021-29154, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [46] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2021-4001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' https://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/ cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name=CVE-2021-4001, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [47] MITRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' CVE-2021-29155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' https://cve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='org/cgi-bin/cvename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='name= CVE-2021-29155, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [48] Mozilla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' The Firefox Projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='mozilla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' org/en-US/firefox/browsers/, 2022.' metadata={'source': 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9781450375542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1145/3381052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='3381328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [50] Luke Nelson, Jacob Van Geffen, Emina Torlak, and Xi Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Specification and verification in the field: Applying formal methods to BPF just-in-time compilers in the linux kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In 14th USENIX Symposium on Op- erating Systems Design and Implementation (OSDI 20), pages 41–61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' USENIX Association, November 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ISBN 978-1-939133-19-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [51] Soyeon Park, Sangho Lee, Wen Xu, HyunGon Moon, and Taesoo Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' libmpk: Software abstraction for intel memory protection keys (Intel MPK).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ISBN 978-1- 939133-17-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [54] Yulei Sui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' SVF References.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' http://svf-tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='io/SVF/.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [59] Harishankar Vishwanathan, Matan Shachnai, Srinivas Narayana, and Santosh Nagarakatte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Sound, pre- cise, and fast abstract interpretation with tristate num- bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In Proceedings of the 20th IEEE/ACM Inter- national Symposium on Code Generation and Opti- mization, CGO ’22, page 254–265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' IEEE Press, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ISBN 9781665405843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='1109/CGO53902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' 9741267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [60] Xi Wang, David Lazar, Nickolai Zeldovich, Adam Chli- pala, and Zachary Tatlock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' Jitk: A trustworthy In-Kernel interpreter infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' In 11th USENIX Sympo- sium on Operating Systems Design and Implementation (OSDI 14), pages 33–47, Broomfield, CO, October 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' USENIX Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' ISBN 978-1-931971-16-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' [61] Miao Yu, Virgil Gligor, and Limin Jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FQT4oBgHgl3EQf1zZe/content/2301.13421v1.pdf'} +page_content=' An i/o separa- tion model for formal verification of kernel implementa- tions.' metadata={'source': 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as A Multilingual Word Aligner +Weikang Wang1∗ Guanhua Chen2∗ +Hanqing Wang1 +Yue Han1 +Yun Chen1† +1Shanghai University of Finance and Economics +2Southern University of Science and Technology +wwk@163.sufe.edu.cn +ghchen08@gmail.com +{whq,hanyue}@163.sufe.edu.cn +yunchen@sufe.edu.cn +Abstract +Multilingual +pretrained +language +models +(mPLMs) have shown their effectiveness in +multilingual word alignment induction. How- +ever, these methods usually start from mBERT +or XLM-R. In this paper, we investigate +whether multilingual sentence Transformer +LaBSE is a strong multilingual word aligner. +This idea is non-trivial as LaBSE is trained +to +learn +language-agnostic +sentence-level +embeddings, while the alignment extraction +task requires the more fine-grained word- +level embeddings to be language-agnostic. +We +demonstrate +that +the +vanilla +LaBSE +outperforms other mPLMs currently used +in the alignment task, and then propose to +finetune LaBSE on parallel corpus for further +improvement. +Experiment results on seven +language pairs show that our best aligner +outperforms previous state-of-the-art models +of all varieties. +In addition, our aligner +supports different language pairs in a single +model, and even achieves new state-of-the-art +on zero-shot language pairs that does not +appear in the finetuning process. +1 +Introduction +Word alignment aims to find the correspondence +between words in parallel texts (Brown et al., 1993). +It is useful in a variety of natural language process- +ing (NLP) applications such as noisy parallel cor- +pus filtering (Kurfalı and Östling, 2019), bilingual +lexicon induction (Shi et al., 2021), code-switching +corpus building (Lee et al., 2019; Lin et al., 2020) +and incorporating lexical constraints into neural +machine translation (NMT) models (Hasler et al., +2018; Chen et al., 2021b). +Recently, neural word alignment approaches +have developed rapidly and outperformed statistical +word aligners like GIZA++ (Och and Ney, 2003) +and fast-align (Dyer et al., 2013). Some works +∗The first two authors contribute equally. +†Corresponding author. +Figure 1: Cosine similarities between subword repre- +sentations in a parallel sentence pair from 8th layer of +mBERT (left) and 6th layer of LaBSE (right). +Red +boxes denote the gold alignments. +(Garg et al., 2019; Li et al., 2019; Zenkel et al., +2019, 2020; Chen et al., 2020b; Zhang and van Gen- +abith, 2021; Chen et al., 2021a) induce alignments +from NMT model or its variants. However, these +bilingual models only support the language pair +involved in the training process. They also treat the +source and target side differently, thus two models +are required for bidirectional alignment extraction. +Another line of works (Jalili Sabet et al., 2020; Dou +and Neubig, 2021) build multilingual word aligners +with contextualized embeddings from the multilin- +gual pretrained language model (Wu and Dredze, +2019; Conneau et al., 2020, mPLM). Thanks to +the language-agnostic representations learned with +multilingual masked language modeling task, these +methods are capable of inducing word alignments +even for language pairs without any parallel corpus. +Different from previous methods, in this pa- +per we present AccAlign, a more accurate mul- +tilingual word aligner with the multilingual sen- +tence Transformer LaBSE (Feng et al., 2022, see +Figure 1). The LaBSE is trained on large scale +parallel corpus of various language pairs to learn +language-agnostic sentence embeddings with con- +trastive learning. However, it is unclear whether +LaBSE has learned language-agnostic word-level +arXiv:2301.12140v1 [cs.CL] 28 Jan 2023 + +0.81 +0.68 +0.55 +0.51 +0.55 +0.50 +0.51 +0.58 +0.89 +0.62 +0.47 +0.29 +0.28 +0.28 +0.31 +0.31 +Das +0.64 +0.87 +0.58 +0.52 +0.52 +0.55 +0.51 +0.57 +0.57 +0.89 +0.53 +0.32 +0.28 +0.33 +0.28 +0.33 +0.61 +0.66 +0.74 +0.53 +0.56 +0.51 +0.55 +0.58 +0.47 +0.56 +0.87 +0.36 +0.33 +0.31 +0.32 +0.25 +0.53 +0.57 +0.65 +0.48 +0.46 +0.50 +0.48 +0.54 +0.39 +0.53 +0.57 +0.34 +0.25 +0.36 +0.32 +0.29 +、 +0.56 +0.56 +0.77 +0.79 +0.64 +0.54 +0.54 +0.54 +0.42 +0.50 +0.67 +0.40 +0.33 +0.42 +0.46 +0.30 +0.48 +0.52 +0.53 +0.73 +0.70 +0.58 +0.50 +0.52 +0.24 +0.28 +0.33 +0.90 +0.52 +0.36 +0.34 +0.28 +e +0.54 +0.52 +0.56 +0.65 +0.85 +0.63 +0.61 +0.59 +0.25 +0.27 +0.31 +0.49 +0.88 +0.38 +0.41 +0.29 +verstehen +0.50 +0.54 +0.52 +0.55 +0.62 +0.69 +0.77 +0.54 +0.28 +0.28 +0.33 +0.34 +0.39 +0.52 +0.89 +0.29 +0.63 +0.62 +0.56 +0.57 +0.60 +0.58 +0.57 +0.94 +0.21 +0.26 +0.19 +0.18 +0.20 +0.23 +0.20 +0.61 +Jno +That +can +That +our +can +nderstand +understand +mBERT +LaBSEembeddings, which is the key for the success of +word alignment extraction. Specifically, we first +direct induce word alignments from LaBSE and +demonstrate that LaBSE outperforms other mPLMs +currently used in the alignment task. This indi- +cates that LaBSE has implicitly learned language- +agnostic word-level embeddings at some intermedi- +ate layer. Then we propose a simple and effective +finetuning method to further improve performance. +Empirical results on seven language pairs show that +our best aligner outperforms previous SOTA mod- +els of all varieties. In addition, our aligner supports +different language pairs in a single model, and even +achieves new SOTA on zero-shot language pairs +that does not appear in finetuning process.1 +2 +AccAlign +2.1 +Background: LaBSE +LaBSE (Feng et al., 2022) is the state-of-the-art +model for the cross-lingual sentence retrieval task. +Given an input sentence, the model can retrieve the +most similar sentence from candidates in a different +language. LaBSE is first pretrained on a combina- +tion of masked language modeling (Devlin et al., +2019) and translation language modeling (Conneau +and Lample, 2019) tasks. After that, it is effec- +tively finetuned with contrastive loss on 6B parallel +sentences across 109 languages. We leave the train- +ing detail of LaBSE in the appendix. However, as +LaBSE does not include any word-level training +loss when finetuning with contrastive loss, it is un- +clear whether the model has learned high-quality +language-agnostic word-level embeddings, which +is the key for a multilingual word aligner. +2.2 +Alignment Induction from LaBSE +To investigate whether LaBSE is a strong multilin- +gual word aligner, we first induce word alignments +from vanilla LaBSE without any modification or +finetuning. This is done by utilizing the contextual +embeddings from LaBSE. Specifically, consider +a bilingual sentence pair x = ⟨x1, x2, ..., xn⟩ and +y = ⟨y1, x2, ..., ym⟩, we denote the contextual em- +beddings from LaBSE as hx = ⟨hx1, ..., hxn⟩ and +hy = ⟨hy1, ..., hym⟩, respectively. Following pre- +vious work (Dou and Neubig, 2021; Jalili Sabet +et al., 2020), we get the similarity matrix from the +contextual embeddings: +S = hxhT +y. +(1) +1Code is available at https://github.com/sufenlp/ +AccAlign. +Figure 2: The framework of adapter-based finetuning. +The blue blocks are kept frozen, while the red adapter +blocks are updated during finetuning. +The similarity matrix is normalized for each row to +get Sxy. Sxy is treated as the probability matrix as +its i-th row represents the probabilities of aligning +xi to all tokens in y. The reverse probability ma- +trix Syx is computed similarly by normalizing each +column of S. Taking intersection of the two prob- +ability matrices yields the final alignment matrix: +A = (Sxy > c) ∗ (ST +yx > c), +(2) +where c is a threshold and Aij = 1 indicates that +xi and yj are aligned. The above method induces +alignments on the subword level, which are con- +verted into word-level alignments by aligning two +words if any of their subwords are aligned follow- +ing (Zenkel et al., 2020; Jalili Sabet et al., 2020). +2.3 +Finetuning LaBSE for Better Alignments +Inspired by (Dou and Neubig, 2021), we propose a +finetuning method to further improve performance +given parallel corpus with alignment labels. +Adapter-based Finetuning +Adapter-based fine- +tuning (Houlsby et al., 2019; Bapna and Firat, 2019; +He et al., 2021) is not only parameter-efficient, +but also benefits model performance, especially +for low-resource and cross-lingual tasks (He et al., +2021). Figure 2 illustrates our overall framework, +where the adapters are adopted from (Houlsby et al., +2019). For each layer of LaBSE, we introduce +an adapter for each sublayer, which maps the in- +put vector of dimension d to dimension m where +m < d, and then re-maps it back to dimension d. +Let h and h′ denote the input and output vector, + +Add & Norm +Adapter +Feed-forward +00000 +Add & Norm +000 +Adapter +00000 +Feed-forward +Self-attention +Adapter +XL +AccAlignerModel +Setting +de-en +sv-en +fr-en +ro-en +ja-en +zh-en +fa-en +avg +Bilingual Statistical Methods +fast-align (Dyer et al., 2013) +scratch +27.0 +- +10.5 +32.1 +51.1 +38.1 +- +- +eflomal (Östling and Tiedemann, 2016) +22.6 +- +8.2 +25.1 +47.5 +28.7 +- +- +GIZA++ (Och and Ney, 2003) +20.6 +- +5.9 +26.4 +48.0 +35.1 +- +- +Bilingual Neural Methods +MTL-FULLC-GZ (Garg et al., 2019) +scratch +16.0 +- +4.6 +23.1 +- +- +- +- +BAO-GUIDE (Zenkel et al., 2020) +16.3 +- +5.0 +23.4 +- +- +- +- +SHIFT-AET (Chen et al., 2020b) +15.4 +- +4.7 +21.2 +- +17.2 +- +- +MASK-ALIGN (Chen et al., 2021a) +14.4 +- +4.4 +19.5 +- +13.8 +- +- +BTBA-FCBO-SST (Zhang and van Genabith, 2021) +14.3 +- +6.7 +18.5 +- +- +- +- +Multilingual Neural Methods +SimAlign (Jalili Sabet et al., 2020) +no ft +18.8 +11.2 +7.6 +27.2 +46.6 +21.6 +32.7 +23.7 +AwesomeAlign (Dou and Neubig, 2021) +no ft +17.4 +9.7 +5.6 +27.9 +45.6 +18.1 +33.0 +22.5 +self-sup ft +15.9 +7.9 +4.4 +26.2 +42.4 +14.9 +27.1 +19.8 +sup ft +15.2 +7.2 +4.0 +25.5 +40.6 +13.4 +25.8 +18.8 +AccAlign +no ft +16.0 +7.3 +4.5 +20.8 +43.3 +16.2 +23.4 +18.8 +self-sup ft +14.3 +5.8 +3.9 +21.6 +39.2 +13.0 +22.6 +17.2 +sup ft +13.6 +5.2 +2.8 +20.8 +36.9 +11.5 +22.2 +16.1 +Table 1: AER comparison between AccAlign and the baselines on test set of 7 language pairs. self-sup and +sup mean finetuning the model with parallel corpus of self-supervised and human-annotated alignment labels, +respectively. All multilingual methods are tested on zero-shot language pairs. +respectively. The output vector h′ is calculated as: +h +′ = Wup · tanh(Wdown · h) + h. +(3) +Note that a skip-connection is employed to approx- +imate an identity function if parameters of the pro- +jection matrices are near zero. During finetuning, +only parameters of the adapters are updated. +Training Objective +Let ˆA denote the alignment +labels for the given sentence pair x and y. We +define the learning objective as: +L = +� +ij +ˆAij +1 +2 +� +(Sxy)ij +n ++ (ST +yx)ij +m +� +, +(4) +where Sxy and Syx are the alignment probabil- +ity matrices, n and m are the length of sentence +x and y, respectively. Intuitively, this objective +encourages the gold aligned words to have closer +contextualized representations. In addition, as both +Sxy and ST +yx are encouraged to be close to ˆA, it im- +plicitly encourages the two alignment probability +matrices to be symmetrical to each other as well. +Our framework can be easily extended to cases +where alignment labels are unavailable, by replac- +ing ˆA with pseudo labels A (Equation 2) and train- +ing in a self-supervised manner. +3 +Experiments +3.1 +Setup +As we aim at building an accurate multilingual +word aligner, we evaluate AccAlign on a di- +verse alignment test set of seven language pairs: +de/sv/ro/fr/ja/zh/fa-en. For finetuning LaBSE, we +use nl/cs/hi/tr/es/pt-en as the training set and cs-en +as the validation set. To reduce the alignment anno- +tation efforts and the finetuning cost, our training +set only contains 3, 362 annotated sentence pairs. +To simulate the most difficult use cases where the +test language pair may not included in training, we +set the test language pairs different from training +and validation. Namely, LaBSE is tested in a zero- +shot manner. We denote this dataset as ALIGN6. +We induce alignments from 6-th layer of LaBSE, +which is selected on the validation set. We use +Alignment Error Rate (AER) as the evaluation met- +ric. Our model is not directly comparable to the +bilingual baselines, as they build model for each +test language pair using large scale parallel corpus +of that language pair. In contrast, our method is +more efficient as it supports all language pairs in +a single model and our finetuning only requires +3, 362 sentence pairs. +Appendix B show more +dataset, model, baselines and other setup details. +3.2 +Main Results +Table 1 shows the comparison of our methods +against baselines. AccAlign-supft achieves new +SOTA on word alignment induction, outperforming +all baselines in 6 out of 7 language pairs. AccAlign +is also simpler than AwesomeAlign, which is the +best existing multilingual word aligner, as Awe- +someAlign finetunes with a combination of five +objectives, while AccAlign only has one objective. +The vanilla LaBSE is a strong multilingual word + +Model +fi-el +fi-he +SimAglin +noft +69.3 +85.8 +AwesomeAlign +noft +69.8 +84.4 +self-sup ft +68.8 +87.7 +sup ft +67.4 +86.1 +AccAlign +noft +47.0 +81.2 +self-sup ft +40.8 +76.1 +sup ft +36.7 +71.7 +Table 2: AER comparison between AccAlign and mul- +tilingual baselines on non-English zero-shot language +pairs. The best AER for each column is bold and un- +derlined. +aligner (see AccAlign-noft). It performs better than +SimAlign-noft and AwesomeAlign-noft, and com- +parable with AwesomeAlign-supft, indicating that +LaBSE has learned high-quality language-agnostic +word embeddings. Our finetuning method is ef- +fective as well, improving AccAlign-noft by 1.6 +and 2.7 AER with self-supervised and supervised +alignment labels, respectively. Our model improves +multilingual baselines even more significantly on +non-English language pairs. See Table 2 of ap- +pendix for detailed results. +3.3 +Analysis +Performance on non-English Language Pair +We conduct experiments to evaluate AccAlign +against multilingual baselines on non-English test +language pairs. The fi-el (Finnish-Greek) and fi-he +(Finnish-Hebrew) test set contains 791 and 2,230 +annotated sentence pairs, respectively. Both test +sets are from ImaniGooghari et al. (2021)2. The +results are shown in Table 2. As can be seen, Ac- +cAlign in all three settings significantly improves +all multilingual baselines. The improvements is +much larger compared with zero-shot English lan- +guage pairs, demonstrating the effectiveness of Ac- +cAlign on non-English language pairs. We also +observe that finetuning better improves AccAlign +than AwesomeAlign. This verifies the strong cross- +lingual transfer ability of LaBSE , even between +English-centric and non-English language pairs. +Adapter-based vs. Full Finetuning +We com- +pare full and adapter-based fine-tuning in Table 3. +Compared with full finetuning, adapter-based fine- +tuning updates much less parameters and obtains +better performance under both supervised and self- +supervised settings, demonstrating its efficiency +and effectiveness for zero-shot word alignments. +2https://github.com/cisnlp/graph-align +Ft type +full +adapter +Ft mode +self-supervised (avg.) +17.4 +17.2 +supervised (avg.) +16.2 +16.1 +Number of ft param. +428M +2.4M +Table 3: +AER comparison of full finetuning and +adapter-based finetuning. +Bilingual Finetuning +To better understand our +method, we compare with AwesomeAlign under +bilingual finetuning setup where the model is fine- +tuned and tested in the same single language pair. +We follow the setup in (Dou and Neubig, 2021) and +use finetuning corpus without human-annotated la- +bels. As shown in Table 4, LaBSE outperforms +AwesomeAlign in the finetuning language pair +(18.8 vs. 18.2). The performance gap becomes +larger for zero-shot language pairs (21.3 vs. 18.8). +The results demonstrate that AccAlign is an effec- +tive zero-shot aligner, as LaBSE has learned more +language-agnostic representations which benefit +cross-lingual transfer. +Different Multilingual Pretrained Models +We +investigate the performance of AccAlign-noft when +replacing LaBSE with other mPLMs, including +XLM-R, mBERT and four other multilingual sen- +tence Transformer from HuggingFace. LaBSE out- +performs other mPLMs by 3.5 to 9.6 averaged AER. +Table 9 in appendix shows more details. +Performance across Layer +We investigate the +performance of AccAlign-noft when extracts align- +ments from different layers. Layer 6, which is the +layer we use for all experiments, outperforms other +layers by 0.1 to 26.0 averaged AER. Please refer to +Table 10 in appendix for more details. +Representation Analysis +To succeed in multi- +lingual word alignment, the contextual embed- +dings should prefer two following properties: (1) +language-agnostic: two aligned bilingual words +should be mapped to nearby features in the +same language-agnostic feature space. (2) word- +identifiable: the embeddings of two random tokens +from the same sentence should be distinguishable. +Therefore, we analyze the embeddings from dif- +ferent layers of AccAlign under different settings +by computing cosine similarity for aligned word +pairs and word pairs randomly sampled from the +same sentence, denoted as sbi and smono (see ap- +pendix for more experiment details). Intuitively, +bigger sbi and smaller smono are preferred as we + +Model +Test lang. +Ft lang. +de-en +fr-en +ro-en +ja-en +zh-en +avg. +AwesomeAlign +ft lang. +14.9 +4.0 +22.9 +38.1 +14.1 +18.8 +zero-shot langs (avg.) +16.3 +4.7 +26.6 +43.7 +15.0 +21.3 +AccAlign +ft lang. +14.2 +3.8 +21.0 +38.0 +13.8 +18.2 +zero-shot langs (avg.) +14.8 +3.9 +20.7 +40.5 +13.8 +18.8 +Table 4: AER results with bilingual finetuning. +Figure 3: sbi (↑) and smono (↓) of AccAlign without +finetuning (noft), with self-supervised finetuning (self- +sup ft) and supervised finetuning (sup ft). +expect the features of aligned words to be similar +while that of two different words to be different. +The results on de-en test set are presented in Fig- +ure 3. For vanilla LaBSE (green curves), we find +that features from 6-th layer, namely the best layer +to induce alignment, successfully trades off these +two properties as it obtains the biggest sbi − smono +among all layers. In addition, adapter-based fine- +tuning improves performance mainly by making +features more word-identifiable, as it significantly +decreases smono while almost maintaining sbi . +4 +Conclusion +In this paper, we introduce AccAlign, a novel multi- +lingual word aligner based on multilingual sentence +Transformer LaBSE. The best proposed approach +finetunes LaBSE on a few thousands of annotated +parallel sentences and achieves state-of-the-art per- +formance even for zero-shot language pairs. Ac- +cAlign is believed to be a valuable alignment tool +that can be used out-of-the-box for other NLP tasks. +Limitations +AccAlign has shown to extract high quality word +alignments when the input texts are two well-paired +bilingual sentences. +However, the condition is +not always met. In lexically constrained decod- +ing of NMT (Hasler et al., 2018; Song et al., 2020; +Chen et al., 2021b), the aligner takes a full source- +language sentence and a partial target-language +translation as the input at each step to determine +the right position to incorporate constraints. In cre- +ating translated training corpus in zero-resource +language for sequence tagging or parsing (Ni et al., +2017; Jain et al., 2019; Fei et al., 2020), the aligner +extracts alignments from the labelled sentence and +its translation to conduct label projection. Both +cases deviate from our current settings as the input +sentence may contain translation error or even be +incomplete. We leave exploring the robustness of +AccAlign as the future work. +At the same time, our proposed method only +supports languages included in LaBSE. This hin- +ders applying AccAlign to more low-resource lan- +guages. Future explorations are needed to rapidly +adapt AccAlign to new languages (Neubig and Hu, +2018; Garcia et al., 2021). +Acknowledgements +This project was supported by National Natural +Science Foundation of China (No. 62106138) and +Shanghai Sailing Program (No. 21YF1412100). +We thank the anonymous reviewers for their in- +sightful feedbacks on this work. +References +Niraj Aswani and Robert Gaizauskas. 2005. Aligning +words in english-hindi parallel corpora. In Proceed- +ings of the ACL Workshop on Building and Using +Parallel Texts, pages 115–118. +Ankur Bapna and Orhan Firat. 2019. +Simple, scal- +able adaptation for neural machine translation. In +Proceedings of the 2019 Conference on Empirical +Methods in Natural Language Processing and the +9th International Joint Conference on Natural Lan- +guage Processing (EMNLP-IJCNLP), pages 1538– +1548, Hong Kong, China. 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As- +sociation for Computational Linguistics. + +A +LaBSE +LaBSE (Feng et al., 2022) is the state-of-the-art +model for the cross-lingual sentence retrieval task. +Given an input sentence, the model can retrieve the +most similar sentence from candidates in a differ- +ent language. It has 471M parameters and supports +109 languages. The model is first pretrained on a +combination of masked language modeling (De- +vlin et al., 2019) and translation language model- +ing (Conneau and Lample, 2019) tasks on the 17B +monolingual data and 6B bilingual translation pairs, +respectively. After that, it is effectively finetuned +with contrastive loss on 6B bilingual translation +pairs across 109 languages. +Specifically, given a bilingual sentence pair +⟨xi, yi⟩, we use exi and eyi to denote their sen- +tence embeddings from LaBSE. Then the model is +finetuned using contrative loss with in-batch nega- +tives (Chen et al., 2020a): +ℓ = − 1 +N +N +� +i=1 +� +log +exp +� +φ(exi, eyi) +� +�N +j=1 exp +� +φ(exi, eyj) +�+ +log +exp +� +φ(exi, eyi) +� +�N +j=1 exp +� +φ(exj, eyi) +� +� +, +(5) +where φ(exi, eyj) measures the similarity of sen- +tence xi and yj in the embedding space: +φ +� +exi, eyj +� += +� +e⊤ +xieyj − b +if i = j +e⊤ +xieyj +if i ̸= j . +(6) +Note that a margin b is introduced to improve the +separation between positive and negative pairs. +B +Experiments Setup +B.1 +Language Code +We refer to the language information in Table 1 of +(Fan et al., 2021). The information of the languages +used in this paper is listed in Table 5. +B.2 +Dataset +Table 6 shows the detailed data statistics of +ALIGN6. The ja and zh sentences are preprocessed +by Dou and Neubig (2021) and Liu and Sun (2015), +respectively. For finetuning AccAlign and multilin- +gual baselines, we use the training and validation +set from ALIGN6. As bilingual baselines are not +capable of zero-shot alignment induction, they are +trained from scratch with parallel corpus of the +test language pair using the same dataset as Dou +ISO +Name +Family +en +English +Germanic +nl +Dutch +Germanic +cs +Czech +Slavic +hi +Hindi +Indo-Aryan +tr +Turkish +Turkic +es +Spanish +Romance +pt +Portuguese +Romance +de +German +Germanic +sv +Swedish +Germanic +fr +French +Romance +ro +Romanian +Romance +ja +Japanese +Japonic +zh +Chinese +Chinese +fa +Persian +Iranian +Table 5: The information of the languages used in this +paper. +and Neubig (2021). The bilingual training data +set of de/fr/ro/ja/zh-en contain 1.9M, 1.1M, 450K, +444K and 40K parallel sentence pairs, respectively, +which are much larger than the training dataset of +ALIGN6. +B.3 +Model Setup +We use the contextual word embeddings from the +6-th layer of the official LaBSE3, which have 768 +dimensions. We set the threshold in Equation 2 to +0.1, which is selected on validation set by manual +tuning among [0, 0.2]. For adapter-based finetun- +ing, we set the hidden dimension of the adapters to +be 128. The adapters have 2.4M parameters, which +account 0.5% of the parameters of LaBSE. We use +the AdamW optimizer with learning rate of 1e-4, +and do not use warmup or dropout. The batch size +is set to 40 and maximum updates number is 1500 +steps. We use a single NVIDIA V100 GPU for all +experiments. +B.4 +Baselines +Besides three statistical baselines fast-align (Dyer +et al., 2013), eflomal (Östling and Tiedemann, +2016) and GIZA++ (Och and Ney, 2003), we com- +pare AccAlign with the following neural baselines: +MTL-FULLC-GZ (Garg et al., 2019). This model +supervises an attention head in Transformer-based +NMT model with GIZA++ word alignments in a +multitask learning framework. +BAO-GUIDE (Zenkel et al., 2020). This model +3https://huggingface.co/sentence-transformers/LaBSE + +Type +Lang. +Source +Link +# Sents +Training set +cs-en +Mareˇcek (2011) +http://ufal.mff.cuni.cz/ +czech-english-manual-word-alignment +2400 +nl-en +Macken (2010) +http://www.tst.inl.nl +372 +hi-en +Aswani and Gaizauskas (2005) +http://web.eecs.umich.edu/~mihalcea/wpt05/ +90 +tr-en +Cakmak et al. (2012) +http://web.itu.edu.tr/gulsenc/resources.htm +300 +es-en +Graca et al. (2008) +https://www.hlt.inesc-id.pt/w/Word_Alignments +100 +pt-en +Graca et al. (2008) +https://www.hlt.inesc-id.pt/w/Word_Alignments +100 +Validation set +cs-en +Mareˇcek (2011) +http://ufal.mff.cuni.cz/ +czech-english-manual-word-alignment +101 +Test set +de-en +Vilar et al. (2006) +http://www-i6.informatik.rwth-aachen.de/ +goldAlignment/ +508 +sv-en +Holmqvist and Ahrenberg (2011) +https://www.ida.liu.se/divisions/hcs/nlplab/ +resources/ges/ +192 +fr-en +Mihalcea and Pedersen (2003) +http://web.eecs.umich.edu/~mihalcea/wpt/ +447 +ro-en +Mihalcea and Pedersen (2003) +http://web.eecs.umich.edu/~mihalcea/wpt05/ +248 +ja-en +Neubig (2011) +http://www.phontron.com/kftt +582 +zh-en +Liu and Sun (2015) +https://nlp.csai.tsinghua.edu.cn/~ly/systems/ +TsinghuaAligner/TsinghuaAligner.html +450 +fa-en +Tavakoli and Faili (2014) +http://eceold.ut.ac.ir/en/node/940 +400 +Table 6: Training, validation and test dataset of ALIGN6. Note that this is a zero-shot setting as the test language +pairs do not appear in training and validation. +adds an extra alignment layer to repredict the to-be- +aligned target token and further improves perfor- +mance with Bidirectional Attention Optimization. +SHIFT-AET (Chen et al., 2020b). This model +trains a separate alignment module in a self- +supervised manner, and induce alignments when +the to-be-aligned target token is the decoder input. +MASK-ALIGN (Chen et al., 2021a). This model +is a self-supervised word aligner which makes use +of the full context on the target side. +BTBA-FCBO-SST (Zhang and van Genabith, +2021). This model has similar idea with Chen +et al. (2021a), but with different model architecture +and training objectives. +SimAlign (Jalili Sabet et al., 2020). This model is a +multilingual word aligner which induces alignment +with contextual word embeddings from mBERT +and XLM-R. +AwesomeAlign (Dou and Neubig, 2021). This +model improves over SimAlign by designing new +alignment induction method and proposing to fur- +ther finetune the mPLM on parallel corpus. +Among them, SimAlign and AwesomeAlign are +multilingual aligners which support multiple lan- +guage pairs in a single model, while others are +bilingual word aligners which require training from +scratch with bilingual corpus for each test lan- +guage pair. We re-implement SimAlign and Awe- +someAlign, while quote the results from (Dou and +Neubig, 2021) for the three statistical baselines and +the corresponding paper for other baselines. +B.5 +Sentence Transformer +We compare LaBSE with four other multilingual +sentence Transformer in HuggingFace. The de- +tailed information of these models are: +distiluse-base-multilingual-cased-v2.4 +This +model is a multilingual knowledge distilled version +of m-USE (Yang et al., 2020), which has 135M +parameters and supports more than 50+ languages. +paraphrase-xlm-r-multilingual-v1.5 This model +is a multilingual version of paraphrase-distilroberta- +base-v1 (Reimers and Gurevych, 2019), which has +278M parameters and supports 50+ languages. It +initializes the student model with an mPLM and +trains it to imitate monolingual sentence Trans- +former on parallel data with knowledge distillation. +paraphrase-multilingual-MiniLM-L12-v2.6 +This model is a multilingual version of paraphrase- +MiniLM-L12-v2 (Reimers and Gurevych, 2019), +which has 118M parameters and supports 50+ +languages. It trains similarly as paraphrase-xlm- +r-multilingual-v1, but with different teacher and +student model initialization. +paraphrase-multilingual-mpnet-base-v2.7 This +model is a multilingual version of paraphrase- +mpnet-base-v2 (Reimers and Gurevych, 2019), +4https://huggingface.co/sentence-transformers/distiluse- +base-multilingual-cased-v2 +5https://huggingface.co/sentence- +transformers/paraphrase-xlm-r-multilingual-v1 +6https://huggingface.co/sentence- +transformers/paraphrase-multilingual-MiniLM-L12-v2 +7https://huggingface.co/sentence- +transformers/paraphrase-multilingual-mpnet-base-v2 + +which has 278M parameters and supports 50+ lan- +guages. It trains similarly as paraphrase-xlm-r- +multilingual-v1, but with different teacher model +initialization. +B.6 +Bilingual Finetuning +We use the same dataset as bilingual baselines for +bilingual finetuning following (Dou and Neubig, +2021). At each time, we finetune LaBSE with one +language pair among de/fr/ro/ja/zh-en and test on +all seven language pairs. For Awesome-align, we +follow the setup in their paper, while for AccAlign, +we use the same hyperparameters as the main ex- +periments. +B.7 +Representation Analysis +We conduct representation analysis on de-en test +set. To compute sbi, we calculate the averaged co- +sine similarity of all gold aligned bilingual word +pairs. To compute smono, we randomly permute a +given sentence x = ⟨x1, x2, ..., xn⟩ to get x′ = +⟨x′ +1, x′ +2, ..., x′ +n⟩ and then create n word pairs as +{⟨xi-x′ +i⟩}n +i=1. We go through all de and en test +sentences and report the averaged cosine similarity +of all created word pairs as smono. +C +Experiment Results +Detailed results for each test language in Sec- +tion 3.3 are shown in Table 7 to Table 10. + +Ft mode +Ft type +de-en +sv-en +fr-en +ro-en +ja-en +zh-en +fa-en +avg +Self-supervised +full +14.7 +5.8 +3.7 +21.6 +39.9 +13.3 +22.7 +17.4 +adapter +14.3 +5.8 +3.9 +21.6 +39.2 +13.0 +22.6 +17.2 +Supervised +full +13.6 +5.3 +2.8 +21.0 +37.1 +11.0 +22.5 +16.2 +adapter +13.6 +5.2 +2.7 +20.8 +36.8 +11.5 +22.2 +16.1 +Table 7: AER comparison of full finetuning and adapter-based finetuning. The best AER for each column is bold +and underlined. +Model +Ft lang. +Test lang. +de-en +fr-en +ro-en +ja-en +zh-en +sv-en +fa-en +AwesomeAlign +de-en +14.9 +4.7 +26.2 +43.6 +14.6 +7.1 +28.2 +fr-en +16.4 +4.0 +26.9 +44.6 +15.7 +7.6 +28.0 +ro-en +15.8 +4.7 +22.9 +44.2 +15.1 +7.8 +27.0 +ja-en +16.8 +4.9 +27.0 +38.1 +15.2 +8.5 +30.0 +zh-en +16.2 +4.6 +26.2 +42.4 +14.1 +8.1 +28.0 +AccAlign +de-en +14.2 +3.8 +20.9 +39.3 +13.1 +5.7 +22.5 +fr-en +14.6 +3.8 +20.8 +41.0 +14.1 +6.0 +22.5 +ro-en +15.2 +4.0 +21.0 +42.1 +14.4 +6.5 +23.2 +ja-en +14.8 +3.9 +20.3 +38.0 +13.5 +6.3 +22.5 +zh-en +14.6 +3.9 +20.7 +38.9 +13.4 +5.9 +22.4 +Table 8: AER results with bilingual finetuning. The results where the model is trained and tested on the same +language pair are bold and underlined. +layer +de-en +sv-en +fr-en +ro-en +ja-en +zh-en +fas-en +avg +mBERT +8 +17.4 +8.7 +5.6 +27.9 +45.6 +18.1 +33.0 +22.3 +XLM-R +8 +23.1 +13.3 +9.2 +28.6 +62.0 +30.3 +28.6 +27.9 +distiluse-base-multilingual-cased-v2 +3 +23.7 +17.2 +9.8 +29.2 +56.3 +29.2 +33.5 +28.4 +paraphrase-xlm-r-multilingual-v1 +6 +17.4 +8.7 +4.9 +24.7 +53.8 +26.1 +26.5 +23.2 +paraphrase-multilingual-MiniLM-L12-v2 +6 +19.4 +9.4 +6.2 +26.0 +57.7 +29.7 +27.4 +25.1 +paraphrase-multilingual-mpnet-base-v2 +5 +18.0 +8.9 +5.4 +24.1 +54.9 +25.7 +25.5 +23.2 +LaBSE +6 +16.0 +7.3 +4.5 +20.8 +43.3 +16.2 +23.4 +18.8 +Table 9: AER comparison of LaBSE and other multilingual pretrained model. All are without finetuning. We +determine the best layer of alignment induction for each model using the validation set. The best AER for each +column is bold and underlined. +Layer +de-en +sv-en +fr-en +ro-en +ja-en +zh-en +fa-en +avg +0 +32.4 +27.7 +20.5 +44.2 +65.5 +40.1 +38.7 +38.4 +1 +27.3 +19.7 +12.8 +35.6 +64.0 +33.9 +35.4 +32.7 +2 +22.3 +14.0 +8.6 +28.8 +58.0 +25.0 +31.3 +26.9 +3 +18.5 +9.9 +6.0 +24.0 +50.3 +17.9 +26.8 +21.9 +4 +17.7 +8.7 +5.9 +23.3 +48.4 +16.3 +25.7 +20.9 +5 +15.8 +7.4 +4.5 +21.5 +43.7 +15.4 +23.8 +18.9 +6 +16.0 +7.3 +4.5 +20.8 +43.3 +16.2 +23.4 +18.8 +7 +16.5 +7.6 +4.8 +22.4 +43.4 +15.0 +23.7 +19.1 +8 +16.2 +7.3 +5.0 +21.6 +42.7 +16.7 +23.4 +19.0 +9 +16.8 +7.6 +5.3 +21.5 +42.7 +17.9 +23.2 +19.3 +10 +17.7 +9.0 +5.6 +23.0 +44.4 +20.4 +24.4 +20.6 +11 +36.7 +27.0 +24.2 +43.6 +61.3 +35.0 +46.2 +39.1 +12 +43.1 +33.2 +30.5 +46.0 +65.7 +42.6 +52.4 +44.8 +Table 10: AER comparison of vanilla LaBSE across layers. Layer 0 is the embedding layer. The best AER for +each column is bold and underlined. + diff --git a/2NFLT4oBgHgl3EQfqi-E/content/tmp_files/load_file.txt b/2NFLT4oBgHgl3EQfqi-E/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..52b5b4bf3bc768ea23092c649738f34bd69d0899 --- /dev/null +++ b/2NFLT4oBgHgl3EQfqi-E/content/tmp_files/load_file.txt @@ -0,0 +1,1214 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf,len=1213 +page_content='Multilingual Sentence Transformer as A Multilingual Word Aligner Weikang Wang1∗ Guanhua Chen2∗ Hanqing Wang1 Yue Han1 Yun Chen1† 1Shanghai University of Finance and Economics 2Southern University of Science and Technology wwk@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='sufe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='cn ghchen08@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='com {whq,hanyue}@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='sufe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='cn yunchen@sufe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='cn Abstract Multilingual pretrained language models (mPLMs) have shown their effectiveness in multilingual word alignment induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' How- ever, these methods usually start from mBERT or XLM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' In this paper, we investigate whether multilingual sentence Transformer LaBSE is a strong multilingual word aligner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This idea is non-trivial as LaBSE is trained to learn language-agnostic sentence-level embeddings, while the alignment extraction task requires the more fine-grained word- level embeddings to be language-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We demonstrate that the vanilla LaBSE outperforms other mPLMs currently used in the alignment task, and then propose to finetune LaBSE on parallel corpus for further improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Experiment results on seven language pairs show that our best aligner outperforms previous state-of-the-art models of all varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' In addition, our aligner supports different language pairs in a single model, and even achieves new state-of-the-art on zero-shot language pairs that does not appear in the finetuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 1 Introduction Word alignment aims to find the correspondence between words in parallel texts (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' It is useful in a variety of natural language process- ing (NLP) applications such as noisy parallel cor- pus filtering (Kurfalı and Östling, 2019), bilingual lexicon induction (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2021), code-switching corpus building (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020) and incorporating lexical constraints into neural machine translation (NMT) models (Hasler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Recently, neural word alignment approaches have developed rapidly and outperformed statistical word aligners like GIZA++ (Och and Ney, 2003) and fast-align (Dyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Some works ∗The first two authors contribute equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Figure 1: Cosine similarities between subword repre- sentations in a parallel sentence pair from 8th layer of mBERT (left) and 6th layer of LaBSE (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Red boxes denote the gold alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' (Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Zenkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Zhang and van Gen- abith, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2021a) induce alignments from NMT model or its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' However, these bilingual models only support the language pair involved in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' They also treat the source and target side differently, thus two models are required for bidirectional alignment extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Another line of works (Jalili Sabet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Dou and Neubig, 2021) build multilingual word aligners with contextualized embeddings from the multilin- gual pretrained language model (Wu and Dredze, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020, mPLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Thanks to the language-agnostic representations learned with multilingual masked language modeling task, these methods are capable of inducing word alignments even for language pairs without any parallel corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Different from previous methods, in this pa- per we present AccAlign, a more accurate mul- tilingual word aligner with the multilingual sen- tence Transformer LaBSE (Feng et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='61 Jno That can That our can nderstand understand mBERT LaBSEembeddings, which is the key for the success of word alignment extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Specifically, we first direct induce word alignments from LaBSE and demonstrate that LaBSE outperforms other mPLMs currently used in the alignment task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This indi- cates that LaBSE has implicitly learned language- agnostic word-level embeddings at some intermedi- ate layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Then we propose a simple and effective finetuning method to further improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Empirical results on seven language pairs show that our best aligner outperforms previous SOTA mod- els of all varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' In addition, our aligner supports different language pairs in a single model, and even achieves new SOTA on zero-shot language pairs that does not appear in finetuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 2 AccAlign 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 Background: LaBSE LaBSE (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2022) is the state-of-the-art model for the cross-lingual sentence retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Given an input sentence, the model can retrieve the most similar sentence from candidates in a different language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' LaBSE is first pretrained on a combina- tion of masked language modeling (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2019) and translation language modeling (Conneau and Lample, 2019) tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' After that, it is effec- tively finetuned with contrastive loss on 6B parallel sentences across 109 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We leave the train- ing detail of LaBSE in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' However, as LaBSE does not include any word-level training loss when finetuning with contrastive loss, it is un- clear whether the model has learned high-quality language-agnostic word-level embeddings, which is the key for a multilingual word aligner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 Alignment Induction from LaBSE To investigate whether LaBSE is a strong multilin- gual word aligner, we first induce word alignments from vanilla LaBSE without any modification or finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This is done by utilizing the contextual embeddings from LaBSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Specifically, consider a bilingual sentence pair x = ⟨x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', xn⟩ and y = ⟨y1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', ym⟩, we denote the contextual em- beddings from LaBSE as hx = ⟨hx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', hxn⟩ and hy = ⟨hy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', hym⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Following pre- vious work (Dou and Neubig, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Jalili Sabet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020), we get the similarity matrix from the contextual embeddings: S = hxhT y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' (1) 1Code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='com/sufenlp/ AccAlign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Figure 2: The framework of adapter-based finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The blue blocks are kept frozen, while the red adapter blocks are updated during finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The similarity matrix is normalized for each row to get Sxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Sxy is treated as the probability matrix as its i-th row represents the probabilities of aligning xi to all tokens in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The reverse probability ma- trix Syx is computed similarly by normalizing each column of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Taking intersection of the two prob- ability matrices yields the final alignment matrix: A = (Sxy > c) ∗ (ST yx > c), (2) where c is a threshold and Aij = 1 indicates that xi and yj are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The above method induces alignments on the subword level, which are con- verted into word-level alignments by aligning two words if any of their subwords are aligned follow- ing (Zenkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Jalili Sabet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 Finetuning LaBSE for Better Alignments Inspired by (Dou and Neubig, 2021), we propose a finetuning method to further improve performance given parallel corpus with alignment labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Adapter-based Finetuning Adapter-based fine- tuning (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Bapna and Firat, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2021) is not only parameter-efficient, but also benefits model performance, especially for low-resource and cross-lingual tasks (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Figure 2 illustrates our overall framework, where the adapters are adopted from (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' For each layer of LaBSE, we introduce an adapter for each sublayer, which maps the in- put vector of dimension d to dimension m where m < d, and then re-maps it back to dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Let h and h′ denote the input and output vector, Add & Norm Adapter Feed-forward 00000 Add & Norm 000 Adapter 00000 Feed-forward Self-attention Adapter XL AccAlignerModel Setting de-en sv-en fr-en ro-en ja-en zh-en fa-en avg Bilingual Statistical Methods fast-align (Dyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2013) scratch 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 eflomal (Östling and Tiedemann, 2016) 22.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 Bilingual Neural Methods MTL-FULLC-GZ (Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2019) scratch 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 BAO-GUIDE (Zenkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 SHIFT-AET (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020b) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 MASK-ALIGN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2021a) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 BTBA-FCBO-SST (Zhang and van Genabith, 2021) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 Multilingual Neural Methods SimAlign (Jalili Sabet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020) no ft 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 AwesomeAlign (Dou and Neubig, 2021) no ft 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 self-sup ft 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 sup ft 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 AccAlign no ft 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 self-sup ft 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 sup ft 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 Table 1: AER comparison between AccAlign and the baselines on test set of 7 language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' self-sup and sup mean finetuning the model with parallel corpus of self-supervised and human-annotated alignment labels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' All multilingual methods are tested on zero-shot language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The output vector h′ is calculated as: h ′ = Wup · tanh(Wdown · h) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' (3) Note that a skip-connection is employed to approx- imate an identity function if parameters of the pro- jection matrices are near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' During finetuning, only parameters of the adapters are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Training Objective Let ˆA denote the alignment labels for the given sentence pair x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We define the learning objective as: L = � ij ˆAij 1 2 � (Sxy)ij n + (ST yx)ij m � , (4) where Sxy and Syx are the alignment probabil- ity matrices, n and m are the length of sentence x and y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Intuitively, this objective encourages the gold aligned words to have closer contextualized representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' In addition, as both Sxy and ST yx are encouraged to be close to ˆA, it im- plicitly encourages the two alignment probability matrices to be symmetrical to each other as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Our framework can be easily extended to cases where alignment labels are unavailable, by replac- ing ˆA with pseudo labels A (Equation 2) and train- ing in a self-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 3 Experiments 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 Setup As we aim at building an accurate multilingual word aligner, we evaluate AccAlign on a di- verse alignment test set of seven language pairs: de/sv/ro/fr/ja/zh/fa-en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' For finetuning LaBSE, we use nl/cs/hi/tr/es/pt-en as the training set and cs-en as the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' To reduce the alignment anno- tation efforts and the finetuning cost, our training set only contains 3, 362 annotated sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' To simulate the most difficult use cases where the test language pair may not included in training, we set the test language pairs different from training and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Namely, LaBSE is tested in a zero- shot manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We denote this dataset as ALIGN6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We induce alignments from 6-th layer of LaBSE, which is selected on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We use Alignment Error Rate (AER) as the evaluation met- ric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Our model is not directly comparable to the bilingual baselines, as they build model for each test language pair using large scale parallel corpus of that language pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' In contrast, our method is more efficient as it supports all language pairs in a single model and our finetuning only requires 3, 362 sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Appendix B show more dataset, model, baselines and other setup details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 Main Results Table 1 shows the comparison of our methods against baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' AccAlign-supft achieves new SOTA on word alignment induction, outperforming all baselines in 6 out of 7 language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' AccAlign is also simpler than AwesomeAlign, which is the best existing multilingual word aligner, as Awe- someAlign finetunes with a combination of five objectives, while AccAlign only has one objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The vanilla LaBSE is a strong multilingual word Model fi-el fi-he SimAglin noft 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 AwesomeAlign noft 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 self-sup ft 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 sup ft 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 AccAlign noft 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 self-sup ft 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 sup ft 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 Table 2: AER comparison between AccAlign and mul- tilingual baselines on non-English zero-shot language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The best AER for each column is bold and un- derlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' aligner (see AccAlign-noft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' It performs better than SimAlign-noft and AwesomeAlign-noft, and com- parable with AwesomeAlign-supft, indicating that LaBSE has learned high-quality language-agnostic word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Our finetuning method is ef- fective as well, improving AccAlign-noft by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 AER with self-supervised and supervised alignment labels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Our model improves multilingual baselines even more significantly on non-English language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' See Table 2 of ap- pendix for detailed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 Analysis Performance on non-English Language Pair We conduct experiments to evaluate AccAlign against multilingual baselines on non-English test language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The fi-el (Finnish-Greek) and fi-he (Finnish-Hebrew) test set contains 791 and 2,230 annotated sentence pairs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Both test sets are from ImaniGooghari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' (2021)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' As can be seen, Ac- cAlign in all three settings significantly improves all multilingual baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The improvements is much larger compared with zero-shot English lan- guage pairs, demonstrating the effectiveness of Ac- cAlign on non-English language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We also observe that finetuning better improves AccAlign than AwesomeAlign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This verifies the strong cross- lingual transfer ability of LaBSE , even between English-centric and non-English language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Adapter-based vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Full Finetuning We com- pare full and adapter-based fine-tuning in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Compared with full finetuning, adapter-based fine- tuning updates much less parameters and obtains better performance under both supervised and self- supervised settings, demonstrating its efficiency and effectiveness for zero-shot word alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='com/cisnlp/graph-align Ft type full adapter Ft mode self-supervised (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=') 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 supervised (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=') 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 Number of ft param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 428M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4M Table 3: AER comparison of full finetuning and adapter-based finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Bilingual Finetuning To better understand our method, we compare with AwesomeAlign under bilingual finetuning setup where the model is fine- tuned and tested in the same single language pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We follow the setup in (Dou and Neubig, 2021) and use finetuning corpus without human-annotated la- bels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' As shown in Table 4, LaBSE outperforms AwesomeAlign in the finetuning language pair (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The performance gap becomes larger for zero-shot language pairs (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The results demonstrate that AccAlign is an effec- tive zero-shot aligner, as LaBSE has learned more language-agnostic representations which benefit cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Different Multilingual Pretrained Models We investigate the performance of AccAlign-noft when replacing LaBSE with other mPLMs, including XLM-R, mBERT and four other multilingual sen- tence Transformer from HuggingFace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' LaBSE out- performs other mPLMs by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 averaged AER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Table 9 in appendix shows more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Performance across Layer We investigate the performance of AccAlign-noft when extracts align- ments from different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Layer 6, which is the layer we use for all experiments, outperforms other layers by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 to 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 averaged AER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Please refer to Table 10 in appendix for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Representation Analysis To succeed in multi- lingual word alignment, the contextual embed- dings should prefer two following properties: (1) language-agnostic: two aligned bilingual words should be mapped to nearby features in the same language-agnostic feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' (2) word- identifiable: the embeddings of two random tokens from the same sentence should be distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Therefore, we analyze the embeddings from dif- ferent layers of AccAlign under different settings by computing cosine similarity for aligned word pairs and word pairs randomly sampled from the same sentence, denoted as sbi and smono (see ap- pendix for more experiment details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Intuitively, bigger sbi and smaller smono are preferred as we Model Test lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Ft lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' de-en fr-en ro-en ja-en zh-en avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' AwesomeAlign ft lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 zero-shot langs (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=') 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 AccAlign ft lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 zero-shot langs (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=') 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 Table 4: AER results with bilingual finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Figure 3: sbi (↑) and smono (↓) of AccAlign without finetuning (noft), with self-supervised finetuning (self- sup ft) and supervised finetuning (sup ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' expect the features of aligned words to be similar while that of two different words to be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The results on de-en test set are presented in Fig- ure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' For vanilla LaBSE (green curves), we find that features from 6-th layer, namely the best layer to induce alignment, successfully trades off these two properties as it obtains the biggest sbi − smono among all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' In addition, adapter-based fine- tuning improves performance mainly by making features more word-identifiable, as it significantly decreases smono while almost maintaining sbi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 4 Conclusion In this paper, we introduce AccAlign, a novel multi- lingual word aligner based on multilingual sentence Transformer LaBSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The best proposed approach finetunes LaBSE on a few thousands of annotated parallel sentences and achieves state-of-the-art per- formance even for zero-shot language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Ac- cAlign is believed to be a valuable alignment tool that can be used out-of-the-box for other NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Limitations AccAlign has shown to extract high quality word alignments when the input texts are two well-paired bilingual sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' However, the condition is not always met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' In lexically constrained decod- ing of NMT (Hasler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2021b), the aligner takes a full source- language sentence and a partial target-language translation as the input at each step to determine the right position to incorporate constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' In cre- ating translated training corpus in zero-resource language for sequence tagging or parsing (Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Fei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020), the aligner extracts alignments from the labelled sentence and its translation to conduct label projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Both cases deviate from our current settings as the input sentence may contain translation error or even be incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We leave exploring the robustness of AccAlign as the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' At the same time, our proposed method only supports languages included in LaBSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This hin- ders applying AccAlign to more low-resource lan- guages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Future explorations are needed to rapidly adapt AccAlign to new languages (Neubig and Hu, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Acknowledgements This project was supported by National Natural Science Foundation of China (No.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='11359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Thomas Zenkel, Joern Wuebker, and John DeNero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' End-to-end neural word alignment outper- forms GIZA++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' In Proceedings of the 58th Annual Meeting of the Association for Computational Lin- guistics, pages 1605–1617, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Jingyi Zhang and Josef van Genabith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' A bidi- rectional transformer based alignment model for un- supervised word alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' In Proceedings of the 59th Annual Meeting of the Association for Compu- tational Linguistics and the 11th International Joint Conference on Natural Language Processing (Vol- ume 1: Long Papers), pages 283–292, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' As- sociation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' A LaBSE LaBSE (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2022) is the state-of-the-art model for the cross-lingual sentence retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Given an input sentence, the model can retrieve the most similar sentence from candidates in a differ- ent language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' It has 471M parameters and supports 109 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The model is first pretrained on a combination of masked language modeling (De- vlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2019) and translation language model- ing (Conneau and Lample, 2019) tasks on the 17B monolingual data and 6B bilingual translation pairs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' After that, it is effectively finetuned with contrastive loss on 6B bilingual translation pairs across 109 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Specifically, given a bilingual sentence pair ⟨xi, yi⟩, we use exi and eyi to denote their sen- tence embeddings from LaBSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Then the model is finetuned using contrative loss with in-batch nega- tives (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020a): ℓ = − 1 N N � i=1 � log exp � φ(exi, eyi) � �N j=1 exp � φ(exi, eyj) �+ log exp � φ(exi, eyi) � �N j=1 exp � φ(exj, eyi) � � , (5) where φ(exi, eyj) measures the similarity of sen- tence xi and yj in the embedding space: φ � exi, eyj � = � e⊤ xieyj − b if i = j e⊤ xieyj if i ̸= j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' (6) Note that a margin b is introduced to improve the separation between positive and negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' B Experiments Setup B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 Language Code We refer to the language information in Table 1 of (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The information of the languages used in this paper is listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 Dataset Table 6 shows the detailed data statistics of ALIGN6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The ja and zh sentences are preprocessed by Dou and Neubig (2021) and Liu and Sun (2015), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' For finetuning AccAlign and multilin- gual baselines, we use the training and validation set from ALIGN6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' As bilingual baselines are not capable of zero-shot alignment induction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' they are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='trained from scratch with parallel corpus of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='test language pair using the same dataset as Dou ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='ISO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='Family ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='Table 5: The information of the languages used in this ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' and Neubig (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The bilingual training data set of de/fr/ro/ja/zh-en contain 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9M, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1M, 450K, 444K and 40K parallel sentence pairs, respectively, which are much larger than the training dataset of ALIGN6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 Model Setup We use the contextual word embeddings from the 6-th layer of the official LaBSE3, which have 768 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We set the threshold in Equation 2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1, which is selected on validation set by manual tuning among [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' For adapter-based finetun- ing, we set the hidden dimension of the adapters to be 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The adapters have 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4M parameters, which account 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5% of the parameters of LaBSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We use the AdamW optimizer with learning rate of 1e-4, and do not use warmup or dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The batch size is set to 40 and maximum updates number is 1500 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We use a single NVIDIA V100 GPU for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 Baselines Besides three statistical baselines fast-align (Dyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2013), eflomal (Östling and Tiedemann, 2016) and GIZA++ (Och and Ney, 2003), we com- pare AccAlign with the following neural baselines: MTL-FULLC-GZ (Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This model supervises an attention head in Transformer-based NMT model with GIZA++ word alignments in a multitask learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' BAO-GUIDE (Zenkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This model 3https://huggingface.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='cn/~ly/systems/ TsinghuaAligner/TsinghuaAligner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='html 450 fa-en Tavakoli and Faili (2014) http://eceold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='ir/en/node/940 400 Table 6: Training, validation and test dataset of ALIGN6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Note that this is a zero-shot setting as the test language pairs do not appear in training and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' adds an extra alignment layer to repredict the to-be- aligned target token and further improves perfor- mance with Bidirectional Attention Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' SHIFT-AET (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This model trains a separate alignment module in a self- supervised manner, and induce alignments when the to-be-aligned target token is the decoder input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' MASK-ALIGN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This model is a self-supervised word aligner which makes use of the full context on the target side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' BTBA-FCBO-SST (Zhang and van Genabith, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This model has similar idea with Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' (2021a), but with different model architecture and training objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' SimAlign (Jalili Sabet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This model is a multilingual word aligner which induces alignment with contextual word embeddings from mBERT and XLM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' AwesomeAlign (Dou and Neubig, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' This model improves over SimAlign by designing new alignment induction method and proposing to fur- ther finetune the mPLM on parallel corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Among them, SimAlign and AwesomeAlign are multilingual aligners which support multiple lan- guage pairs in a single model, while others are bilingual word aligners which require training from scratch with bilingual corpus for each test lan- guage pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We re-implement SimAlign and Awe- someAlign, while quote the results from (Dou and Neubig, 2021) for the three statistical baselines and the corresponding paper for other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 Sentence Transformer We compare LaBSE with four other multilingual sentence Transformer in HuggingFace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The de- tailed information of these models are: distiluse-base-multilingual-cased-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 This model is a multilingual knowledge distilled version of m-USE (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', 2020), which has 135M parameters and supports more than 50+ languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' paraphrase-xlm-r-multilingual-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 This model is a multilingual version of paraphrase-distilroberta- base-v1 (Reimers and Gurevych, 2019), which has 278M parameters and supports 50+ languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' It initializes the student model with an mPLM and trains it to imitate monolingual sentence Trans- former on parallel data with knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' paraphrase-multilingual-MiniLM-L12-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 This model is a multilingual version of paraphrase- MiniLM-L12-v2 (Reimers and Gurevych, 2019), which has 118M parameters and supports 50+ languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' It trains similarly as paraphrase-xlm- r-multilingual-v1, but with different teacher and student model initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' paraphrase-multilingual-mpnet-base-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 This model is a multilingual version of paraphrase- mpnet-base-v2 (Reimers and Gurevych, 2019), 4https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='co/sentence-transformers/distiluse- base-multilingual-cased-v2 5https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='co/sentence- transformers/paraphrase-xlm-r-multilingual-v1 6https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='co/sentence- transformers/paraphrase-multilingual-MiniLM-L12-v2 7https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='co/sentence- transformers/paraphrase-multilingual-mpnet-base-v2 which has 278M parameters and supports 50+ lan- guages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' It trains similarly as paraphrase-xlm-r- multilingual-v1, but with different teacher model initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 Bilingual Finetuning We use the same dataset as bilingual baselines for bilingual finetuning following (Dou and Neubig, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' At each time, we finetune LaBSE with one language pair among de/fr/ro/ja/zh-en and test on all seven language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' For Awesome-align, we follow the setup in their paper, while for AccAlign, we use the same hyperparameters as the main ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 Representation Analysis We conduct representation analysis on de-en test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' To compute sbi, we calculate the averaged co- sine similarity of all gold aligned bilingual word pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' To compute smono, we randomly permute a given sentence x = ⟨x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', xn⟩ to get x′ = ⟨x′ 1, x′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=', x′ n⟩ and then create n word pairs as {⟨xi-x′ i⟩}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We go through all de and en test sentences and report the averaged cosine similarity of all created word pairs as smono.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' C Experiment Results Detailed results for each test language in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 are shown in Table 7 to Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Ft mode Ft type de-en sv-en fr-en ro-en ja-en zh-en fa-en avg Self-supervised full 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 adapter 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 Supervised full 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 adapter 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='1 Table 7: AER comparison of full finetuning and adapter-based finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The best AER for each column is bold and underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Model Ft lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Test lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' de-en fr-en ro-en ja-en zh-en sv-en fa-en AwesomeAlign de-en 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} 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results where the model is trained and tested on the same language pair are bold and underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' layer de-en sv-en fr-en ro-en ja-en zh-en fas-en avg mBERT 8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='6 18.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 LaBSE 6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 Table 9: AER comparison of LaBSE and other multilingual pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' All are without finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' We determine the best layer of alignment induction for each model using the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The best AER 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content='8 Table 10: AER comparison of vanilla LaBSE across layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' Layer 0 is the embedding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} +page_content=' The best AER for each column is bold and underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf'} diff --git a/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf b/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..18a3276870f9bc03533cd4e353aef36cb411c272 --- /dev/null +++ b/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Three Gorges University, Yichang 443002, China +2Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, +Central China Normal University, Wuhan 430079, China +3Center for Astronomy and Space Sciences, +China Three Gorges University, Yichang 443002, China +(Dated: January 5, 2023) +Abstract +Abstract: We investigate the magnetized QCD matter and chiral phase transition in a (2 + 1)- +flavor Nambu–Jona-Lasinio (NJL) model at finite temperature and chemical potential by comparing +the contributions from the tensor spin polarization (TSP) and anomalous magnetic moment (AMM) +of quarks. For light u and d quarks, when TSP and AMM are not considered, the magnetized system +is characterized by magnetic catalysis. The introduction of TSP will further enhance the magnetic +catalytic characteristics. +On the other hands, when AMM is introduced, the phase transition +temperature decreases with the magnetic field, which is the feature of inverse magnetic catalysis. +The phase diagram of u and d quarks will change from the crossover phase transition to the first +order phase transition with the increase of magnetic field and chemical potential when AMM is +induced. The phase diagram will not change from the crossover phase transition to the first order +phase transition when TSP is induced. For the phase diagram of strange s quark, whether TSP +or AMM is induced, the phase diagram will keep a crossover phase transition with the increase of +magnetic field and chemical potential. +∗ Corresponding author: fengsq@ctgu.edu.cn +1 + +I. +INTRODUCTION +Comprehending properties of QCD matter under a strong magnetic field is of essential +importance to further investigate the evolution of the early universe [1], non-central heavy- +ion collisions [2–5], neutron-star merges [6, 7], and the interior of magnestar [8, 9]. The +exploration of the QCD vacuum and strongly interacting matter under external strong mag- +netic fields has fascinated much attention, see reviews, e.g., Refs. [10–14]. Here we stress +the study of the magnetic field of non-central heavy-ion collisions, which comes from the +laboratory of mankind. The magnetic field reaches up to +√ +eB ∼ 0.1GeV for RHIC and +√ +eB ∼ 0.5 GeV for LHC in non-central heavy-ion collisions. This magnetic field is external +since it is generated by the spectators, and though it has a very short lifetime(of the order of +1 fm/c). However, as taken in Refs. [15–18], the presence of the quark-gluon plasma (QGP) +medium response effect, substantially delays the decay of these time-dependent magnetic +fields. This is why in the most cases, the effect of constant and uniform magnetic fields on +quark matter is discussed in the literature. The magnetic field coincides with the produc- +tion of the QGP and thus may have a fairly important effect on the properties of the phase +transition. For example, the chiral magnetic effect (CME) [16, 19–22], magnetic cataly- +sis (MC) in the vacuum [23–25], inverse magnetic catalysis (IMC) around the chiral phase +transition [26–29]. +The magnetic field can lead to spin polarization, that is, the condensation of quark +anti-quark (¯qq) pairs with spin parallel. Ref.[30] shows that a tensor-type interaction ∼ +� ¯ψΣ3ψ +�2 + +� ¯ψiγ5Σ3ψ +�2 produces a spin polarization (SP) +� ¯ψiγ1γ2ψ +� +, which is very similar +to the anomalous magnetic moment (AMM) produced by quarks in a magnetic field. The +tensor polarization operator ¯ψσµνψ can also be named as the spin polarization operator, or +the spin density since ¯ψσ12ψ = ψγ0Σ3ψ. If the quark spinor ψ is projected into the sub-spin +space ψ = ψ↑ + ψ↓ , corresponding to ¯ψσ12ψ ∼ +� ¯ψ↑ψ↑ +� +− +� ¯ψ↓ψ↓ +� +, which can be used to +measure the difference between the spin-up quark pair and the spin-down quark pair. +We investigate the magnetized QCD matter in a (2 + 1)-flavor Nambu–Jona-Lasinio +(NJL) model at finite temperature and chemical potential by comparing the contributions +from the tensor spin polarization (TSP) and AMM of quarks. For a particle with charge +e, mass m and spin ⃗s, its corresponding magnetic moment (MM) is µ. Corresponding to +¯qq pair with antiparallel spin pairs, it has a net magnetic moment (MM), so the chiral +2 + +condensation triggers a dynamic AMM. Under the action of the magnetic field, the net MM +tends to be parallel to the magnetic field. For SP with ¯qq pair parallel spin pairing, the +MM of spin-aligned quarks and anti-quarks cancel each other, and the spin polarization +pairing does not present a net MM. Therefore, compared with the chiral condensation with +a nonzero net MM, the total MM of the system considering SP condensation will reduce. +Therefore, systems with spin polarization are expected to exhibit relative diamagnetism. At +high temperatures, the pair of ¯qq dissociates, and all charged quarks become a single small +magnet, which is arranged in turn along the magnetic field; Therefore, QCD matter at high +temperature manifests paramagnetism. +The catalysis of chiral symmetry breaking induced by a magnetic field, namely the MC +effect, can be easily understood from dimension reduction. On the other hand, IMC effect, +the critical temperature of the chiral phase transition decreases with the increasing mag- +netic field, which is intuitively contradictory to the MC effect and is still a puzzle. Although +there are many publications trying to explain IMC by considering running coupling constant +generated by the magnetic field [31] and chiral imbalance caused by sphaleron transition or +instanton anti-instanton pairing [32]. Some interesting and novel properties of magnetized +QCD materials have recently been presented by lattice calculations, for example, magne- +tized materials exhibit paramagnetism (positive susceptibility) at high temperatures and +diamagnetism (negative susceptibility) at low temperatures [33, 34]. +The effect of an AMM of quark has drawn quite a lot of interest recently [35–41] in order +to investigate the IMC effect. The dynamical chiral symmetry broken is known as one of +the most important characteristics of QCD, which makes quarks achieve a dynamical mass +of QCD. Refs. [42, 43] pointed out that quarks’ AMM can also be dynamically produced +like the dynamic quark mass. Therefore, once quarks achieve dynamic mass, they should +also achieve dynamical AMM [42, 44–46]. The coefficient κ of quarks’ AMM in the magnetic +field by the effective interaction 1 +2qκFµν ¯ψσµνψ = 1 +2 [γµ, γν] is introduced and the IMC effect +at finite temperature is proposed by Ref. [47]. For QCD, both explicit and spontaneous +chiral symmetry breaking is dedicated to the AMM of quarks, which is also called dynamical +AMM [43]. +In this paper, we investigate the magnetism of QCD matter and chiral phase transition +under a magnetic field with the contribution from the TSP and the AMM of quarks re- +spectively. This paper is organized as follows: in Sec. II, we introduce the (2 + 1)-flavor +3 + +NJL models by including the AMM and the TSP in the external magnetic field respectively. +in Sec. III, we investigate MC and IMC by the AMM and TSP, respectively. Then the +dependencies of dynamical mass, entropy, sound-velocity, and critical point on the magnetic +field by comparing the contributions from the TSP and the AMM of quarks are studied in +Sec. III. Finally, we make the summaries and conclusions in Sec. IV. +II. +THE 2 + 1 FLAVORS NJL MODEL UNDER A MAGNETIC FIELD +The Lagrangian density of the (2 + 1)-flavor NJL model [48, 49] in the presence of an +external magnetic field is given as: +L = ¯ψ +� +iγµDµ + γ0µ − m +� +ψ + Gs +8 +� +a=0 +�� ¯ψλaψ +�2 + +� ¯ψiγ5λaψ +�2� +− K +� +det ¯ψ (1 + γ5) ψ + det ¯ψ (1 − γ5) ψ +� +, +(1) +where the quark field ψ carries three flavors (f = u, d, s) and three colors (c = r, g, b ), and +λa(a = 1, · · ·N2 +f − 1) represents the SU(3) Gell-Mann matrices in the three flavor space. +Current quark mass m is considered as mu = md for isospin symmetry of light quarks, +strange quark mass ms is different from the other light quark (mu and md) masses. The +difference between the strange and non-strange quark masses obviously breaks the SU(3) +flavor symmetry. µ is the quark chemical potential, and we assume that the quark chemical +potentials of the strange and non-strange quarks are the same. A covariant derivative with +magnetic field is introduced as Du = ∂µ + i QAext +µ , and the charge matrix in flavor space is +Q = diag (qu, qd, qs) = diag +�2 +3, −1 +3, −1 +3 +� +. +(2) +In general, if one chooses the gauge field Aext +µ += (0, 0, Bx1, 0), a constant magnetic field +should point at the x3-direction. The K term of Eq. (1) is the term of Kobayashi-Maskawa- +t’Hooft interaction [49–51]. +A. +The introduction of a (2 + 1)- flavors NJL model with TSP +It is shown that [30, 35] the breaking of the rotational symmetry by a uniform magnetic +field induces a separation between longitudinal and transverse fermion modes along the +direction of the magnetic field. This separation gives rise to the effective splitting of the +4 + +couplings in the one-gluon exchange interactions on which the NJL models are usually based. +This splitting is therefore reported in the four-fermion couplings of a QCD-inspired NJL +model in a magnetic field, and we can use the Fierz identities in a magnetic field [30, 31, 52] +to propose the interactions of scalar and tensor of the (2 + 1)-flavor NJL Lagrangian: +LTSP = ¯ψ +� +iγµDµ + γ0µ − m +� +ψ + Gs +8 +� +a=0 +�� ¯ψλaψ +�2 + +� ¯ψiγ5λaψ +�2� ++ Gt +8 +� +a=0 +�� ¯ψΣ3λaψ +�2 + +� ¯ψΣ3iγ5λaψ +�2� +− K +� +det +� ¯ψ (1 + γ5) ψ +� ++ det +� ¯ψ (1 − γ5) ψ +�� +. +(3) +The coupling constant Gs in the scalar/pseudo-scalar channel is closely related to the +spontaneously chiral symmetry breaking, which produces a dynamical quark mass, and +the tensor/ pseudo-tensor channels term Gt +8� +a=0 +�� ¯ψc +fΣ3λaψc +f +�2 + +� ¯ψc +fiΣ3γ5λaψc +f +�2� +is closely +related to the spin-spin interaction, which causes spin polarization condensation. +For the (2 + 1)-flavor NJL model, tensor-type interaction at the mean field level leads to +two types of spin polarization as +F3 = −2Gt +� ¯ψΣ3λ3ψ +� +, +F8 = −2Gt +� ¯ψΣ3λ8ψ +� +. +(4) +In general, F3 contains only u and d quark spin polarization condensates, on the other +hand, F8 is associated with the strange quark spin polarization condensate. The running +coupling constants are divided into longitudinal (g∥) and transverse (g⊥) components due +to the existence of the magnetic field. In our current study, the couplings of the above NJL +interactions relevant to quark gluon vertex coupling are expressed as Gs = +� +g2 +|| + g2 +⊥ +� +/Λ2 +and Gt = +� +g2 +|| − g2 +⊥ +� +/Λ2. The distinguishing transverse and parallel Fierz identities auto- +matically create a new channel of four-fermion interaction term with second order tensor +structure in Lagrangian density during the transformation from splitting quark-gluon cou- +pling to the scalar and pseudoscalar bilinear quantity [30]. Gs and Gt can be considered as +the scalar and tensor channel interaction couplings, respectively. +5 + +The effective potential by using standardized process is given +ΩTSP =Gs +� +f=u,d,s +� +ψψ +�2 +f + Gt +� +ψλ3Σ3ψ +�2 + Gt +� +ψλ8Σ3ψ +�2 − Nc +2π +� +f=u,d,s +|qfB| +∞ +� +l=0 +αl +∞ +� +−∞ +dpz +2π +× +� +εf,l,η + T ln +� +1 + exp +�−εf,l,η − µ +T +�� ++ T ln +� +1 + exp +�−εf,l,η + µ +T +��� ++ 4K +� +ψψ +� +u +� +ψψ +� +d +� +ψψ +� +s +(5) +where l= 0, 1, 2 ... +represents the quantum number of Landau level,and η = ±1 cor- +responds to the two kinds of spin direction of quark-antiquark(¯qq) pair. Contribution of +non-degenerate particles due to spin difference at non-lowest Landau energy levels can be +taken into account with the definition of this new operator αl = δ0,l +∆ (l) � +η=±1 +, where ∆ (l) +is denoted by +∆ (l) = + + + +0 +1 +l = 0 +l > 0 +(6) +and the energy spectrum of the lowest Landau Level ( LLL) (l = 0) and non-LLL (l ̸= 0) +are given as +ε2 +u,l=0 = p2 +z + +� +Mf + +� +F3 + F8 +√ +3 +��2 +, +ε2 +u,l̸=0,η=±1 = p2 +z + +�� +Mf +2 + 2|qfB|l + η +� +F3 + F8 +√ +3 +��2 +, +ε2 +d,l=0 = p2 +z + +� +Mf + +� +F3 − F8 +√ +3 +��2 +, +ε2 +d,l̸=0,η=±1 = p2 +z + +�� +Mf +2 + 2|qfB|l + η +� +F3 − F8 +√ +3 +��2 +, +ε2 +s,l=0 = p2 +z + +� +Mf + +�2F8 +√ +3 +��2 +, +ε2 +s,l̸=0,η=±1 = p2 +z + +�� +Mf +2 + 2|qfB|l + η +�2F8 +√ +3 +��2 +. +(7) +Note that the breaking of energy spectrum degeneracy caused by spin known as Zeeman +effect. Therefore, the contributions of spin come not only from the ground state of Landau +level, but also from the whole excited states of Landau level. The tensor condensate param- +eter F3 and F8 are self-consistently satisfied the minimum of the thermodynamic potential, +6 + +which are similar to dynamical quark mass Mf. At first, one can obtain three gap equations +for Mf (f = u, d, s) +∂ΩTSP (Mf, F3, F8) +∂Mf += 0, +(8) +and the other two gap equations for F3 and F8 is given as +∂ΩTSP (Mf, F3, F8) +∂F3 += 0, +∂ΩTSP (Mf, F3, F8) +∂F8 += 0. +(9) +To ensure that the thermodynamic potential in vacuum returns to zero, we define the +normalized thermodynamic potential as effective potential +Ωeff (T, µ, eB) = Ω (T, µ, eB) − Ω (0, 0, eB) . +(10) +Some of the relevant thermodynamical quantities can be evaluated by the effective po- +tential. The quark number density is +ρf = +� +l,η +Nc |qfeB| +4π2 +∞ +� +-∞ +dpz +� +n+ − n−� +, +(11) +where n± = 1/(exp [(εf,l,η ∓ µ) /T] + 1) is quark (antiquark) number distribution. +The +entropy density Sf = −∂Ωeff +∂T +is given as +Sf = − +� +l,η +Nc |qfeB| +4π2 +∞ +� +-∞ +dpz +� +ln +� +1 − n+� ++ ln +� +1 − n−� +− εf,l,η +T +� +n+ + n−� ++ µ +T (n+ − n−) +� +. +(12) +The energy density is given as +ε = T ∂P +∂T +µ∂P +∂µ − P, +(13) +where P is pressure. The square of sound-speed are defined as +c2 +s = ∂P +∂ε = +� µ +Sf +∂ρf +∂T + T +Sf +∂Sf +∂T +�-1 +. +(14) +7 + +B. +the introduction of the (2 + 1)- flavor NJL model with AMM +The effective Lagrangian density of the (2 + 1)- flavor with AMM [48, 49] is given as +LAMM = ¯ψ +� +iγµDµ + γ0µ − m+1 +2qfκσµνFµν +� +ψ ++ Gs +8 +� +a=0 +�� ¯ψλaψ +�2 + +� ¯ψiγ5λaψ +�2� +− K +� +det ¯ψ (1 + γ5) ψ + det ¯ψ (1 − γ5) ψ +� +. +(15) +The effective potential with AMM can be taken as +ΩAMM =Gs +� +f=u,d,s +� +ψψ +�2 +f + 4K +� +ψψ +� +u +� +ψψ +� +d +� +ψψ +� +s − Nc +2π +� +f=u,d,s +|qfB| +∞ +� +l=0 +� +t=±1 +∞ +� +−∞ +dpz +2π +× +� +Ef,l,t + T ln +� +1 + exp +�−Ef,l,t − µ +T +�� ++ T ln +� +1 + exp +�−Ef,l,t + µ +T +��� +, +(16) +where +Ef,l,t = +� +p2 +z + +�� +Mf +2 + 2|qfB|l +�1/2 − tκfqfeB +�2 +(17) +is the energy spectrum under different Landau energy levels, and t = ±1 corresponds to the +two kinds of spin direction of ¯qq pair. One can obtain three coupling gap equations for each +order parameter as +∂ΩAMM +∂Mf += 0, +(18) +where f = u, d, s for the three different flavors. Thus we can obtain three dynamical quark +masses of u, d, and s as +Mu = mu − 4Gs +� ¯ψψ +� +u + 2K +� +ψψ +� +d +� +ψψ +� +s, +Md = md − 4Gs +� ¯ψψ +� +d + 2K +� +ψψ +� +u +� +ψψ +� +s, +Ms = ms − 4Gs +� ¯ψψ +� +s + 2K +� +ψψ +� +u +� +ψψ +� +d, +(19) +where +� ¯ψψ +� +f = NcGs +2π +∞ +� +l=0 +αl|qfB| ++∞ +� +−∞ +dpz +2π +Mf +εf,l,t +� +1 − sκfqfB +ˆ +Mf,l,t +� � +1 − +1 +e +εf,l,t+µ +T ++ 1 +− +1 +e +εf,l,t−µ +T ++ 1 +� +(20) +corresponds to chiral condensation of different quark flavors. +8 + +III. +RESULTS AND DISCUSSIONS +To calibrate sets of parameters to applicable observable, parameters are referred [49, 53] +to be chosen as: Λ = 631.4 MeV, mu = md = 5.6 MeV, ms = 135.7 MeV, Λ2Gs = 1.835 +and KΛ5 = 9.29. +The empirical values are given as fπ = 93 MeV, mπ = 138 MeV, +mK = 495.7 MeV, and mη′ = 957.5 MeV. +The tensor channel coupling constant Gt restricted by the magnetic fields ought to be +zero in the case of the vanished magnetic field, and equals the value of Gs when eB → ∞. +At the following study, the value of Gt is taken as Gt = Gs/2. +FIG. 1. +The dependence of dynamical quark mass (M) on temperature (T) for four different +magnetic fields ( eB = 0.05, 0.10, 0.15 and 0.20 GeV2 ) , which does not consider TSP and AMM. +Fig 1.(a) is for µ = 0.0 GeV; and Fig 1.(b) is for µ = 0.25 GeV. +In order to investigate the effect of AMM on the phase transition, we make comparisons +between the two AMM sets. The compatible results obtained in [54] we define it as AMM1 +set as κu = κd = 0.38, κs = 0.25, while the defined AMM2 set chosen as κu = 0.123, κd = +9 + +0.6 +Ms +eB = 0.05GeV2 +-eB = 0.10GeV2 +.. eB = 0.15GeV2 +0.5 +eB = 0.20GeV2 +(GeV) +0.4 +M +0.3 +0.2 +eB = 0.05GeV2 + eB = 0.10GeV2 +Mu +eB = 0.15GeV2 +0.1 +-eB = 0.20GeV2 +0.6 +(b) +Ms +eB = 0.05GeV2 +eB = 0.10GeV2 +0.5 +..eB = 0.15GeV2 +-eB = 0.20GeV2 +(GeV) +0.4 +M +0.3 +0.2 +eB = 0.05GeV2 + -eB = 0.10GeV2 +Mu +0.1 +...eB = 0.15GeV2 +---eB = 0.20GeV2 +0 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +T (GeV)0.555, κs = 0.329 fixed by [55]. +Due to the NJL model is non-renormalizable, the divergent vacuum terms merged in gap +equation regularized by using the magnetic-field-independent regularization (MIFR) scheme +[56, 57], which gets rid of the nonphysical part by separating the vacuum term form the +integrals. The scheme dealing with the sums of all Landau level within the integrals by +means of Hurwitz zeta function are presented. +FIG. 2. +The dependence of dynamical quark mass (M) on temperature (T) for four different +magnetic fields ( eB = 0.05, 0.10, 0.15 and 0.20 GeV2 ) by considering TSP. Fig 2.(a) is for +µ = 0.0 GeV; and Fig 2.(b) is for µ = 0.25 GeV. +The dynamical mass or the quark condensate plays as an order parameter for the chiral +phase transition. Chiral restoration happens at high temperatures and/or high chemical +potentials. In Fig. 1(a, b), the dynamical quark masses M of u, d and s quarks without +considering AMM and TSP are manifested as decreasing smooth functions of temperatures +at µ = 0 and µ = 0.25 GeV, which indicates a chiral crossover. The dynamical mass M +is apparently enhanced by increasing the magnetic field. The magnetic field is shown at +eB = 0.05, 0.1, 0.15, and 0.2 GeV2 with µ = 0 and µ = 0.25 GeV respectively. Since we +10 + +(a) +eB = 0.05GeV2 +0.6 +Ms +eB =0.10GeV2 +•eB = 0.15GeV2 +0.5 +ieB = 0.20GeV2 +(GeV) +0.4 +M +0.3 +0.2 +eB = 0.05GeV2 +Mu + eB= 0.10GeV2 +... eB = 0.15GeV2 +0.1 +-eB = 0.20GeV2 +0 +(q) +0.6 +Ms +eB = 0.05GeV2 +- eB = 0.10GeV2 +. eB = 0.15GeV2 +0.5 +eB = 0.20GeV2 +(GeV) +0.4 +M +0.3 +0.2 +eB = 0.05GeV2 +Mu +-eB=0.10GeV2 +. eB = 0.15GeV2 +0.1 +---eB = 0.20GeV2 +0 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +T (GeV)have considered non-vanishing current quark mass, the chiral symmetry is never restored +fully. Since the dynamical mass is proportional to chiral condensate, it can be seen from +Fig.1 that the larger the magnetic field is, the larger the corresponding chiral condensation +is. This phenomenon is manifested as magnetic catalysis [19, 23, 24, 58], which accounts for +the magnetic field has a strong tendency to enhance (or catalyze) spin-zero ¯qq condensates. +By considering TSP of quarks, we investigate the temperature dependence of constituent +quark mass for eB = 0.05, 0.10, 0.15 and 0.20 +GeV2 respectively shown in Fig.2(a, b). +The dynamical mass M by considering TSP of quarks is manifested as a decreasing smooth +function of temperatures for different magnetic fields and chemical potentials, which cor- +responds to a chiral crossover. The dynamical mass M is apparently enhanced with the +increase of magnetic field, It is suggested that the introduction of TSP will enhance the +magnetic catalysis effect. +FIG. 3. Fig.3(a, b) shows the contour plots of the F3 and F8 distributions with zero chemical +potential in the T − eB plane, and Fig.3(c, d) shows similar plots of the F3 and F8 distributions +but with non-zero chemical potential µ = 0.25 GeV. +11 + +(a) F, with u = O Gev +(b) F。with u = 0 GeV +0.5 +0.5 +0.4 +0.12 +0.35 +0.4 +0.4 +0.1 +0.3 +(GeV3) +0.3 +0.08 +(GeV +0.3 +0.25 +0.2 +eB +eB +0.06 +0.2 +0.2 +0.15 +0.04 +0.1 +0.1 +0.1 +0.02 +0.05 +0 +0 +0 +0 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +T (GeV) +T (GeV) +(c) F, with μ = 0.25 GeV +(d) F。with u = 0.25 GeV +0.5 +0.5 +0.14 +0.06 +0.12 +0.4 +0.4 +0.05 +0.1 + (Gev2) +0.04 +0.3 +0.3 +0.08 +0.03 +8 +8 +0.06 +0.2 +0.2 +0.02 +0.04 +0.1 +0.01 +0.1 +0.02 +0 +0 +0 +0 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +T (GeV) +T (GeV)In the T − eB plane of the Fig.3, the corresponding temperature range is +0 ≤ T ≤ +0.3 GeV, and the magnetic field range is 0 ≤ eB ≤ 0.5 GeV2. Fig.3 (a, b) displays the +contour plots of the F3 and F8 distributions with zero chemical potential in the T − eB +plane, and Fig.3 (c, d) shows similar plots of the F3 and F8 distributions but with non-zero +chemical potential µ = 0.25 GeV. The (2 + 1)-flavor spin polarization is different from +that of two flavor spin polarization because of an additional term F8 = −2Gt +� ¯ψΣ3λ8ψ +� +associated with the λ8 flavor generator. +The spin condensates affect dynamical quark masses and quark dispersion relation. It is +found that the nonzero values of the two spin condensates F3 and F8 exist in the restored +chiral symmetry phase with high temperature and large magnetic field, but F3 and F8 are +almost zero in the chiral symmetry broken phase. We also noticed that F8 decreases sharply +with the increase of chemical potential, but F3 changes slightly with the chemical potential. +FIG. 4. The dynamical quark mass (M) as a function of temperature (T) for four different magnetic +fields (eB = 0.05, 0.10, 0.15 and 0.20 GeV2) by considering the different sets of AMM. Fig.4(a, +b) are for µ = 0 and µ = 0.25 GeV respectively with AMM1 set as κu = κd = 0.38, κs = 0.25. +Fig.4(c, d) is same as Fig.4 (a, b) but for AMM2 set as κu = 0.123, κd = 0.555, κs = 0.329. +Figure 4. displays the dependence of dynamical quark mass (M) on temperature (T) +12 + +0.6F (b) +(a) +eB = 0.05GeV2 +0.6 +Ms +M. +eB = 0.05 GeV2 +- eB = 0.10GeV2 +-eB = 0.10GeV2 +...eB = ..5 GeV2. +0.5 +0.5 +-eB = 0.20GeV2 +eB = 0.15GeV2 +M 0.4 +(GeV) +0.4 +(Gev +M +0.3 +0.3 +M +Mu +eB = 0.05GeV2 +0.2 +0.2 +eB = 0.05GeV2 +- eB = 0.10GeV2 +eB = 0.15GeV2 +- -eB = 0.10GeV2 +Mu +0.1 +0.1 +- eB = 0.20GeV2 +:eB = 0.15GeV2 +0 +0 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.05 +0.1 +0.15 +0.2 +0 +T (GeV) +T (GeV) +0.6 +0.6 +(d) +(c) +M: +M. +eB = 0.05 GeV2 +eB = 0.05 GeV2 +eB = 0.10GeV2 +0.5 +-eB = 0.10GeV2 +0.5 +eB = 0.15GeV2 +eB = 0.20GeV2 +eB = 0.15GeV2 +0.4 +(GeV +0.3 +M +M +0.3 +Mu +一 +0.2 +0.2 +Mu +eB = 0.05GeV2 +eB = 0.05GeV2 +- eB = 0.10GeV2 +-eB = 0.10GeV2 +0.1 +0.1 +-eB = 0.15GeV2 +- +-eB = 0.15GeV2 +.eB = 0.20GeV2 +0 +0 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0 +0.05 +0.1 +0.15 +0.2 +T (GeV) +T (GeV)for four different magnetic fields (eB = 0.05, 0.10, 0.15 and 0.20 GeV2) by considering the +two AMM’s sets. Fig.4(a, b) are for µ = 0 GeV and µ = 0.25 GeV with AMM1 set as +κu = κd = 0.38 and κs = 0.25. Fig.4(c, d) is same as Fig.4(a, b) but with AMM2 set as +κu = 0.123, κd = 0.555 and κs = 0.329. Contrary to the behavior of the zero AMM in Fig.1, +the mass-decreasing behavior of u and d quarks in the chiral restoration is not a smooth +slope but a sudden drop, which indicates the existence of a first-order transition. However, +the smooth slope of the dynamical mass for the crossover can be still present in the weak +field eB = 0.05 GeV2 for the non-zero AMM. The mass-decreasing behavior of s quark in the +chiral restoration is still a smooth slope, which suggests a chiral crossover for s quark. From +Fig.4, it is found that the dynamical quark mass of u and d quarks have the characteristics +of inverse magnetic catalysis in the chiral restoration phase (T ≥ TC) by using the AMM +sets. +FIG. 5. The critical temperature of u and d quarks as a function of the magnetic field at µ = 0 +(a) and = 0.25 GeV (b). +In Fig. 5, the critical temperature is shown as a function of the magnetic field with the +chemical potentials µ = 0 and 0.25 GeV respectively. It is found that the critical temperature +decreases with the magnetic field for the AMM1 and AMM2 sets, which indicates an inverse +magnetic catalysis which qualitatively agrees with lattice result in [33]. +On the contrary, with the TSP, TC enhances as a function of the magnetic field, which +is the extension of the magnetic catalysis effect from vacuum to finite temperature. The +different effects of AMM and TSP on chiral condensate can be easily understood from the +dispersion relations in Eq. (7) and Eq. (17), the AMM reduces the LLL energy and the +TSP lifts up the LLL energy, which causes the different effects. +13 + +(a) +(b) +0.13 +0.19 +- no AMM&TSP +0.11 +(GeV) + no AMM&TSP +.TSP +.TSP +AMM1 +-AMM1 +-AMM2 +- AMM2 +0.15 +0.07 +0.13 +0.05 +0.05 +0.1 +0.15 +0.2 +0.05 +0.1 +0.15 +0.2 +eB (GeV2) +eB (GeV2)FIG. 6. The same as Fig. 5, but for the s-quark. +The critical temperature of chiral phase transition of s quark as a function of eB is man- +ifested in Fig.6. Compared with light quarks of u and d, the phase transition temperature +TC of s quark with TSP increases significantly with the increase of magnetic field, which +corresponds to the characteristics of magnetic catalysis. +The introduction of AMM sets +corresponds to inverse magnetic catalytic characteristics. +Figure 7 displays the dependencies of the entropy density of u , d and s quarks on +temperature at zero chemical potential. It can be noted that the introduction of the AMM +makes the crossover phase transition sharp. +It is worth noting that the AMM in Fig.7 +corresponds to three different settings, which are AMM0, AMM1 and AMM2, respectively. +AMM0 means that the AMM is not considered, that is, all κ values in Eq. (17) are set +to zero. AMM1 and AMM2 sets have been mentioned above. When eB = 0.05 GeV2, the +magnetic field is not big enough to excite the effect on entropy. When eB = 0.2 GeV2, some +of the effects of the magnetic field on entropy for different AMM sets and TSP can be excited. +It is found that the entropy shows a sharp change near the phase transition temperature +after adding AMM sets, and this sharp change is more obvious with the magnetic field +increases and chemical potential, showing a first-order phase characteristic. The change of +entropy with temperature near the phase transition temperature is relatively smooth after +adding TSP, and it behaves like the crossover transition. +14 + +0.34 +0.34 F +(b) +0.3 +- no AMM&TSP + no AMM&TSP +0.3 +(GeV) +(GeV) +.TSP +. TSP +AMM1 +0.26 +-AMM1 +c +C +AMM2 +-AMM2 +0.26 +0.22 +0.22 +0.18 +0.05 +0.1 +0.15 +0.2 +0.05 +0.1 +0.15 +0.2 +eB (GeV2) +eB (GeV2)FIG. 7. The dependence of S/T 3 on temperature T at µ = 0GeV with different magnetic field. +Fig.7 (a) is for eB = 0.05 GeV2 and Fig.7 (b) is for eB = 0.2 GeV2. +The dependence of square of sound-velocity c2 +s on temperature T is manifested in Fig.8. +Fig.8(a) and Fig.8(b) are for zero chemical potential µ = 0 and µ = 0.25 GeV respectively. +The square of sound-velocity shows a sudden rapid rise inflection near the phase transition +after adding AMM sets, and this rapid rise is more obvious with the magnetic field increases, +showing a obviously first-order phase characteristic. +On the other hands, the change of +square of sound-velocity with temperature near the phase transition is relatively smooth +inflection after adding TSP, showing a obviously cross-over transition characteristic. The +result obtained by using the square of sound velocity is completely consistent with the result +of entropy analysis. +Compared with u and d quarks, the square of sound-velocity of s quark with temperature +is relatively smooth inflection after adding TSP and AMM sets. It is proposed that s quarks +have always maintained obvious cross-over characteristics. In the high-temperature region, +the square of sound-velocity c2 +s increases with temperature and obtains the saturation value +15 + +16 +(a) +12 +S +8 +no AMM&TSP, eB = 0.05GeV2 +" AMM1,eB = 0.05GeV2 +4 +AMM2,eB = 0.05GeV2 +--TSP,eB = 0.05GeV2 +Stefan-Boltzmann limit +0 +(b) +16 +12 +2 +S +8 + no AMM&TSP,eB = 0.20GeV2 +. AMM1, eB = 0.20GeV2 +4 +-AMM2,eB = 0.20GeV2 +--TSP,eB = 0.20GeV2 +Stefan-Boltzmann limit +0 +0.05 +0.1 +0.15 +0.2 +0.25 +T (GeV)c2 +s = 1/3 to satisfy the relativistic requirement. This suggests that the equation of state +in the chiral restoration phase at high temperatures is close to the Stefan-Boltzmann limit +ε = 3P. +FIG. 8. The sound-velocity square C2 +s of u and d with s quarks as a function of the temperature +T with different chemical potential. Fig.8 (a, b) is for u and d quarks with zero chemical potential +µ = 0, and µ = 0.25 GeV, and Fig.8 (c, d) is for s quarks +IV. +SUMMARY AND CONCLUSIONS +In this work, we thoroughly study the effect from TSP and AMM on the vacuum, phase +transition and thermal magnetized QCD in the (2 + 1)-flavor Nambu-Jona-Lasinio (NJL) +model with nonzero current quark masses at finite temperature and chemical potential. An +unified physical mechanism to illustrate the novel consequences from recent lattice QCD as +magnetic catalysis and inverse magnetic catalysis effect proposed in the paper. +We focus on two topics: the AMM and TSP. For these two topics, we should pay special +attention to the dispersion relation, especially the lowest Landau level, which determines +16 + +(a) u and d quarks with u = O GeV +(b) u and d quarks with u = 0.25 GeV +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +2s +TSP, eB = 0.05GeV2 +0.2 +0.2 +TSP,eB = 0.05 GeV2 +- AMM1, eB = 0.05GeV2 +AMM1, eB = 0.05GeV2 +AMM2, eB = 0.05GeV2 +- AMM2, eB = 0.05GeV2 +0.1 +—TSP,eB = 0.20GeV2 +0.1 ++ -AMM1,eB = 0.20GeV2 ++- AMM1,eB = 0.20GeV2 +- AMM2, eB = 0.20GeV2 ++-- AMM2, eB = 0.20GeV2 +0: +01 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0 +0.05 +0.1 +0.15 +0.2 +T (GeV) +T (GeV) +(c) s quarks with u = O GeV +(d) s quarks with u = 0.25 GeV +0.4 +0.4 +TSP,eB = 0.05GeV2 +TSP,eB = 0.05GeV2 +- AMM1,eB = 0.05GeV2 +- AMM1,eB = 0.05GeV2 + AMM2, eB = 0.05GeV2 +- AMM2,eB = 0.05GeV2 +0.3 +0.3 +TSP, eB = 0.20GeV2 +-TSP,eB = 0.15GeV2 ++ AMM1,eB = 0.15GeV2 +- AMM1,eB = 0.20GeV2 +2s +2s +- AMM2,eB = 0.20GeV2 ++- AMM2, eB = 0.15GeV2 +0.2 +0.2 +0.1 +0.1 ++0 +0+ +0.05 +0.1 +0.15 +0.2 +0 +0.25 +0 +0.05 +0.1 +0.15 +0.2 +T (GeV) +T (GeV)the properties of the magnetized quark matter system. +The TSP lifts up the LLL en- +ergy: ELLL = +� +p2 +z + (M + F3 + F8 +√ +3)2� 1 +2, while the AMM effect diminishes the LLL energy: +ELLL = +� +p2 +z + (M − κ |qf| B)2�1/2 therefore, the TSP and the AMM take almost opposite +effects on magnetized quark matter. When the AMM and TSP contributions are not con- +sidered, the corresponding phase transition temperature increases with the magnetic field, +showing the characteristics of magnetic catalysis. When considering only the contribution of +TSP, the phase transition temperature also increases with the magnetic field, showing the +characteristics of magnetic catalysis. On the other hand, when AMM are introduced, the +phase transition temperature decreases with the magnetic field, showing the characteristics +of inverse magnetic catalysis. +It is found that the square of sound-velocity shows a sudden rapid rise inflection near +the phase transition after adding AMM sets, and this rapid rise is more obvious with the +magnetic field increases, showing a obviously first-order phase characteristic. On the other +hands, after adding TSP, the change of square of sound-velocity with temperature near the +phase transition is relatively smooth inflection, showing a obviously cross-over transition +characteristic. +The result obtained by using the square of sound velocity is completely +consistent with the result of entropy analysis. +The (2 + 1)-flavor spin polarization is different from that of two flavor because of an ad- +ditional F8 = −2Gt +� ¯ψΣ3λ8ψ +� +associated with the λ8 flavor generator. The spin condensates +affect the dynamical quark masses, chiral phase transition,and quark dispersion relation. 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Lett. 73, 3499 (1994). +20 + diff --git a/69AzT4oBgHgl3EQfgPxu/content/tmp_files/load_file.txt b/69AzT4oBgHgl3EQfgPxu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d35855c2ee2757868ccbdb9656645c920f9c4d3 --- /dev/null +++ b/69AzT4oBgHgl3EQfgPxu/content/tmp_files/load_file.txt @@ -0,0 +1,1126 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf,len=1125 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='01465v1 [hep-ph] 4 Jan 2023 Spin Polarization and Anomalous Magnetic Moment in a (2 + 1)-flavor Nambu-Jona-Lasinio model in the thermomagnetic background Yi-Wei Qiu1 and Sheng-Qin Feng1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' ∗ 1College of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' China Three Gorges University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Yichang 443002,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' China 2Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Central China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Wuhan 430079,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' China 3Center for Astronomy and Space Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' China Three Gorges University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Yichang 443002,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' China (Dated: January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 2023) Abstract Abstract: We investigate the magnetized QCD matter and chiral phase transition in a (2 + 1)- flavor Nambu–Jona-Lasinio (NJL) model at finite temperature and chemical potential by comparing the contributions from the tensor spin polarization (TSP) and anomalous magnetic moment (AMM) of quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' For light u and d quarks, when TSP and AMM are not considered, the magnetized system is characterized by magnetic catalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The introduction of TSP will further enhance the magnetic catalytic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' On the other hands, when AMM is introduced, the phase transition temperature decreases with the magnetic field, which is the feature of inverse magnetic catalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The phase diagram of u and d quarks will change from the crossover phase transition to the first order phase transition with the increase of magnetic field and chemical potential when AMM is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The phase diagram will not change from the crossover phase transition to the first order phase transition when TSP is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' For the phase diagram of strange s quark, whether TSP or AMM is induced, the phase diagram will keep a crossover phase transition with the increase of magnetic field and chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' ∗ Corresponding author: fengsq@ctgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='cn 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' INTRODUCTION Comprehending properties of QCD matter under a strong magnetic field is of essential importance to further investigate the evolution of the early universe [1], non-central heavy- ion collisions [2–5], neutron-star merges [6, 7], and the interior of magnestar [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The exploration of the QCD vacuum and strongly interacting matter under external strong mag- netic fields has fascinated much attention, see reviews, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Here we stress the study of the magnetic field of non-central heavy-ion collisions, which comes from the laboratory of mankind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The magnetic field reaches up to √ eB ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1GeV for RHIC and √ eB ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 GeV for LHC in non-central heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' This magnetic field is external since it is generated by the spectators, and though it has a very short lifetime(of the order of 1 fm/c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' However, as taken in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' [15–18], the presence of the quark-gluon plasma (QGP) medium response effect, substantially delays the decay of these time-dependent magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' This is why in the most cases, the effect of constant and uniform magnetic fields on quark matter is discussed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The magnetic field coincides with the produc- tion of the QGP and thus may have a fairly important effect on the properties of the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' For example, the chiral magnetic effect (CME) [16, 19–22], magnetic cataly- sis (MC) in the vacuum [23–25], inverse magnetic catalysis (IMC) around the chiral phase transition [26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The magnetic field can lead to spin polarization, that is, the condensation of quark anti-quark (¯qq) pairs with spin parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' [30] shows that a tensor-type interaction ∼ � ¯ψΣ3ψ �2 + � ¯ψiγ5Σ3ψ �2 produces a spin polarization (SP) � ¯ψiγ1γ2ψ � , which is very similar to the anomalous magnetic moment (AMM) produced by quarks in a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The tensor polarization operator ¯ψσµνψ can also be named as the spin polarization operator, or the spin density since ¯ψσ12ψ = ψγ0Σ3ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' If the quark spinor ψ is projected into the sub-spin space ψ = ψ↑ + ψ↓ , corresponding to ¯ψσ12ψ ∼ � ¯ψ↑ψ↑ � − � ¯ψ↓ψ↓ � , which can be used to measure the difference between the spin-up quark pair and the spin-down quark pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' We investigate the magnetized QCD matter in a (2 + 1)-flavor Nambu–Jona-Lasinio (NJL) model at finite temperature and chemical potential by comparing the contributions from the tensor spin polarization (TSP) and AMM of quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' For a particle with charge e, mass m and spin ⃗s, its corresponding magnetic moment (MM) is µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Corresponding to ¯qq pair with antiparallel spin pairs, it has a net magnetic moment (MM), so the chiral 2 condensation triggers a dynamic AMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Under the action of the magnetic field, the net MM tends to be parallel to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' For SP with ¯qq pair parallel spin pairing, the MM of spin-aligned quarks and anti-quarks cancel each other, and the spin polarization pairing does not present a net MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Therefore, compared with the chiral condensation with a nonzero net MM, the total MM of the system considering SP condensation will reduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Therefore, systems with spin polarization are expected to exhibit relative diamagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' At high temperatures, the pair of ¯qq dissociates, and all charged quarks become a single small magnet, which is arranged in turn along the magnetic field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Therefore, QCD matter at high temperature manifests paramagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The catalysis of chiral symmetry breaking induced by a magnetic field, namely the MC effect, can be easily understood from dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' On the other hand, IMC effect, the critical temperature of the chiral phase transition decreases with the increasing mag- netic field, which is intuitively contradictory to the MC effect and is still a puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Although there are many publications trying to explain IMC by considering running coupling constant generated by the magnetic field [31] and chiral imbalance caused by sphaleron transition or instanton anti-instanton pairing [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Some interesting and novel properties of magnetized QCD materials have recently been presented by lattice calculations, for example, magne- tized materials exhibit paramagnetism (positive susceptibility) at high temperatures and diamagnetism (negative susceptibility) at low temperatures [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The effect of an AMM of quark has drawn quite a lot of interest recently [35–41] in order to investigate the IMC effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The dynamical chiral symmetry broken is known as one of the most important characteristics of QCD, which makes quarks achieve a dynamical mass of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' [42, 43] pointed out that quarks’ AMM can also be dynamically produced like the dynamic quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Therefore, once quarks achieve dynamic mass, they should also achieve dynamical AMM [42, 44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The coefficient κ of quarks’ AMM in the magnetic field by the effective interaction 1 2qκFµν ¯ψσµνψ = 1 2 [γµ, γν] is introduced and the IMC effect at finite temperature is proposed by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' For QCD, both explicit and spontaneous chiral symmetry breaking is dedicated to the AMM of quarks, which is also called dynamical AMM [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' In this paper, we investigate the magnetism of QCD matter and chiral phase transition under a magnetic field with the contribution from the TSP and the AMM of quarks re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' This paper is organized as follows: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' II, we introduce the (2 + 1)-flavor 3 NJL models by including the AMM and the TSP in the external magnetic field respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' III, we investigate MC and IMC by the AMM and TSP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Then the dependencies of dynamical mass, entropy, sound-velocity, and critical point on the magnetic field by comparing the contributions from the TSP and the AMM of quarks are studied in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Finally, we make the summaries and conclusions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' THE 2 + 1 FLAVORS NJL MODEL UNDER A MAGNETIC FIELD The Lagrangian density of the (2 + 1)-flavor NJL model [48,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 49] in the presence of an external magnetic field is given as: L = ¯ψ � iγµDµ + γ0µ − m � ψ + Gs 8 � a=0 �� ¯ψλaψ �2 + � ¯ψiγ5λaψ �2� − K � det ¯ψ (1 + γ5) ψ + det ¯ψ (1 − γ5) ψ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (1) where the quark field ψ carries three flavors (f = u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' s) and three colors (c = r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' b ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' and λa(a = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' · · ·N2 f − 1) represents the SU(3) Gell-Mann matrices in the three flavor space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Current quark mass m is considered as mu = md for isospin symmetry of light quarks, strange quark mass ms is different from the other light quark (mu and md) masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The difference between the strange and non-strange quark masses obviously breaks the SU(3) flavor symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' µ is the quark chemical potential, and we assume that the quark chemical potentials of the strange and non-strange quarks are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' A covariant derivative with magnetic field is introduced as Du = ∂µ + i QAext µ , and the charge matrix in flavor space is Q = diag (qu, qd, qs) = diag �2 3, −1 3, −1 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (2) In general, if one chooses the gauge field Aext µ = (0, 0, Bx1, 0), a constant magnetic field should point at the x3-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The K term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (1) is the term of Kobayashi-Maskawa- t’Hooft interaction [49–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The introduction of a (2 + 1)- flavors NJL model with TSP It is shown that [30, 35] the breaking of the rotational symmetry by a uniform magnetic field induces a separation between longitudinal and transverse fermion modes along the direction of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' This separation gives rise to the effective splitting of the 4 couplings in the one-gluon exchange interactions on which the NJL models are usually based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' This splitting is therefore reported in the four-fermion couplings of a QCD-inspired NJL model in a magnetic field, and we can use the Fierz identities in a magnetic field [30, 31, 52] to propose the interactions of scalar and tensor of the (2 + 1)-flavor NJL Lagrangian: LTSP = ¯ψ � iγµDµ + γ0µ − m � ψ + Gs 8 � a=0 �� ¯ψλaψ �2 + � ¯ψiγ5λaψ �2� + Gt 8 � a=0 �� ¯ψΣ3λaψ �2 + � ¯ψΣ3iγ5λaψ �2� − K � det � ¯ψ (1 + γ5) ψ � + det � ¯ψ (1 − γ5) ψ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (3) The coupling constant Gs in the scalar/pseudo-scalar channel is closely related to the spontaneously chiral symmetry breaking, which produces a dynamical quark mass, and the tensor/ pseudo-tensor channels term Gt 8� a=0 �� ¯ψc fΣ3λaψc f �2 + � ¯ψc fiΣ3γ5λaψc f �2� is closely related to the spin-spin interaction, which causes spin polarization condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' For the (2 + 1)-flavor NJL model, tensor-type interaction at the mean field level leads to two types of spin polarization as F3 = −2Gt � ¯ψΣ3λ3ψ � , F8 = −2Gt � ¯ψΣ3λ8ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (4) In general, F3 contains only u and d quark spin polarization condensates, on the other hand, F8 is associated with the strange quark spin polarization condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The running coupling constants are divided into longitudinal (g∥) and transverse (g⊥) components due to the existence of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' In our current study, the couplings of the above NJL interactions relevant to quark gluon vertex coupling are expressed as Gs = � g2 || + g2 ⊥ � /Λ2 and Gt = � g2 || − g2 ⊥ � /Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The distinguishing transverse and parallel Fierz identities auto- matically create a new channel of four-fermion interaction term with second order tensor structure in Lagrangian density during the transformation from splitting quark-gluon cou- pling to the scalar and pseudoscalar bilinear quantity [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Gs and Gt can be considered as the scalar and tensor channel interaction couplings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 5 The effective potential by using standardized process is given ΩTSP =Gs � f=u,d,s � ψψ �2 f + Gt � ψλ3Σ3ψ �2 + Gt � ψλ8Σ3ψ �2 − Nc 2π � f=u,d,s |qfB| ∞ � l=0 αl ∞ � −∞ dpz 2π × � εf,l,η + T ln � 1 + exp �−εf,l,η − µ T �� + T ln � 1 + exp �−εf,l,η + µ T ��� + 4K � ψψ � u � ψψ � d � ψψ � s (5) where l= 0, 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' represents the quantum number of Landau level,and η = ±1 cor- responds to the two kinds of spin direction of quark-antiquark(¯qq) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Contribution of non-degenerate particles due to spin difference at non-lowest Landau energy levels can be taken into account with the definition of this new operator αl = δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='l +∆ (l) � η=±1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' where ∆ (l) is denoted by ∆ (l) = \uf8f1 \uf8f2 \uf8f3 0 1 l = 0 l > 0 (6) and the energy spectrum of the lowest Landau Level ( LLL) (l = 0) and non-LLL (l ̸= 0) are given as ε2 u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='l=0 = p2 z + � Mf + � F3 + F8 √ 3 ��2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' ε2 u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='l̸=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='η=±1 = p2 z + �� Mf 2 + 2|qfB|l + η � F3 + F8 √ 3 ��2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' ε2 d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='l=0 = p2 z + � Mf + � F3 − F8 √ 3 ��2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' ε2 d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='l̸=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='η=±1 = p2 z + �� Mf 2 + 2|qfB|l + η � F3 − F8 √ 3 ��2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' ε2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='l=0 = p2 z + � Mf + �2F8 √ 3 ��2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' ε2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='l̸=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='η=±1 = p2 z + �� Mf 2 + 2|qfB|l + η �2F8 √ 3 ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (7) Note that the breaking of energy spectrum degeneracy caused by spin known as Zeeman effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Therefore, the contributions of spin come not only from the ground state of Landau level, but also from the whole excited states of Landau level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The tensor condensate param- eter F3 and F8 are self-consistently satisfied the minimum of the thermodynamic potential, 6 which are similar to dynamical quark mass Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' At first, one can obtain three gap equations for Mf (f = u, d, s) ∂ΩTSP (Mf, F3, F8) ∂Mf = 0, (8) and the other two gap equations for F3 and F8 is given as ∂ΩTSP (Mf, F3, F8) ∂F3 = 0, ∂ΩTSP (Mf, F3, F8) ∂F8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (9) To ensure that the thermodynamic potential in vacuum returns to zero, we define the normalized thermodynamic potential as effective potential Ωeff (T, µ, eB) = Ω (T, µ, eB) − Ω (0, 0, eB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (10) Some of the relevant thermodynamical quantities can be evaluated by the effective po- tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The quark number density is ρf = � l,η Nc |qfeB| 4π2 ∞ � ∞ dpz � n+ − n−� , (11) where n± = 1/(exp [(εf,l,η ∓ µ) /T] + 1) is quark (antiquark) number distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The entropy density Sf = −∂Ωeff ∂T is given as Sf = − � l,η Nc |qfeB| 4π2 ∞ � ∞ dpz � ln � 1 − n+� + ln � 1 − n−� − εf,l,η T � n+ + n−� + µ T (n+ − n−) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (12) The energy density is given as ε = T ∂P ∂T +µ∂P ∂µ − P, (13) where P is pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The square of sound-speed are defined as c2 s = ∂P ∂ε = � µ Sf ∂ρf ∂T + T Sf ∂Sf ∂T �-1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (14) 7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' the introduction of the (2 + 1)- flavor NJL model with AMM The effective Lagrangian density of the (2 + 1)- flavor with AMM [48, 49] is given as LAMM = ¯ψ � iγµDµ + γ0µ − m+1 2qfκσµνFµν � ψ + Gs 8 � a=0 �� ¯ψλaψ �2 + � ¯ψiγ5λaψ �2� − K � det ¯ψ (1 + γ5) ψ + det ¯ψ (1 − γ5) ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (15) The effective potential with AMM can be taken as ΩAMM =Gs � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='s � ψψ �2 f + 4K � ψψ � u � ψψ � d � ψψ � s − Nc 2π � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='s |qfB| ∞ � l=0 � t=±1 ∞ � −∞ dpz 2π × � Ef,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='t + T ln � 1 + exp �−Ef,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='t − µ T �� + T ln � 1 + exp �−Ef,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='t + µ T ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (16) where Ef,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='t = � p2 z + �� Mf 2 + 2|qfB|l �1/2 − tκfqfeB �2 (17) is the energy spectrum under different Landau energy levels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' and t = ±1 corresponds to the two kinds of spin direction of ¯qq pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' One can obtain three coupling gap equations for each order parameter as ∂ΩAMM ∂Mf = 0, (18) where f = u, d, s for the three different flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Thus we can obtain three dynamical quark masses of u, d, and s as Mu = mu − 4Gs � ¯ψψ � u + 2K � ψψ � d � ψψ � s, Md = md − 4Gs � ¯ψψ � d + 2K � ψψ � u � ψψ � s, Ms = ms − 4Gs � ¯ψψ � s + 2K � ψψ � u � ψψ � d, (19) where � ¯ψψ � f = NcGs 2π ∞ � l=0 αl|qfB| +∞ � −∞ dpz 2π Mf εf,l,t � 1 − sκfqfB ˆ Mf,l,t � � 1 − 1 e εf,l,t+µ T + 1 − 1 e εf,l,t−µ T + 1 � (20) corresponds to chiral condensation of different quark flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 8 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS To calibrate sets of parameters to applicable observable, parameters are referred [49, 53] to be chosen as: Λ = 631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 MeV, mu = md = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='6 MeV, ms = 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='7 MeV, Λ2Gs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='835 and KΛ5 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The empirical values are given as fπ = 93 MeV, mπ = 138 MeV, mK = 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='7 MeV, and mη′ = 957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The tensor channel coupling constant Gt restricted by the magnetic fields ought to be zero in the case of the vanished magnetic field, and equals the value of Gs when eB → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' At the following study, the value of Gt is taken as Gt = Gs/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The dependence of dynamical quark mass (M) on temperature (T) for four different magnetic fields ( eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20 GeV2 ) , which does not consider TSP and AMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (a) is for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='0 GeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' and Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (b) is for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' In order to investigate the effect of AMM on the phase transition, we make comparisons between the two AMM sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The compatible results obtained in [54] we define it as AMM1 set as κu = κd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='38, κs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25, while the defined AMM2 set chosen as κu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='123, κd = 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='6 Ms eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='. eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 Mu eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='6 (b) Ms eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='.eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 Mu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 ---eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 T (GeV)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='555, κs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='329 fixed by [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Due to the NJL model is non-renormalizable, the divergent vacuum terms merged in gap equation regularized by using the magnetic-field-independent regularization (MIFR) scheme [56, 57], which gets rid of the nonphysical part by separating the vacuum term form the integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The scheme dealing with the sums of all Landau level within the integrals by means of Hurwitz zeta function are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The dependence of dynamical quark mass (M) on temperature (T) for four different magnetic fields ( eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20 GeV2 ) by considering TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (a) is for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='0 GeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' and Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (b) is for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The dynamical mass or the quark condensate plays as an order parameter for the chiral phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Chiral restoration happens at high temperatures and/or high chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 1(a, b), the dynamical quark masses M of u, d and s quarks without considering AMM and TSP are manifested as decreasing smooth functions of temperatures at µ = 0 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV, which indicates a chiral crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The dynamical mass M is apparently enhanced by increasing the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The magnetic field is shown at eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 GeV2 with µ = 0 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Since we 10 (a) eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='6 Ms eB =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 ieB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 Mu eB= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 0 (q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='6 Ms eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 Mu eB=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 ---eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 T (GeV)have considered non-vanishing current quark mass, the chiral symmetry is never restored fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Since the dynamical mass is proportional to chiral condensate, it can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 that the larger the magnetic field is, the larger the corresponding chiral condensation is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' This phenomenon is manifested as magnetic catalysis [19, 23, 24, 58], which accounts for the magnetic field has a strong tendency to enhance (or catalyze) spin-zero ¯qq condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' By considering TSP of quarks, we investigate the temperature dependence of constituent quark mass for eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20 GeV2 respectively shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The dynamical mass M by considering TSP of quarks is manifested as a decreasing smooth function of temperatures for different magnetic fields and chemical potentials, which cor- responds to a chiral crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The dynamical mass M is apparently enhanced with the increase of magnetic field, It is suggested that the introduction of TSP will enhance the magnetic catalysis effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3(a, b) shows the contour plots of the F3 and F8 distributions with zero chemical potential in the T − eB plane, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3(c, d) shows similar plots of the F3 and F8 distributions but with non-zero chemical potential µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 11 (a) F, with u = O Gev (b) F。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='with u = 0 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 (GeV3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='08 (GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 eB eB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 T (GeV) T (GeV) (c) F, with μ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV (d) F。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='with u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 (Gev2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='03 8 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='02 0 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 T (GeV) T (GeV)In the T − eB plane of the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3, the corresponding temperature range is 0 ≤ T ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 GeV, and the magnetic field range is 0 ≤ eB ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 (a, b) displays the contour plots of the F3 and F8 distributions with zero chemical potential in the T − eB plane, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 (c, d) shows similar plots of the F3 and F8 distributions but with non-zero chemical potential µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The (2 + 1)-flavor spin polarization is different from that of two flavor spin polarization because of an additional term F8 = −2Gt � ¯ψΣ3λ8ψ � associated with the λ8 flavor generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The spin condensates affect dynamical quark masses and quark dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' It is found that the nonzero values of the two spin condensates F3 and F8 exist in the restored chiral symmetry phase with high temperature and large magnetic field, but F3 and F8 are almost zero in the chiral symmetry broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' We also noticed that F8 decreases sharply with the increase of chemical potential, but F3 changes slightly with the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The dynamical quark mass (M) as a function of temperature (T) for four different magnetic fields (eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20 GeV2) by considering the different sets of AMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4(a, b) are for µ = 0 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV respectively with AMM1 set as κu = κd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='38, κs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4(c, d) is same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 (a, b) but for AMM2 set as κu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='123, κd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='555, κs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' displays the dependence of dynamical quark mass (M) on temperature (T) 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='6F (b) (a) eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='6 Ms M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='eB = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='.5 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 (Gev M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 M Mu eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 -eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 Mu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 :eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0 T (GeV) T (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='6 (d) (c) M: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 (GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 M M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 Mu 一 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 Mu eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 T (GeV) T (GeV)for four different magnetic fields (eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20 GeV2) by considering the two AMM’s sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4(a, b) are for µ = 0 GeV and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV with AMM1 set as κu = κd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='38 and κs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4(c, d) is same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4(a, b) but with AMM2 set as κu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='123, κd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='555 and κs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Contrary to the behavior of the zero AMM in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1, the mass-decreasing behavior of u and d quarks in the chiral restoration is not a smooth slope but a sudden drop, which indicates the existence of a first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' However, the smooth slope of the dynamical mass for the crossover can be still present in the weak field eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 GeV2 for the non-zero AMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The mass-decreasing behavior of s quark in the chiral restoration is still a smooth slope, which suggests a chiral crossover for s quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4, it is found that the dynamical quark mass of u and d quarks have the characteristics of inverse magnetic catalysis in the chiral restoration phase (T ≥ TC) by using the AMM sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The critical temperature of u and d quarks as a function of the magnetic field at µ = 0 (a) and = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 5, the critical temperature is shown as a function of the magnetic field with the chemical potentials µ = 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' It is found that the critical temperature decreases with the magnetic field for the AMM1 and AMM2 sets, which indicates an inverse magnetic catalysis which qualitatively agrees with lattice result in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' On the contrary, with the TSP, TC enhances as a function of the magnetic field, which is the extension of the magnetic catalysis effect from vacuum to finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The different effects of AMM and TSP on chiral condensate can be easily understood from the dispersion relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (17), the AMM reduces the LLL energy and the TSP lifts up the LLL energy, which causes the different effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 13 (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='19 no AMM&TSP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='11 (GeV) no AMM&TSP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='TSP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='TSP AMM1 AMM1 AMM2 AMM2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 eB (GeV2) eB (GeV2)FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 5, but for the s-quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The critical temperature of chiral phase transition of s quark as a function of eB is man- ifested in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Compared with light quarks of u and d, the phase transition temperature TC of s quark with TSP increases significantly with the increase of magnetic field, which corresponds to the characteristics of magnetic catalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The introduction of AMM sets corresponds to inverse magnetic catalytic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Figure 7 displays the dependencies of the entropy density of u , d and s quarks on temperature at zero chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' It can be noted that the introduction of the AMM makes the crossover phase transition sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' It is worth noting that the AMM in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='7 corresponds to three different settings, which are AMM0, AMM1 and AMM2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' AMM0 means that the AMM is not considered, that is, all κ values in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' (17) are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' AMM1 and AMM2 sets have been mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' When eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 GeV2, the magnetic field is not big enough to excite the effect on entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' When eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 GeV2, some of the effects of the magnetic field on entropy for different AMM sets and TSP can be excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' It is found that the entropy shows a sharp change near the phase transition temperature after adding AMM sets, and this sharp change is more obvious with the magnetic field increases and chemical potential, showing a first-order phase characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The change of entropy with temperature near the phase transition temperature is relatively smooth after adding TSP, and it behaves like the crossover transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='34 F (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 no AMM&TSP no AMM&TSP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 (GeV) (GeV) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='TSP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' TSP AMM1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='26 AMM1 c C AMM2 AMM2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 eB (GeV2) eB (GeV2)FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The dependence of S/T 3 on temperature T at µ = 0GeV with different magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='7 (a) is for eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 GeV2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='7 (b) is for eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The dependence of square of sound-velocity c2 s on temperature T is manifested in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='8(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='8(b) are for zero chemical potential µ = 0 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The square of sound-velocity shows a sudden rapid rise inflection near the phase transition after adding AMM sets, and this rapid rise is more obvious with the magnetic field increases, showing a obviously first-order phase characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' On the other hands, the change of square of sound-velocity with temperature near the phase transition is relatively smooth inflection after adding TSP, showing a obviously cross-over transition characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The result obtained by using the square of sound velocity is completely consistent with the result of entropy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Compared with u and d quarks, the square of sound-velocity of s quark with temperature is relatively smooth inflection after adding TSP and AMM sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' It is proposed that s quarks have always maintained obvious cross-over characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' In the high-temperature region, the square of sound-velocity c2 s increases with temperature and obtains the saturation value 15 16 (a) 12 S 8 no AMM&TSP, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 " AMM1,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 4 AMM2,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 --TSP,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 Stefan-Boltzmann limit 0 (b) 16 12 2 S 8 no AMM&TSP,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' AMM1, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 4 AMM2,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 --TSP,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 Stefan-Boltzmann limit 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 T (GeV)c2 s = 1/3 to satisfy the relativistic requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' This suggests that the equation of state in the chiral restoration phase at high temperatures is close to the Stefan-Boltzmann limit ε = 3P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The sound-velocity square C2 s of u and d with s quarks as a function of the temperature T with different chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='8 (a, b) is for u and d quarks with zero chemical potential µ = 0, and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='8 (c, d) is for s quarks IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS In this work, we thoroughly study the effect from TSP and AMM on the vacuum, phase transition and thermal magnetized QCD in the (2 + 1)-flavor Nambu-Jona-Lasinio (NJL) model with nonzero current quark masses at finite temperature and chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' An unified physical mechanism to illustrate the novel consequences from recent lattice QCD as magnetic catalysis and inverse magnetic catalysis effect proposed in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' We focus on two topics: the AMM and TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' For these two topics, we should pay special attention to the dispersion relation, especially the lowest Landau level, which determines 16 (a) u and d quarks with u = O GeV (b) u and d quarks with u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 2s TSP, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 TSP,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 GeV2 AMM1, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 AMM1, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 AMM2, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 AMM2, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 —TSP,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 + -AMM1,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 +- AMM1,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 AMM2, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 +-- AMM2, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 0: 01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 T (GeV) T (GeV) (c) s quarks with u = O GeV (d) s quarks with u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='4 TSP,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 TSP,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 AMM1,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 AMM1,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 AMM2, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 AMM2,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='3 TSP, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 TSP,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 + AMM1,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 AMM1,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 2s 2s AMM2,eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='20GeV2 +- AMM2, eB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 +0 0+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='2 T (GeV) T (GeV)the properties of the magnetized quark matter system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The TSP lifts up the LLL en- ergy: ELLL = � p2 z + (M + F3 + F8 √ 3)2� 1 2, while the AMM effect diminishes the LLL energy: ELLL = � p2 z + (M − κ |qf| B)2�1/2 therefore, the TSP and the AMM take almost opposite effects on magnetized quark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' When the AMM and TSP contributions are not con- sidered, the corresponding phase transition temperature increases with the magnetic field, showing the characteristics of magnetic catalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' When considering only the contribution of TSP, the phase transition temperature also increases with the magnetic field, showing the characteristics of magnetic catalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' On the other hand, when AMM are introduced, the phase transition temperature decreases with the magnetic field, showing the characteristics of inverse magnetic catalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' It is found that the square of sound-velocity shows a sudden rapid rise inflection near the phase transition after adding AMM sets, and this rapid rise is more obvious with the magnetic field increases, showing a obviously first-order phase characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' On the other hands, after adding TSP, the change of square of sound-velocity with temperature near the phase transition is relatively smooth inflection, showing a obviously cross-over transition characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The result obtained by using the square of sound velocity is completely consistent with the result of entropy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The (2 + 1)-flavor spin polarization is different from that of two flavor because of an ad- ditional F8 = −2Gt � ¯ψΣ3λ8ψ � associated with the λ8 flavor generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' The spin condensates affect the dynamical quark masses, chiral phase transition,and quark dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' It is found that the nonzero values of the two spin condensates F3 and F8 exist in the restored chiral symmetry phase with high temperature and large magnetic field, but F3 and F8 are almost zero in the chiral symmetry broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by National Natural Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 11875178, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Tatsumi, AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 847, 171 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Duncan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Kharzeev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Liao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Voloshin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Wang, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 88, 1 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' [12] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Huang, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' 79, 076302 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Andersen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Naylor, and 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' D 97, 066015 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' [19] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' Kharzeev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfgPxu/content/2301.01465v1.pdf'} +page_content=' McLerran, 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--git a/9tAzT4oBgHgl3EQfSvsB/content/tmp_files/2301.01235v1.pdf.txt b/9tAzT4oBgHgl3EQfSvsB/content/tmp_files/2301.01235v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0588e905731ace3dda148e0376faa36db4fc7454 --- /dev/null +++ b/9tAzT4oBgHgl3EQfSvsB/content/tmp_files/2301.01235v1.pdf.txt @@ -0,0 +1,1528 @@ +An Empirical Investigation into the Reproduction of +Bug Reports for Android Apps +Jack Johnson∗, Junayed Mahmud†, Tyler Wendland∗, Kevin Moran†, Julia Rubin‡, Mattia Fazzini∗ +∗University of Minnesota, MN, USA; joh19267@umn.edu, wendl155@umn.edu, mfazzini@umn.edu +†George Mason University, VA, USA; jmahmud@gmu.edu, kpmoran@gmu.edu +‡University of British Columbia, BC, Canada; mjulia@ece.ubc.ca +Abstract—One of the key tasks related to ensuring mobile +app quality is the reporting, management, and resolution of +bug reports. As such, researchers have committed considerable +resources toward automating various tasks of the bug man- +agement process for mobile apps, such as reproduction and +triaging. However, the success of these automated approaches is +largely dictated by the characteristics and properties of the bug +reports they operate upon. As such, understanding mobile app +bug reports is imperative to drive the continued advancement +of report management techniques. While prior studies have +examined high-level statistics of large sets of reports, we currently +lack an in-depth investigation of how the information typically +reported in mobile app issue trackers relates to the specific details +generally required to reproduce the underlying failures. +In this paper, we perform an in-depth analysis of 180 re- +producible bug reports systematically mined from Android apps +on GitHub and investigate how the information contained in +the reports relates to the task of reproducing the described +bugs. In our analysis, we focus on three pieces of information: +the environment needed to reproduce the bug report, the steps +to reproduce (S2Rs), and the observed behavior. Focusing on +this information, we characterize failure types, identify the +modality used to report the information, and characterize the +quality of the information within the reports. We find that bugs +are reported in a multi-modal fashion, the environment is not +always provided, and S2Rs often contain missing or non-specific +enough information. These findings carry with them important +implications on automated bug reproduction techniques as well as +automated bug report management approaches more generally. +I. INTRODUCTION +The importance of the quality of mobile applications (collo- +quially referred to as apps) has grown in recent years as smart- +phones and tablets have become deeply integrated into users’ +daily lives. Once an application has been released to users, its +quality is largely ensured by continuing maintenance activities, +which have been shown to consume considerable amounts of +engineering effort [1]. These important maintenance activities +are typically centered around bug report management and +include activities related to understanding, reproducing, and +resolving bug reports. +A number of unique development constraints related to +mobile apps, such as pressure for frequent releases [2], [3], +the need to cope with constantly evolving platform APIs [4], +[5], a large volume of user feedback [6], [7], [8], [9], +[10], and testing challenges [11] complicate the bug report +management process. Software engineering researchers have +recognized these domain-specific challenges and have worked +toward providing automated solutions across several bug report +management activities for mobile apps, including bug report +quality assessment [12], reproduction [13], [14], triaging [15], +and bug localization [16], [17]. +One common thread among these various automated solu- +tions is that they operate directly upon the information con- +tained within bug reports and, as such, are directly affected by +the characteristics and quality of various report components, +such as environmental information (e.g., device, software ver- +sion), reproduction steps (S2Rs), and observed behavior (OB). +Thus, researchers and practitioners require a solid empirical +foundation that delineates common characteristics of mobile +app bug reports to build effective automated techniques. +In prior work, researchers have examined high-level statis- +tics (e.g., number and type of report, fix rates, fix time) +of large sets of bug reports. For example, Battacharya et +al. [18] performed an empirical study on bugs submitted to +the Android platform on 24 widely-used open source apps. +Others have compared high-level bug characteristics between +mobile apps and desktop apps [19]. However, to the best +of our knowledge, no study has yet provided an in-depth +characterization of how the information contained in mobile +bug reports might impact the task of bug reproduction. One +likely reason that past studies have not examined this relation +is that as it requires manually reproducing real bug reports, +which is a time-consuming and difficult task. Despite the +difficulty of this analysis, understanding this information is +critical as both developers and automated bug analysis tech- +niques may need to (i) understand the type of reported failure, +(ii) understand multiple modalities of information, such as +text, images, or screen-recordings, and (iii) identify or infer +information that is either vague or missing from the reports. +In short, empirically analyzing both the characteristics and +quality of the information reported in mobile app bugs is +critical for both the practical and scientific advancement bug +report management for mobile apps. +In this paper, we conduct and in-depth characterization of +reproducible bug reports for Android apps. To this end, we +significantly extend ANDROR2 [20] – a dataset of reproducible +bug reports for Android apps which contains bugs representing +a range of failure types. We augmented the dataset with addi- +tional, manually verified and fully reproduced bug reports from +open source Android apps hosted on GitHub [21] and available +on the Google Play store [22], obtaining a dataset of 180 bug +reports. In this work, we focus on bug reports for Android +arXiv:2301.01235v1 [cs.SE] 3 Jan 2023 + +apps as Android is the most widely used operating system for +mobile apps [23]. To the best of our knowledge, ours is the +largest dataset of (i) fully reproduced bug reports for Android +apps, which (ii) contains both user-submitted and developer- +submitted reports, and (iii) in contrast to related work, focuses +on different types of failures beyond app crashes. Given this +dataset, we focused our in-depth analysis on three sources +of information: the description of the environment needed +to reproduce the bug report, the steps to reproduce, and the +observed behavior. +Leveraging the fact that our studied reports are considered +fully reproducible, we perform an in-depth analysis of both the +report characteristics—including the failure types and modal- +ities of reported information—and the quality of reported +information. In relation to the quality of reported information, +we focus on three aspects: the types and prevalence of missing +information, whether report discussion threads contain helpful +information for reproducing the reports, and the specificity +of reported information (which investigates whether reported +information can be directly used for reproducing the reports). +Although these aspects are only some of ones that describe the +quality of reported information, we believe that the analysis +of these aspects provides useful insights into the reproduction +of bug reports and hence focus on them. +Our analysis shows that (i) reported failures can be grouped +into four types, three of which are not yet considered by +existing automated reproduction techniques, (ii) different in- +formation modalities are used to report the details related to +the environment, steps to reproduce, and observed behavior, +(iii) a large number of reports (74%) have at least one step +to reproduce that requires multiple operations in the app +indicating that the information provided for the step is not +always specific enough, (iv) the great majority of reports +(92%) have at least one missing reproduction step, illustrating +that the operations required to reproduce the reports must +often be inferred, and (v) bug report discussions can, in some +cases (19%), provide additional information useful for the +reproduction of the reports. Finally, we discuss implications +of our findings, which can help guide future research on +automated reproduction of bug reports and, more generally, +bug report management activities. +In summary, the main contributions of this paper are: +• A large set of 180 manually mined and reproduced +bug reports for Android apps that contains user- and +developer-submitted bug reports of multiple failure types. +• A study that examines bug characteristics and information +quality in reproducible mobile app bug reports. This +advances upon prior studies which do not manually verify +and collect reproducible bug reports. +• A discussion on the implications of our findings, which +illustrates the need for future research on non-crashing +oracles, multi-modal understanding of report information, +mocking environments, and missing and non-specific +reproduction steps. +• A replication package [24] that contains our dataset of +bug reports, data analysis reports, and scripts to perform +Bug Report +Title: +Bug: Long pressing the amount input brings up QWERTY keyboard +Content: +Software specifications: +• GnuCash Android version: 2.2.0 +• System Android version: 6.0 +• Device type: Motorola Moto G (2nd Generation) +Steps to reproduce the behaviour: +1. Navigate to Transactions screen +2. Tap the Add button +3. Enter Description (optional) +4. Focus the Amount input +5. Long press to bring up the context menu +Expected behaviour: +See the context menu +Actual behaviour: +Fig. 1: Bug report for the GNUCASH app. +the study analyses, which can facilitate future replications +and extensions of this work. +II. BACKGROUND AND TERMINOLOGY +Given a bug report that describes a failure in an app, we +use the term reporter to identify the person submitting the +bug report. A reporter can be either a user or a developer. In +this study, we consider a person who never contributed to the +source code of an app to be a user and all other reporters to +be developers. +We conceptually group the information contained in a bug +report into multiple parts, each of which detail a particular +aspect of the report. The parts and aspects of interest in this +study are the ones providing details on how to reproduce the +failure described in a report. These aspects are: the environ- +ment, the steps to reproduce (S2Rs), and the observed behavior +(OB). The environment includes information on the software +and hardware necessary to reproduce the failure described in +a report. This part can contain information such as the app +version, the operating system (OS) version, and the device +where the failure occurred. The S2Rs provide details on the +operations that should be performed on a device in order +to reproduce the failure. We use the terms GUI action (or +simply action) and GUI interaction (or simply interaction) +interchangeably to indicate the operations performed on the +GUI of a device. An S2R (which are the unit of information +composing the S2Rs) can be mapped to one or more GUI +actions. The OB describes the failure and can be used to check +that the failure was successfully reproduced. In practice, the +information from these conceptual parts can be interleaved +across the paragraphs and sections of a bug report. Bug +reports can also have a discussion thread. A discussion thread +contains discussion messages and these messages can provide +2 + +12:22 +X +New transaction +SAVE +Heating/Utilities +7 +8 +9 +X +C +4 +5 +6 +* +1 +2 +3 ++ +0 +2 +3 +8 +0 +9 +W +e +V +u +a +S +d +g +b +X +X +n +m +?123 +English +V +口additional information on the environment, the S2Rs, and the +OB associated with the report. +Figure 1 provides an example of a user-submitted bug re- +port [25]. This bug report is taken from the report management +system of GNUCASH, an app for finance tracking, and is +slightly modified for presentation purposes. The bug report +contains information related to the environment, the S2Rs, and +the OB, which are located in the Software specifications, Steps +to reproduce the behaviour, and Actual behaviour sections of +the report, respectively. +To exercise the bug, the user navigated to the transactions +screen, started adding a new transaction, and long-clicked on +the GUI element representing the amount of the transaction. +The failure manifests as a wrong screen being displayed to the +user: screen with a keyboard view instead of the context menu. +The OB describing the failure is reported using text (in the +title) and using an image (in the Actual behaviour section). We +refer to the way in which a piece of information is reported as +the reporting modality (or modality in short) and reporters can +provide the same information multiple times using different +modalities. Because the user did not reach the desired screen, +we identify this failure as a navigation failure. We use the +terms failure type and failure category interchangeably to refer +to the categorization of the failure. +The report has five S2Rs (numbered items under the Steps +to reproduce the behaviour section) and 13 GUI actions are +necessary to reproduce the failure. An example of GUI action +is performing a click on the add button in the transaction +screen of the app as indicated by 2. Tap the Add button. An +S2R can map to one or more GUI actions. In this example, +the first S2R (1. Navigate to Transactions screen) maps to +three GUI actions. We refer to S2Rs that map to multiple +GUI actions as non-specific S2Rs. Of the remaining four S2Rs, +three map to one GUI action and one S2R is optional (3. Enter +Description (optional).) This optional S2R is not included in +13 GUI actions necessary to reproduce the failure. Seven (13- +3-3) of the GUI actions in this example are not described by +any of the S2Rs. We refer to such GUI actions as unmapped +GUI actions and say that they correspond to missing S2Rs. +We refer to the remaining actions as mapped GUI actions. +If an unmapped GUI action occurs before the first mapped +GUI action, we call the missing S2R that corresponds to +the unmapped action a missing context S2R, indicating that +some contextual information is missing from the bug report. +Otherwise, if a missing S2R is associated with a GUI action +occurring after the first mapped GUI action, we refer to the +S2R as a missing inline S2R. +III. METHODOLOGY +To characterize reproducible bug reports, inform research on +automated bug reproduction, and, more generally, provide in- +sights for research on bug report management, we formulated +and answered the following research questions (RQs): +• RQ1: What are the failure types associated with +reproducible bug reports? In this RQ, we analyzed +and categorized failures associated with reproducible bug +reports. With the findings from this RQ we aim to inform +research on automatic failure recognition. +• RQ2: What information modalities are used to report +the information contained in reproducible bug re- +ports? This RQ categorizes the modalities used to report +environment, S2Rs, and OB information. The findings +from this RQ aim to inform research in bug triaging, +report reproduction, and report quality assessment. +• RQ3: Do reproducible bug reports have missing in- +formation? We answer this question by analyzing the +information contained in reproducible bug reports w.r.t. +operations required to reproduce the failures described in +the reports. This RQ aims to direct efforts on research +for identifying and inferring missing information in bug +reports, necessary for bug report reproduction. +• RQ4: Do discussion threads of reproducible bug re- +ports contain helpful information for reproducing the +reports? In this RQ, we analyzed the information gain +obtained by interpreting the bug report discussions. This +RQ aims to evaluate the need for approaches that combine +content from bug reports and their discussions. +• RQ5: How specific is the information reported in +reproducible bug reports? In this RQ, we investigated +whether the information contained in reproducible bug +reports can be directly mapped onto the operations need +to reproduce the reports. This RQ aims to provide insights +on how to leverage the information in bug reports for +reproducing the failures. +Figure 2 provides a high-level outline of the methodology +we used to answer the RQs. In a nutshell, we first assembled +a dataset of reproducible bug reports and then analyzed the +characteristics of the bug reports through qualitative and +quantitative analyses. We describe these steps in detail next. +A. Dataset Creation +The Dataset Creation component of Figure 2 provides an +overview of our data collection workflow, which consisted of +two phases: bug reports filtering and failure reproduction. +1) Bug Reports Filtering: The objective of this phase was +to identify a set of bug reports that we could try to reproduce +and ultimately include in our dataset. In this study, we are +interested in both user-submitted and developer-submitted bug +reports that are reproducible and describe different types of +failures. To the best of our knowledge, ANDROR2 [20] is +the largest dataset of reproducible bug reports for Android +apps that does not exclusively focus on crashes. This dataset +contains 90 user-submitted bug reports, which are associated +with apps available on the Google Play store [22] and hosted +on GitHub [21]. The 90 bug reports are GitHub issues [26] +and are associated with reproduction scripts created by the +ANDROR2’s authors. This set of 90 bug reports was extracted +from a larger set of 6,365 issues that was systematically +mined from GitHub. The set of 6,365 issues contains issues +that: (i) are part of repositories that use Java, (ii) have +the label “bug”, (iii) are in repositories that contain an +AndroidManifest.xml file (as Android apps require this +3 + +Bug Reports +Filtering +AndroR2 +Filtered +Bug Reports +Failure +Reproduction +Reproduced +Bug Reports +RQ1: Failure Type +RQ2: Reporting Modality +RQ3: Missing Information +Dataset Creation +Bug Reports Analysis +RQ4: Discussion Information +RQ5: Information Specificity +Reproduction +Scripts +Bug Reports +Preparation +Annotated +Bug Reports +E,OB,S2Rs +Fig. 2: Overview on the methodology used in the study. +file to properly compile [27]), (iv) contain the word “step” +in them, and (v) are associated with apps also available on the +Google Play store. +Because we are also interested in developer-submitted bug +reports, we started from the set of 6,365 GitHub issues pro- +vided by ANDROR2 and identified 90 reproducible, developer- +submitted bug reports (to match the number of already avail- +able user-submitted bug reports). To identify the 90 developer- +submitted bug reports, we used a methodology similar to that +of ANDROR2. Specifically, we first refined the set of 6,365 +issues to only contain those created by GitHub users that +had contributed to the repositories associated with the issues, +resulting in 2,523 issues. Second, we selected issues that were +closed at the time the issues were mined (November 2020) so +that we could more easily identify whether the issues were also +originally reproduced by the developers. This filtering resulted +in 2,045 reports. Third, after analyzing the set of issues, we +found that some repositories had a much larger number of +issues compared to others. To avoid overfitting the bug report +dataset to a specific app, we considered at most ten issues +per repository. When a repository had more than ten issues, +we randomly selected ten from this set resulting in 645 bug +reports for 164 apps. +2) Failure Reproduction Phase: In the second phase of our +dataset creation process, we randomly selected bug reports +from the set of 645 developer-submitted bug reports until we +reproduced 90 of them. In this process, we disregarded trivially +reproducible bug reports, i.e., those we could reproduce by +simply opening the app. +Two authors tried to reproduce the failures described in the +bug reports. To reproduce a failure, the authors followed the +S2Rs contained in the bug report by mapping the steps to GUI +actions on the screen of the device running the app associated +with the report. If a report had missing S2Rs, the authors +manually explored the functionality of the app to identify the +minimal sequence of GUI actions that would account for those +missing steps, using a trial-and-error approach. When a bug +report could be successfully reproduced by one of the two +authors, the other author also tried to reproduced the same +report to ensure that the reproduced failure was the same as +the one described in the report. For all 90 bug reports, the +authors also encoded the GUI actions in reproduction scripts +using the UIAutomator framework [28]. +To validate whether user-submitted bug reports were still +reproducible, we ran the scripts associated with these reports +in the ANDROR2 dataset. Four reports were not reproducible +as the servers associated with the apps were no longer running. +To replace these bug reports, we identified and reproduced four +additional user-submitted reports from the set of 6,365 GitHub +issues provided by ANDROR2. At the end of this process, +we obtained a set of 90 user-submitted and 90 developer- +submitted reproducible bug reports, which we considered for +the rest of the study. +B. Bug Reports Analysis +In this section, we present the analyses we performed to +characterize aspects related to the reproducibility of Android +bug reports. The Bug Reports Analysis Creation part of +Figure 2 provides a summary of the analyses we performed. +The analyses were driven by two of the paper’s authors and +were performed one at a time to reduce cognitive load. +1) Bug Reports Preparation: Before performing the analy- +ses associated with the RQs, we annotated the information +contained in the bug reports and their discussion threads, +to identify the portions of each report that provide infor- +mation about the environment, S2Rs, and OB. This step +was performed by the two authors together and in multiple +sessions; the authors associated each sentence in the report’s +textual description, as well as each link, image, recording, +and execution logs, with it designated purpose: to describe +environment, S2Rs, and OB. Some elements received multiple +annotations, e.g., a sentence can provide both S2Rs and OB. +2) Analysis for RQ1 (What are the failure types associated +with reproducible bug reports?): To answer RQ1, we per- +formed a qualitative analysis that combines inductive and axial +coding [29], [30]. Inductive coding is a systematic approach +for categorizing data by manually coding (i.e., labeling) the +data. Axial coding relates codes to one another and finds +higher-level codes that represent abstractions of the original +codes. In our analysis, a code is a label that categorizes the +type of a failure and we assigned the code to the bug report +describing the failure. +The analysis was performed by two raters, who analyzed +the description of the failure in the bug report and used the +reproduction scripts to observe how the failure manifested. +The analysis was divided into two parts. In the first part, the +two raters analyzed a sample of the bug reports to define +the analysis codebook – a document detailing the rules for +assigning a specific code to a failure. For each code, the set +4 + +Dμ+ +μ-)× 1010M1 +4.0 +109 +x( +3.5 +B +3.0 +B +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +μ)x 1010M16 +4.0 +109 +x( +3.5 + SM: IVebl=3.921 10-2 +T + SM: IVebl=4.26 10-2 +B +3.0 +B +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +B(Ba→μ+ +μ-)x 1010REFERENCES +14 +Figure 8: Contour Plots of ¯B(Bs → µ+µ−) (left column) and B(Bd → µ+µ−) (right column) versus +|Vcb| and |Vub| in M1 (upper plots) and in M16 (lower plots). +[3] P. H. Frampton, Chiral dilepton model and the flavor question, Phys. Rev. Lett. 69 +(1992) 2889–2891. +[4] V. Pleitez, Challenges for the 3-3-1 models, in 5th Colombian Meeting on High Energy +Physics, 12, 2021. arXiv:2112.10888. +[5] A. J. Buras and E. Venturini, The exclusive vision of rare K and B decays and of the +quark mixing in the standard model, Eur. Phys. J. 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The red region corresponds to |Vcb| ∈ [0.0386, 0.0398] +while the cyan region corresponds to |Vcb| ∈ [0.0422, 0.043]. The SM results in correspondence of +two values of |Vcb| are displayed, as specified in the legenda. The light gray region corresponds to +the experimental range for B(K+ → π+ν¯ν) reported in Table 1. + +B(Bs →μt μ)x 10°, SM +0.00380 +3.7 +3.6 +0.00375 +3.5 +3.4 +0.00370 +3.3 +3.2 +3.1 +0.00365 +0.039 +0.040 +0.041 +0.042 +IVcblB(Bd → μ+ μ-)x 10l0 , SM +0.00380 +1.02 +1.00 +0.98 +0.00375 +0.96 +0.94 +0.92 +0.00370 +0.90 +0.88 +0.86 +0.84 +0.00365 +0.039 +0.040 +0.041 +0.042 +IVcblM1 +14 +12 +10 +8 +↑ +B(K+ +9 +1 +2 +3 +4 +5 +B(KL →°) × 1011M16 +14 +12 +x(4 +10 +SM: IVebl=3.921 10-2 +↑ +8 +SM: IVebl=4.26 10-2 +B(K+ +6 +2 +3 +4 +B(K →°) × 10l1M3 +14 +12 +10 +8 +B(K+ +9 +1 +2 +3 +4 +5 +B(KL →°) × 1011M13 +14 +12 +×(4 +10 +SM: IVebl=3.921 10-2 +↑ +8 + SM: IVcbl=4.26 10-2 +B(K+ +6 +2 +3 +4 +B(KL →°) × 1011REFERENCES +17 +Figure 12: Contour Plots of B(K+ → π+ν¯ν) (left column) and B(KL → π0ν¯ν) (right column) versus +|Vcb| and |Vub| in M1 (upper plots) and in M16 (lower plots). +[12] A. J. Buras and E. 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The red region corresponds to |Vcb| ∈ [0.0386, 0.0398] +while the cyan region corresponds to |Vcb| ∈ [0.0422, 0.043]. The SM results in correspondence of two +values of |Vcb| are displayed, as specified in the legenda. 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Amhis et al., Averages +of b-hadron, c-hadron, and τ-lepton properties as of summer 2016, arXiv:1612.07233. + diff --git a/AtE0T4oBgHgl3EQfxwIb/content/tmp_files/load_file.txt b/AtE0T4oBgHgl3EQfxwIb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4dfd4524bed2279eed3551cf52f2da8e9f2a0859 --- /dev/null +++ b/AtE0T4oBgHgl3EQfxwIb/content/tmp_files/load_file.txt @@ -0,0 +1,1325 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf,len=1324 +page_content='AJB-23-1 BARI-TH/23-743 331 Model Predictions for Rare B and K Decays, and ∆F = 2 Processes: an Update Andrzej J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Burasa,b and Fulvia De Fazioc aTUM Institute for Advanced Study, Lichtenbergstr.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Germany cIstituto Nazionale di Fisica Nucleare,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Sezione di Bari,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Via Orabona 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' I-70126 Bari,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Italy Abstract Motivated by the improved results from the HPQCD lattice collaboration on the hadronic matrix elements entering ∆Ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='d in B0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='d − ¯B0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='d mixings and the increase of the ex- perimental branching ratio for Bs → µ+µ−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' we update our 2016 analysis of various flavour observables in four 331 models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' M13 and M16 based on the gauge group SU(3)C ×SU(3)L ×U(1)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' These four models, which are distinguished by the quantum numbers, are selected among 24 331 models through their consistency with the elec- troweak precision tests and simultaneously by the relation CNP 9 = −b CNP 10 with b ≥ 2, which after new result on Bs → µ+µ− from CMS is favoured over the popular relation CNP 9 = −CNP 10 predicted by several leptoquark models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In this context we investigate in particular the dependence of various observables on |Vcb|, varying it in the broad range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0386, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043], that encompasses both its inclusive and exclusive determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Im- posing the experimental constraints from εK, ∆Ms, ∆Md and the mixing induced CP asymmetries SψKS and SψKS, we investigate for which values of |Vcb| the four models can be made compatible with these data and what is the impact on B and K branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In particular we analyse NP contributions to the Wilson coefficients C9 and C10 and the decays Bs,d → µ+µ−, K+ → π+ν¯ν and KL → π0ν¯ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' This allows us to illustrate how the value of |Vcb| determined together with other parameters of these models is infected by NP contributions and compare it with the one obtained recently under the assumption of the absence of NP in εK, ∆Ms, ∆Md and SψKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='02649v1 [hep-ph] 6 Jan 2023 1 Introduction 1 Contents 1 Introduction 1 2 Flavour Structure of 331 Models 4 3 Selecting the 331 Models 5 4 Numerical Analysis 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 Determining the parameter space .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 CNP 9 and CNP 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='3 ¯B(Bs → µ+µ−) and B(Bd → µ+µ−) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='4 Rare Kaon decays .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 9 5 Summary 10 1 Introduction The Standard Model (SM) describes globally the existing data on quark-flavour violating processes rather well [1] but with the reduction of experimental errors and increased precision in non-perturbative and perturbative QCD and electroweak calculations a number of tensions at the level of 2 − 5 σ seem to emerge in various seemingly unrelated observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' While some of these tensions could turn out to be the result of statistical fluctuations, underestimate of systematical and theoretical errors, it is not excluded that eventually they all signal the presence of some kind of new physics (NP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Therefore, it is interesting to investigate what this NP could be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In the present paper we will address some of these tensions in four particular 331 models based on the gauge group SU(3)C × SU(3)L × U(1)X [2, 3] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' As these models have much smaller number of new parameters than supersymmetric models, Randall-Sundrum scenar- ios and Littlest Higgs models, it is not evident that they can remove all present tensions simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Our paper has been motivated by the following recent facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' As demonstrated in [5] most recent lattice QCD results from HPQCD collaboration [6], based on 2 + 1 + 1 simulations, imply simultaneous agreement of |εK|, ∆Ms, ∆Md, SψKS Sψφ (1) within the SM with the data for rather precise values of |Vcb|, |Vub| and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' This should be contrasted with the situation at the time of our previous analysis 2016 [7], when significant tensions between εK and ∆Ms,d within the SM have been found [8] and the room for NP in the quark mixing sector was much larger than it is now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 1A recent critical reanalysis of 331 models and a collection of references can be found in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 1 Introduction 2 The most recent data on Bs → µ+µ− from CMS imply that in the case of the dominance of left-handed quark currents, as is the case of the 331 models, CNP 9 = −b CNP 10 , b ≥ 2, (2) where CNP 9 , CNP 10 represent the shifts in the Wilson coefficients C9, C10 of the b → sℓ+ℓ− effective Hamiltonian in the presence of NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The relation (2) is in contrast to the pre- viously favoured case b = 1 found in several leptoquark models, in particular in the U1 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Recent messages from the LHCb [9, 10], that the lepton flavour universality violation (LFUV) in b → sℓ+ℓ−, which for many years dominated the B-physics anomalies, prac- tically disappeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' This is good news for 331 models for which LFUV anomalies were problematic, although these models could provide some shifts in the Wilson coefficients C9 and C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Such shifts, in particular in C9, are still required to describe suppressed branching ratios in b → sµ+µ− transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The most recent value for γ obtained by the LHCb collaboration from tree-level decays that reads [11] γ = (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='8+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='7)◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' (3) It is significantly more precise than the LHCb values of γ in 2016 that could be as large as 75◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The question then arises how 331 models face this new situation relative to the 2016 input and what are the implications for many flavour observables, in particular for the decays Bd → K(K∗)µ+µ−, B+ → K+µ+µ− and Bs → φµ+µ− related to the B physics anomalies that imply the need for significant NP contributions to the Wilson coefficient C9 and smaller to C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' But it is also of interest to see what are the implications for rare decays Bs,d → µ+µ−, K+ → π+ν¯ν and KL → π0ν¯ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' It is known from many analyses, and stressed recently in particular in [5, 12] that the tensions between inclusive and exclusive determinations of |Vcb| and |Vub| preclude precise predictions for rare decay observables in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' However, eliminating these parameters with the help of εK, ∆Ms,d and SψKS and setting the latter observables to their experimental values allowed to obtain SM predictions for many flavour observables that are most precise to date [5,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The motivation for this strategy has been strengthened recently by one of us [13] as the one which could minimize the impact of NP on the determination of the CKM parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Indeed, as demonstrated in [5], presently no NP is required to describe precise experimental data on ∆F = 2 observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' This allows in turn to determine the CKM parameters on the basis of ∆F = 2 observables alone without being involved in the issue of |Vcb| and |Vub| tensions and minimizing possible impact of NP on their values that otherwise would infect SM predictions for rare decay branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The resulting values of the CKM parameters read [5] |Vcb| = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='6(4) × 10−3, |Vub| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='72(11) × 10−3, γ = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='6(16)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' (4) While in this manner one can obtain rather precise SM predictions for numerous branching ratios [5, 12, 13], the absence of NP in the ∆F = 2 observables, if confirmed with higher 1 Introduction 3 Decay EXCLUSIVE HYBRID DATA B(K+ → π+ν¯ν) × 1011 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='88(38) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='44(41) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='9(38) [15] B(KL → π0ν¯ν) × 1011 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='37(15) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='74(14) < 300 [16] B(KS → µ+µ−) × 1013 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='49(10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='72(8) 104 [17] B(Bs → µ+µ−) × 109 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='18(12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='67(12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='45(29)[18–21] B(Bd → µ+µ−) × 1010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='864(34) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='999(34) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='05 [18] |εK| × 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='78(11) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='14(12) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='228(11) [22] SψKS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='731(24) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='688(22) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='699(17) [22] ∆Ms ps−1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='02(87) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='35(94) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='749(20) [22] ∆Md ps−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='434(28) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='502(31) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5065(19) [22] Table 1: Predictions (second column) for selected observables within the SM obtained in [5] using the EXCLUSIVE strategy for |Vcb| and |Vub| and γ = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='4◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In the third column we show the results for the HYBRID choice of |Vcb| and |Vub| as given in (6) and in the fourth the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' precision, would be a nightmare scenario for many NP models that attempt to explain the B physics anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' While the ones related to lepton flavour universality violation have been dwarfed recently through new LHCb data [9,10], sizable anomalies remained in several branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In particular using the strategy of [5,12] large anomalies in the low q2 bin in B+ → K+µ+µ− (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1σ) and Bs → φµ+µ− (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='8σ) have been found [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Explaining such anomalies without practically no NP contributions to ∆F = 2 processes is in principle possible but would require significant tuning of NP parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Now, the value of γ in (4) agrees very well with the most recent value from LHCb in (3) and experimental value of β from SψKS is already used in obtaining the CKM parameters in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' It is evident then that the most efficient and transparent strategy to allow NP to enter the ∆F = 2 sector is to modify the value of |Vcb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In this context in [5], two scenarios for the parameters |Vcb| and |Vub| have been analysed within the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The EXCLUSIVE one based on determinations of these parameters in exclusive decays |Vcb| = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='21(62) × 10−3, |Vub| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='61(13) × 10−3, (EXCLUSIVE), (5) and the HYBRID scenario in which the value for |Vcb| is the inclusive one from [14] and the exclusive one for |Vub| as above: |Vcb| = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='16(50) × 10−3, |Vub| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='61(13) × 10−3, (HYBRID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' (6) In Table 1 we show selected results obtained in [5] in these two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The results obtained in the HYBRID scenario do not differ by much from those obtained using the CKM 2 Flavour Structure of 331 Models 4 parameters in (4) [5, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' With exclusive values of |Vcb| that are much lower than given in (4), anomalies in ∆Ms (3σ), ∆Md (4σ) and εK (5σ) are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' But in [5] no analysis of a NP scenario has been presented which would explain these anomalies and whether a model explaining them would also be able to explain anomalies in semi-leptonic B decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In the present paper we investigate whether the 331 models could provide some insight in these issues and what would be the implications for rare branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' As a byproduct our analysis illustrates in simple settings how the determination of |Vcb| in a global fit that includes observables exposing anomalies can be infected by NP contributions [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' It is a concrete illustration of the points made in section 2 of the latter paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In Section 2 we recall briefly the flavour structure of the 331 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In Section 3 we select four 331 models that perform best on the basis of electroweak precision tests and the present experimental values of the ratio CNP 9 /CNP 10 in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In fact these are the only models among the 24 ones considered in [23], that can successfuly face the new relation (2) when other contraints like electroweak precision tests are taken into account [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In Section 4 we present numerical analysis of these models addressing the issues mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' We conclude in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 2 Flavour Structure of 331 Models Let us recall that in the 331 models new flavour-violating effects are governed by tree-level Z′ exchanges with a subdominant but non-negligible role played by tree-level Z exchanges generated through Z − Z′ mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' All the formulae for flavour observables in these models can be found in [23–26] and will not be repeated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In particular the collection of formulae for Z′ couplings to quarks and leptons are given in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' New sources of flavour and CP violation in 331 models are parametrized by new mixing parameters and phases ˜s13, ˜s23, δ1, δ2 (7) with ˜s13 and ˜s23 positive definite and smaller than unity and 0 ≤ δ1,2 ≤ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' They can be constrained by flavour observables as demonstrated in detail in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The non-diagonal Z′ couplings relevant for K, Bd and Bs meson systems can be then parametrized respectively within an excellent approximation through v∗ 32v31 = ˜s13˜s23ei(δ2−δ1), v∗ 33v31 = −˜s13e−iδ1, v∗ 33v32 = −˜s23e−iδ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' (8) ˜s13 and δ1 can be determined from ∆Md and CP-asymmetry SψKS while ˜s23 and δ2 from ∆Ms and CP-asymmetry Sψφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Then the parameters in the K system are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' It is a remarkable feature of 331 models that also FCNC processes in the charm sector can be described without introducing no new free parameters beyond those already present in the beauty and kaon meson systems [27,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' These correlations constitute important tests of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The remaining two parameters, except for MZ′ mass, are β and tan ¯β defined through2 Q = T3 + Y 2 = T3 + βT8 + X, tan ¯β = vρ vη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' (9) 2The parameter β should not be confused with the angle β in the unitarity triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 3 Selecting the 331 Models 5 MI scen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' β tan ¯β MI scen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' β tan ¯β MI scen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' β tan ¯β M1 F1 −2/ √ 3 1 M9 F2 −2/ √ 3 1 M17 F1 −2/ √ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 M2 F1 −2/ √ 3 5 M10 F2 −2/ √ 3 5 M18 F2 −2/ √ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 M3 F1 −1/ √ 3 1 M11 F2 −1/ √ 3 1 M19 F1 −1/ √ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 M4 F1 −1/ √ 3 5 M12 F2 −1/ √ 3 5 M20 F2 −1/ √ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 M5 F1 1/ √ 3 1 M13 F2 1/ √ 3 1 M21 F1 1/ √ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 M6 F1 1/ √ 3 5 M14 F2 1/ √ 3 5 M22 F2 1/ √ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 M7 F1 2/ √ 3 1 M15 F2 2/ √ 3 1 M23 F1 2/ √ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 M8 F1 2/ √ 3 5 M16 F2 2/ √ 3 5 M24 F2 2/ √ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 Table 2: Definition of the various 331 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Here T3,8 and X are the diagonal generators of SU(3)L and U(1)X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Y represents U(1)Y and vi are the vacuum expectation values of scalar triplets responsible for the generation of down- and up-quark masses in these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Different 331 models can also be distinguished by the way quarks transform under SU(3)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In [23] two classes of such models have been analyzed to be denoted by F1 and F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' F1 stands for the case in which the first two generations of quarks belong to triplets of SU(3)L, while the third generation of quarks to antitriplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' F2 stands for the case in which the first two generations of quarks belong to antitriplets of SU(3)L, while the third generation of quarks to triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' A detailed analysis of 24 331 models corresponding to different values of β and tan ¯β for the representations F1 and F2 has been presented in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' They are collected in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' With the values of β and tan ¯β being fixed, flavour phenomenology depends only on the parameters in (7), MZ′ and the CKM parameters which distinguish EXCLUSIVE and HYBRID scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 3 Selecting the 331 Models A detailed analysis of electroweak precision tests in the 24 models in Table 2 has been per- formed in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Interested readers are asked to look at Section 5 of that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Here we just summarize the main outcome of that study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Requiring that the 24 models in question perform well in these tests and are simultaneously consistent with the ratio C9/C10 in (2) selects, as shown in Table 3, the following models M1, M3, M13, M16, (favoured).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' (10) Note that the Z − Z′ mixing plays in some cases an important role and that the two favoured models M8 and M9 analysed by us in [7] are ruled out by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 4 Numerical Analysis 6 MI Full no Mixing MI Full no Mixing MI Full no Mixing M1 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='25 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='87 M9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='60 M17 −175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='6 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='87 M2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='68 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='87 M10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='60 M18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='60 M3 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='07 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='98 M11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='004 M19 −63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='48 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='98 M4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='09 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='98 M12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='004 M20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='004 M5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='004 M13 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='47 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='98 M21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='004 M6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='004 M14 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='56 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='98 M22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='25 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='98 M7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='60 M15 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='3 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='87 M23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='60 M8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='60 M16 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='59 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='87 M24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='44 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='87 Table 3: CNP 9 /CNP 10 in various 331 models with and without Z − Z′ mixing for MZ′ = 3 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' mBs = 5366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='8(2) MeV [22] mBd = 5279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='58(17) MeV [22] ∆Ms = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='749(20) ps−1 [22] ∆Md = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5065(19) ps−1 [22] ∆MK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='005292(9) ps−1 [22] mK0 = 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='61(1) MeV [22] SψKS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='699(17) [22] FK = 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='7(3) MeV [29] |Vus| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2253(8) [22] |ϵK| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='228(11) · 10−3 [22] FBs = 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='3(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='3) MeV [30] FBd = 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='3) MeV [30] FBs � ˆBs = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='7) MeV [6] FBd � ˆBd = 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='6(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5) MeV[6] ˆBs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='232(53) [6] ˆBd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='222(61) [6] mt(mt) = 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='83(67) GeV[31] mc(mc) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='279(13) GeV Stt(xt) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='303 Sut(xc, xt) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='983 × 10−3 ηtt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='55(2) [32] ηut = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='402(5) [32] κε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='94(2) [33] ηB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='55(1) [34,35] τBs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='515(4) ps [36] τBd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='519(4) ps [36] Table 4: Values of the experimental and theoretical quantities used as input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' For future updates see FLAG [30], PDG [22] and HFLAV [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 4 Numerical Analysis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 Determining the parameter space Despite the fact that NP is not required to obtain within the SM simultaneous agreement with data for the ∆F = 2 observables in (1) [5], the present uncertainties in hadronic parameters still allow for some NP contributions, whose size depends strongly on the value of |Vcb| [5,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Therefore in order to constrain the parameters in (7) and subsequently obtain predictions for various observables, we will proceed in each of the four considered 331 models as follows: We will vary ∆Md, SψKs, ∆Ms, Sψφ, ϵK within 5% of the central value of their experi- mental datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 4 Numerical Analysis 7 Concerning CKM parameters, we adopt here a different strategy with respect to our previous analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' We vary |Vub| as in (4), while |Vcb| is varied in such a way to encompass both its inclusive and exclusive determinations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' |Vcb| ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0386, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' For each of the four 331 models considered in this paper we then determine the allowed values of the 331 parameters ˜s13, δ1, ˜s23, δ2 as well as a range for |Vcb| for which a given model satisfies the constraints from ∆F = 2 observables in (1) within 5% as stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' We predict several observables in each model and discuss their dependence on |Vcb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' We compare the outcome in the four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The remaining parameters used in our analysis are collected in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Among the parameters that define the various scenarios, ∆F = 2 observables depend only on |β|, so that the resulting parameter space will be the same for M1 and M16 as well as for M3 and M13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In the two cases we have constructed the tables of the allowed parameters in the form of 6-vectors of the kind (˜s13, δ1, ˜s23, δ2, |Vcb|, |Vub|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Of course it is not possible to display the space of all the variables simultaneously and therefore we do not show these plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Instead, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 1 we show the allowed (|Vcb|, |Vub|) ranges in the two resulting parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' It should be understood that each point corresponds to a set of 331 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In these figures the green points are obtained after imposing the constraints on ∆Md, SψKs, ∆Ms, Sψφ and show that even though such observables select the 331 parameters ˜s13, δ1, ˜s23, δ2 they do not have an impact on the allowed ranges for |Vub| and |Vcb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' On the contrary, when the constraint on εK is imposed, a limitation is found for |Vcb| that is the consequence of the stronger dependence of εK on this parameter than in the case of ∆Ms and ∆Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' However, we can observe that, while in the case of M1 and M16, |Vcb| cannot be smaller than ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0405, no similar constraint is found in the case of M3, M13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 CNP 9 and CNP 10 We have already remarked the nice feature of 331 models that the ratio CNP 9 /CNP 10 depends only on the considered scenario but not on the parameters ˜s13, δ1, ˜s23, δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' However, the separate values of CNP 9 and CNP 10 depend on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 2 we show the correlation between their real parts in the four scenarios, while in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 3 the correlation between their imaginary parts is displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In order to understand which values of |Vcb| correspond to the largest deviations in CNP 9 we consider Max ��Re[CNP 9 ] �� setting |Vub| at its central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' These plots display that, consistently with the result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 1 in the case of M1 and M16 only the values |Vcb| ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0405 are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Moreover, the deviation in |Re[C9]| is a decreasing function of |Vcb|, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 4, together with the plots for the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The situation for |Re[CNP 10 ]| and |Im[CNP 10 ]| is displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' It can be noticed that CNP 9 is to an excellent approximation the same in M1 and M16 on the one hand and in M3 and M13 on the other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' for this reason we have shown the corresponding plots in a single figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' CNP 10 is instead different in all the four considered cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' We observe that while the pattern of NP contributions signalled by the data is correctly described by these models, the absolute values of CNP 9 are likely to turn out to be too small to explain the observed suppression of the branching ratios for B+ → K+µ+µ− and Bs → φµ+µ−, 4 Numerical Analysis 8 Figure 1: Allowed (|Vcb|, |Vub|) ranges in the parameter space of M1 and M16 (upper plot) and in that of M3 and M13 (lower plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Each point corresponds to a set of 331 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The green points are obtained after imposing the constraints on ∆Md, SψKs, ∆Ms, Sψφ, while the light blue points derive from imposing the constraint on εK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' in particular if the final value for |Vcb| from tree-level decays will turn out to be in the ballpark of its inclusive determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='3 ¯B(Bs → µ+µ−) and B(Bd → µ+µ−) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 7 we plot the correlation between the rare decays ¯B(Bs → µ+µ−) and B(Bd → µ+µ−) in the four considered 331 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In these plots, the gray region is obtained considering all the allowed parameter space in each scenario, while the red region corresponds to |Vcb| ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0386, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0398] and the cyan region to |Vcb| ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0422, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The SM results for |Vcb| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='03921 and |Vcb| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0426 are also displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Comparing the four models, we can observe that if |Vcb| is fixed consistenlty with the exclusive determinations, a possible suppression of both branching ratios with respect to their SM values, that is not yet excluded in view of large experimental errors, could be explained only in M3 and M13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' On the other hand, inclusive M1&M16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00375 AMd,SuKs,AMs,Su 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00370 EK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043 IVcblM3 & M13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00375 AMd , Suk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=', AMs, Sud 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00370 EK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043 IVcbl4 Numerical Analysis 9 Figure 2: Correlation between the real parts of CNP 9 and CNP 10 in the four considered 331 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' values of |Vcb| do not define a clear situation in any of the four models: other correlations should be explored in order to discriminate among these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' We detail the dependence of the considered branching fractions on the CKM elements in the contour plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 8 for M1 and M16 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 9 for M3 and M13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Since in each scenario the parameter space involves 6 variables it is possible that fixing (|Vcb|, |Vub|) different values for the considered branching ratios are obtained, because these depend also on the other four parameters of the 331 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Therefore, what is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 8 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 9 is the value of the branching ratios that, for a given pair (|Vcb|, |Vub|), mostly deviates from the corresponding SM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The resulting value of the branching fractions can be read from the legenda on the right of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The benefit of these plots with respect to those already shown is that it is possible to relate a given value of the branching fractions to the entries for (|Vcb|, |Vub|), an information that is hidden in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The SM result as function of (|Vcb|, |Vub|) can be read from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 10: comparison between these plots and the corresponding one in a given 331 model would give an idea of the possible deviation as a function of (|Vcb|, |Vub|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In particular, one can observe that M3 and M13 perform rather similarly to the SM, with values of the branching fractions that increase with |Vcb| almost independently on |Vub|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' On the other hand, this pattern is not followed in M1 and M16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='4 Rare Kaon decays In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 11 we display the correlation between B(K+ → π+ν¯ν) and B(KL → π0ν¯ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The gray points span all the allowed parameter space in each scenario, while the red region corresponds M1 M16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 Re[CP] Re[CND] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 Re[CgP] Re[CgP] M3 M13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 Re[CN] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 Re[C)P]5 Summary 10 Figure 3: Correlation between the imaginary parts of CNP 9 and CNP 10 in the four considered 331 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' to |Vcb| ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0386, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0398] and the cyan region to |Vcb| ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0422, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The SM results for |Vcb| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='921 10−2 and |Vcb| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='26 10−2 are also displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' In all the four models, the largest deviation from SM is possible in the case of B(KL → π0ν¯ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Contour plots analogous to those presented for Bs, Bd decays are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 12 and 13, to be compared with the corresponding SM case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' We observe again that M3 and M13 behave similarly to the SM, while M1 and M16 show a differnt pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Correlation between B(K+ → π+ν¯ν) and ¯B(Bs → µ+µ−) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' It can be observed that in all the four cases the inclusive values of |Vcb| correspond to points that can be compatible with the experimental result for ¯B(Bs → µ+µ−) performing slightly better than the SM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' such points correspond to B(K+ → π+ν¯ν) ≤ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Exclusive values of |Vcb| that are not allowed in M1 and M16, can produce in M3 and M13 also values of ¯B(Bs → µ+µ−) and B(K+ → π+ν¯ν) simultaneously smaller than the experimental range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 5 Summary Motivated by several changes both on experimental and theoretical frontiers we updated our 2016 analysis of various flavour observables in the 331 model based on the gauge group SU(3)C × SU(3)L × U(1)X for MZ′ = 3 TeV, that is still in the LHC reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Among 24 331 models considered in our 2016 analysis only four, namely M1, M3, M13 and M16 are simultaneously consistent with the electroweak precision tests and the relation between CNP 9 and CNP 10 signalled by the most recent data on the B → µ+µ− decay from the M1 M16 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 Im[C)P]5 Summary 11 Figure 4: Maximal deviation of ��Re[CNP 9 ] �� and ��Im[CNP 9 ] �� in the four considered 331 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Figure 5: Maximal deviation of ��Re[CNP 10 ] �� in the four considered 331 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043 IVcbl IVcbl5 Summary 12 Figure 6: Maximal deviation of ��Im[CNP 10 ] �� in the four considered 331 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The lessons from this analysis are as follows: The 331 models allow for the values of the ratio CNP 9 /CNP 10 that are consistent with the most recent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' M13 and M16 are performing best but this can only be decided when new overall fits will be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' However, only models M1 and M16 can reach the values Re[CNP 9 ] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='7, which although likely not quite sufficient to explain properly the the suppression of b → sµ+µ− branching ratios, they reproduce a significant portion of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' For M3 and M13 models only the corresponding values of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 can be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Moreover, we notice that while in the case M1 and M16 models the maximal negative shifts of Re[C9] can still be obtained for inclusive values in the ballpark of |Vcb| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0415, in the case of M3 and M13 the shift of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 can only be obtained for exclusive values of |Vcb| as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' We conclude then that models M1 and M16 perform best in this context but as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 4 for the case of the HYBRID scenario for CKM parameters none of the models can provide suppression of Re[C9] by more than −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 which appears too small from present perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Concerning Re[CNP 10 ] all models show only a small shift which is consisten with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' This is also the case of of the imaginary parts of both CNP 9 and CNP 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' As seen in Fig 11, NP effects in K+ → π+ν¯ν turn out to be small but could be signifi- cantly larger in KL → π0ν¯ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' M1 M16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='15 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043 IVebl IVcblREFERENCES 13 Figure 7: Correlation between ¯B(Bs → µ+µ−) and B(Bd → µ+µ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The gray points span all the allowed parameter space in each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The red region corresponds to |Vcb| ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0386, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0398] while the cyan region corresponds to |Vcb| ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0422, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The SM results in correspondence of two values of |Vcb| are displayed, as specified in the legenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' We are looking forward to improved data on all observables to be able to judge better the ability of the 331 models in explaining signs of NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Acknowledgements A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='B would like to thank Andreas Crivellin for the discussion on the present status of lepto- quark models after new LHCb and CMS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' This research was done in the context of the Excellence Cluster ORIGINS, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Excellence Strategy, EXC-2094, 390783311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' It has also been carried out within the INFN project (Iniziativa Specifica) QFT-HEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Buras, Gauge Theory of Weak Decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Cambridge University Press, 6, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Pisano and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Pleitez, An SU(3) x U(1) model for electroweak interactions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' D46 (1992) 410–417, [hep-ph/9206242].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' M3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 109 x( 3.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='3 B(Ba→μ+ μ-)x 1010REFERENCES 14 Figure 8: Contour Plots of ¯B(Bs → µ+µ−) (left column) and B(Bd → µ+µ−) (right column) versus |Vcb| and |Vub| in M1 (upper plots) and in M16 (lower plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00375 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00 qA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00370 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043 IVeblREFERENCES 16 Figure 10: Contour Plots of ¯B(Bs → µ+µ−) (left column) and B(Bd → µ+µ−) (right column) versus |Vcb| and |Vub| in the SM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Figure 11: Correlation between B(K+ → π+ν¯ν) and B(KL → π0ν¯ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The gray points span all the allowed parameter space in each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The red region corresponds to |Vcb| ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0386, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0398] while the cyan region corresponds to |Vcb| ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0422, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The SM results in correspondence of two values of |Vcb| are displayed, as specified in the legenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The light gray region corresponds to the experimental range for B(K+ → π+ν¯ν) reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' B(Bs →μt μ)x 10°, SM 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='042 IVcblB(Bd → μ+ μ-)x 10l0 , SM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00380 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00370 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='042 IVcblM1 14 12 10 8 ↑ B(K+ 9 1 2 3 4 5 B(KL →°) × 1011M16 14 12 x(4 10 SM: IVebl=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='921 10-2 ↑ 8 SM: IVebl=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='26 10-2 B(K+ 6 2 3 4 B(K →°) × 10l1M3 14 12 10 8 B(K+ 9 1 2 3 4 5 B(KL →°) × 1011M13 14 12 ×(4 10 SM: IVebl=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='921 10-2 ↑ 8 SM: IVcbl=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='26 10-2 B(K+ 6 2 3 4 B(KL →°) × 1011REFERENCES 17 Figure 12: Contour Plots of B(K+ → π+ν¯ν) (left column) and B(KL → π0ν¯ν) (right column) versus |Vcb| and |Vub| in M1 (upper plots) and in M16 (lower plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Buras and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Venturini, Searching for New Physics in Rare K and B Decays without |Vcb| and |Vub| Uncertainties, Acta Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Polon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' B 53 (9, 2021) A1, [arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='11032].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Buras, Standard Model Predictions for Rare K and B Decays without New Physics Infection, arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='03968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Bordone, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Capdevila, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Gambino, Three loop calculations and inclusive |Vcb|, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' B 822 (2021) 136679, [arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00604].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' [15] NA62 Collaboration, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Zamkovsk´y et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=', Measurement of the very rare K+ → π+ν¯ν decay, PoS DISCRETE2020-2021 (2022) 070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' [16] KOTO Collaboration, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=', Search for the KL →π0νν and KL →π0X0 decays at the J-PARC KOTO experiment, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 122 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' 2 021802, [arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='09655].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' B(K+→+) × 10ll , M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00380 12 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00375 10 9 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='042 IVcb!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='REFERENCES 19 Figure 14: Contour Plots of B(K+ → π+ν¯ν) (left column) and B(KL → π0ν¯ν) (right column) versus |Vcb| and |Vub| in the SM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Figure 15: Correlation between B(K+ → π+ν¯ν) and ¯B(Bs → µ+µ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The gray points span all the allowed parameter space in each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The red region corresponds to |Vcb| ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0386, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0398] while the cyan region corresponds to |Vcb| ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0422, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='043].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The SM results in correspondence of two values of |Vcb| are displayed, as specified in the legenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' The light gray region and the blue range correspond to the experimental range for B(K+ → π+ν¯ν) and ¯B(Bs → µ+µ−), respectively, reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' B(K+ -→元+ ) × 10l1 , SM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00380 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='75 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00375 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00370 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='042 IVcblB(KL -→元 ) × 1011 , SM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00380 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00375 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00370 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='00365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='042 IVcblM1 14 12 10 8 B(K+ 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0M16 14 12 10 SM: IVcbl=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='921 10-2 8 SM: IVcbl=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='26 10-2 B(K+ 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 B(Bs → μ+ μ)x 109M3 14 12 10 8 B(K+ 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0M13 14 12 10 SM: IVebl=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='921 10-2 8 SM: IVebl=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='26 10-2 B(K+ 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='0 B(Bs → μ+ μ)x 109REFERENCES 20 [22] Particle Data Group Collaboration, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=' Zyla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content=', Review of Particle Physics, PTEP 2020 (2020), no.' metadata={'source': 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+page_content=', Averages of b-hadron, c-hadron, and τ-lepton properties as of summer 2016, arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} +page_content='07233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfxwIb/content/2301.02649v1.pdf'} diff --git a/B9E0T4oBgHgl3EQfyAKb/content/tmp_files/2301.02654v1.pdf.txt b/B9E0T4oBgHgl3EQfyAKb/content/tmp_files/2301.02654v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fdc1acf74ec8e6b6393125616a2677ad44343b14 --- /dev/null +++ b/B9E0T4oBgHgl3EQfyAKb/content/tmp_files/2301.02654v1.pdf.txt @@ -0,0 +1,2295 @@ +DOES COMPRESSING ACTIVATIONS HELP MODEL PARALLEL TRAINING? +Song Bian * 1 Dacheng Li * 2 Hongyi Wang 2 Eric P. Xing 2 3 4 Shivaram Venkataraman 1 +ABSTRACT +Large-scale Transformer models are known for their exceptional performance in a range of tasks, but training +them can be difficult due to the requirement for communication-intensive model parallelism. One way to improve +training speed is to compress the message size in communication. Previous approaches have primarily focused on +compressing gradients in a data parallelism setting, but compression in a model-parallel setting is an understudied +area. We have discovered that model parallelism has fundamentally different characteristics than data parallelism. +In this work, we present the first empirical study on the effectiveness of compression methods for model parallelism. +We implement and evaluate three common classes of compression algorithms - pruning-based, learning-based, +and quantization-based - using a popular Transformer training framework. We evaluate these methods across +more than 160 settings and 8 popular datasets, taking into account different hyperparameters, hardware, and both +fine-tuning and pre-training stages. We also provide analysis when the model is scaled up. Finally, we provide +insights for future development of model parallelism compression algorithms. +1 +INTRODUCTION +Transformer models have become the dominant model for +many machine learning tasks (Devlin et al., 2018; Radford +et al., 2018; Yang et al., 2019; Dosovitskiy et al., 2020; Gong +et al., 2021; Sharir et al., 2021; Gong et al., 2021). However, +state-of-the-art Transformer models have a large number +of parameters, making it difficult for a single GPU to hold +the entire model. As a result, training large Transformer +models often requires partitioning the model parameters +among multiple GPUs, a technique known as model paral- +lelism (Shoeybi et al., 2019; Rasley et al., 2020). Model +parallelism strategies often introduce significant commu- +nication overhead, as demonstrated in Figure 1 (Li et al., +2022). For instance, the most commonly used tensor model +parallelism strategy requires two all-reduce operations over +a large tensor in each Transformer encoder block per iter- +ation. This can greatly increase the overall computational +cost of training the model (Shoeybi et al., 2019). +To address the issue of high communication overhead in +model parallelism, one approach is to compress the mes- +sages communicated among GPUs, such as activation val- +ues. In the data-parallel setting, several prior works have +explored compressing gradients to reduce the communica- +tion cost of training (Seide et al., 2014; Bernstein et al., +2018; Dettmers, 2015; Lin et al., 2017; Wang et al., 2018b; +*Equal contribution +1Department of Computer Science, Uni- +versity of Wisconsin-Madison 2Machine Learning Department, +Carnegie Mellon University 3MBZUAI 4Petuum Inc.. Correspon- +dence to: Song Bian . +(8, 128) +(32, 128) +(32, 512) +Hyper-parameters +0 +5 +10 +15 +20 +25 +30 +35 +40 +Comm. Overhead + (% of total time) +TP=1, PP=4 +TP=2, PP=2 +TP=4, PP=1 +Figure 1. Communication overhead of model parallelism with dif- +ferent batch sizes and sequence lengths on BERTLarge using Py- +torch 1.12, NCCL, fp16 and 4 GPUs. The x-axis is (batch size, +sequence length) +Vogels et al., 2019). However, there has been limited ex- +ploration of compression methods specifically designed for +model parallelism. Furthermore, it is important to note that +compression in model parallelism is fundamentally different +from compression in data parallelism for two main reasons. +Firstly, as shown in Figure 2, gradients tend to be low-rank, +while activations do not. Therefore, low-rank gradient com- +pression methods, which have been shown to provide state- +of-the-art end-to-end speedup in communication-efficient +data-parallel training, may not directly apply to model paral- +lelism (Vogels et al., 2019). Secondly, the performance ben- +efits of gradient compression methods can be significantly +affected by system optimizations in data parallelism (Agar- +wal et al., 2022). However, model parallelism has a different +arXiv:2301.02654v1 [cs.LG] 6 Jan 2023 + +Does compressing activations help model parallel training? +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Dimension Percentage +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sigma Value Percentage +Activation +Gradient +Figure 2. Low-Rank analysis: Curves are drawn by ordering the +singular values of the SVD decomposition. The result shows that +the gradient is low-rank but the activation is not. The activation is +the output of the 12th transformer layer in BERTLarge model. +set of system optimization techniques than data parallelism, +so it is unclear how these optimizations would impact the +performance of compression methods in model parallelism. +In this paper, we present the first systematic study of model +parallelism compression for large Transformer models. We +evaluate the impact of different compression methods in +terms of both throughput and accuracy. We conduct ex- +periments for both pre-training and fine-tuning tasks. (De- +vlin et al., 2018; Gururangan et al., 2020). In particular, +we implement and evaluate popular gradient compression +methods, e.g., Top-K and Random-K as well as a learning- +based compression method, i.e., auto-encoders (Hinton & +Zemel, 1993), which can not directly be applied to gradi- +ent compression but is compatible with activation compres- +sion. To assist researchers and practitioners training new +Transformer-based models (Liu et al., 2019; Izsak et al., +2021), we study compression methods using different train- +ing hyper-parameters and hardware setups. We also develop +a performance model that can be conveniently used to under- +stand how compression methods would affect throughput +at larger scales. In total, we evaluate compression methods +across over 160 different settings with various compression +algorithms, training stages, hyper-parameters, and hardware, +and over 8 datasets (Wang et al., 2018a). Our findings in- +clude the following takeaways. +Our takeaways. 1. Learning-based compression meth- +ods are most suitable for model-parallelism. On the fine- +tuning stage(§4.2, §4.3), only auto-encoders (AEs) can pro- +vide end-to-end speedup (upto 18%) while preserving the +model’s accuracy (within ∼3 GLUE score (Wang et al., +2018a)). Top-K, Random-K, and quantization methods +slow down training because their message encoding and de- +coding overhead is larger than the communication time they +reduce. Top-K and Random-K also hurt model’s accuracy. +For the pre-training stage (§4.4), only AE provides speedup +(upto 16%) while preserving the model’s accuracy (similar +GLUE score). Top-K marginally improves training time, +but degrades the accuracy. Quantization slows down the +training time, and degrades the accuracy. +2. Training hyper-parameters affect the performance +benefits of compression methods. None of the compres- +sion methods can improve performance when the batch size +and sequence length are small because the cost of message +encoding and decoding becomes relatively higher (as dis- +cussed in section §4.6). In practice, we have found that +the batch size and sequence length need to be at least 32 +and 512, respectively, for the compression methods to pro- +vide throughput gains. The same is true when fine-tuning is +performed on a machine with high-bandwidth NVLink con- +nections between all GPUs (as described in section §4.2). +3. Early model layers are more sensitive to compression. +Our observations show that compressing the early layers or +too many layers significantly decreases the model’s accuracy +(as discussed in section §4.5), which is consistent with the +findings of previous research (Wang et al., 2021). In practice, +we have found that compressing the final 12 layers of a 24- +layer Transformer model is an effective approach. +Contributions. We make the following contributions: +• We conduct the first empirical study on model paral- +lelism compression methods for Transformer models, +considering different compression methods, training +stages, hyper-parameters, and hardware configurations. +• We implement several popular compression algorithms, +including Top-K, Random-K, quantization, and auto- +encoders (AEs), and integrate them into an existing +distributed training system. +• We extensively evaluate these algorithms across over +160 different settings and eight popular datasets. Based +on our experimental results, we provide several take- +aways for future model parallelism compression stud- +ies. We also analyze the speedup when the model size +and cluster size are scaled up. +2 +BACKGROUND AND CHALLENGES +In this section, we first introduce data parallelism and model +parallelism (§2.1). Then we introduce the challenges in +model parallelism compression (§2.2). +2.1 +Data Parallelism and Model Parallelism +Data Parallelism (DP). +DP divides the training examples +among multiple workers (Li et al., 2014; Ho et al., 2013) and +replicates the model at each worker. During each iteration, + +Does compressing activations help model parallel training? +each worker calculates the model gradient based on its as- +signed examples and then synchronizes the gradient with the +other workers (Sergeev & Del Balso, 2018). However, DP +requires each worker to compute and synchronize gradients +for the entire model, which can become challenging as the +model size increases. One issue is that the large gradients +can create a communication bottleneck, and several previous +studies have proposed gradient compression methods (Seide +et al., 2014; Bernstein et al., 2018; Dettmers, 2015; Lin +et al., 2017; Wang et al., 2018b) to address this. Addition- +ally, the worker may not have enough memory to train with +the entire model using even one example, in which case +model parallelism may be necessary. +Model Parallelism (MP). +Model parallelism (MP) di- +vides the model among multiple workers, allowing large +models to be trained by only requiring each worker to main- +tain a portion of the entire model in memory. There are two +main paradigms for MP: inter-layer pipeline model paral- +lelism (PP) and intra-layer tensor model parallelism (TP). PP +divides the layers among workers, with each worker execut- +ing the forward and backward computations in a pipelined +fashion across different training examples (Narayanan et al., +2019; Li et al., 2021). +For example, a mini-batch of +training examples can be partitioned into smaller micro- +batches (Huang et al., 2019), with the forward computation +of the first micro-batch taking place on one worker while +the forward computation of the second micro-batch hap- +pens on another worker in parallel. TP (Lu et al., 2017; +Shazeer et al., 2018; Kim et al., 2016) divides the tensor +computations among workers. In particular, we consider +a specialized strategy developed for Transformer models +that divides the two GEMM layers in the attention module +column-wise and then row-wise, with the same partitioning +applied to the MLP module (Shoeybi et al., 2019). However, +TP still involves a communication bottleneck due to the +need for two all-to-all collective operations in each layer, +motivating the use of compression to reduce the communi- +cation overhead of MP (Shoeybi et al., 2019). two all-to-all +collective operations in each layer (Shoeybi et al., 2019). +This bottleneck motivates our study to use compression for +reducing the communication of model parallelism. +2.2 +Challenges in Model Parallelism Compression +In data parallelism, synchronizing gradients in large models +is a major bottleneck, and several gradient compression al- +gorithms have been proposed (Seide et al., 2014; Bernstein +et al., 2018; Dettmers, 2015; Lin et al., 2017; Wang et al., +2018b) to reduce the communication volume. These algo- +rithms often rely on the observation that the gradient matrix +is low-rank. In model parallelism, we have observed that +communicating activations becomes the bottleneck. How- +ever, we have identified three challenges when adapting +gradient compression algorithms for use in model paral- +lelism. +First, the low-rank observation for gradient matrices does +not hold for activation matrices, as shown in Figure 2. The +sigma value percentage for activation matrices increases +nearly linearly with the dimension percentage, indicating +that the activation matrix is not low-rank. Therefore, ap- +plying gradient compression techniques to activations is +likely to result in a significant loss of accuracy. Second, the +performance of compression methods is heavily influenced +by system optimizations (Li et al., 2020), and many gradi- +ent compression methods do not lead to speed-ups for data +parallelism (Zhang et al., 2017; Agarwal et al., 2022) due +to competition for GPU resources between gradient encod- +ing computation and backward computation. However, the +impact of these optimizations on compression methods in +model parallelism has not been studied. Third, model par- +allelism introduces the possibility of using learning-based +compression methods, such as autoencoders (AE) (Hinton & +Zemel, 1993), which have not been examined in the gradient +compression literature because they require gradient com- +putations and raise new considerations. Given these three +challenges, there is a need for a thorough study of the effects +of different compression methods in model parallelism. +3 +IMPLEMENTATION +In this section, we first introduce the compression algo- +rithms we evaluate in this work (§ 3.1). Then, we discuss +implementation details in Sections 3.2 and 3.3. +3.1 +Compression Algorithms +In this work, we evaluate a range of popular compres- +sion algorithms, including sparsification-based approaches, +learning-based approaches, and quantization-based ap- +proaches (as illustrated in Figure 3). We use Top-K and +Random-K as sparsification-based approaches, as they have +been well-studied in gradient compression (Stich et al., +2018). We also implement AEs, which compress messages +using a small neural network (Hinton & Zemel, 1993). For +quantization, we use the same scheme as in previous re- +search (Wang et al., 2022), but compare its performance to +other compression algorithms in the context of model paral- +lelism, as the prior work only considered pipeline compres- +sion over slow networks. Since the activation matrices for +models are not low-rank (as shown in Figure 2), low-rank +based compression algorithms (such as PowerSGD (Vo- +gels et al., 2019)) are not suitable for model parallelism +compression, and we do not evaluate any low-rank based +compression algorithms in this work. + +Does compressing activations help model parallel training? +3.2 +Tensor Parallelism Compression +We base our implementation on Megatron-LM (Shoeybi +et al., 2019), a popular Transformer models training system +that supports tensor and pipeline model parallelism. To +integrate the compression algorithms into Megatron-LM, +we make the following modifications. For AE, we compress +the activation before the all-reduce step and invoke the +all-reduce function as usual. The implementation of AE +is shown here: for each layer, we have a learnable matrix +w ∈ Rh×c, and the activation X ∈ Rb×s×h, where b is the +batch size, s is the sequence length, h is the hidden size, +and c < h is the compressed size. By using the matrix +w, we output the compressed activation Xw ∈ Rb×s×c. +Then, we use a similar technique(a decoder as opposed to +an encoder) to decompress the compressed activation and +propagate it to the next layer. However, since the Top-K, +Random-K, and quantization can output two independent +tensors with different types (e.g., for Top-K values and +their indices), we cannot use torch.distributed.all-reduce +to sum the tensors up directly. +In light of this, we +replace the all-reduce step with the all-gather function: +gather-from-tensor-model-parallel-region, +which +is +implemented +by +Megatron-LM. +We +use +torch.topk function to select the k largest absolute +values of the activation and random.sample function to +randomly select k values from the activation. Finally, our +implementation of quantization is based on code released +by (Wang et al., 2022). +3.3 +Pipeline Parallelism Compression +Megatron-LM can only send one tensor to the next pipeline +stage per round, so we modify its communication functions +to allow for the transmission of multiple tensors per round +in order to integrate Top-K, Random-K, and quantization. +Since we compress the activation in the forward step, us- +ing compression also reduces the size of the gradient for +activation and thus the communication cost in the backward +step. However, this is not the case when using quantization +to compress the activation for models. This is because, as +previously noted (Wang et al., 2022), the Pytorch backward +engine only supports gradients for floating point tensors, +and therefore the size of the gradient is the same as the size +of the decompressed activation. Our implementation also +allows the integration of error-feedback compression algo- +rithms by retaining the error information from the previous +compression step. +4 +EXPERIMENTS +We next perform experiments using our implementation to +answer the following questions: +• What is the impact of activation compression on system +throughput and which compression method achieves +the best throughput? +• What is the impact on the model’s accuracy? +• How different network bandwidths affect the best com- +pression method? +• How do hyper-parameters such as the batch size and +sequence length affect the benefits of compression? +We answer these questions in the context of two commonly +used scenarios in language modeling: fine-tuning on the +GLUE benchmark (Wang et al., 2018a), and pre-training +on the Wikipedia (Devlin et al., 2018) dataset and the +BooksCorpus (Zhu et al., 2015) dataset. +4.1 +Experimental Setup +In this section, we briefly describe the hardware, model, and +other experiment settings. +System Configuration. To measure the performance of +compression algorithms over different hardware, our ex- +periments are conducted on two different setups. +Our +first setup uses AWS p3.8xlarge machines which have 4 +Tesla V100 GPUs with all GPUs connected by NVLink. +AWS p3.8xlarge instances have 10 Gbps network band- +width across instances. Our second setup uses a local ma- +chine which also has 4 Tesla V100 GPUs but does not have +NVLink. All the GPUs are connected by a single PCIe +bridge. The local server runs Ubuntu 18.04 LTS and the +server has 125GB of memory. +Model. +We use the BERTLARGE model provided by +Megatron-LM (Shoeybi et al., 2019) which has 345M pa- +rameters. We configure the model to have 24 layers with +each layer having a hidden size of 1024 and 16 attention +heads. We use fp16 training to train the BERTLARGE model. +Experimental Settings. For fine-tuning, we follow the +previous work (Devlin et al., 2018; Liu et al., 2019), and +use micro-batch size 32 and sequence length 512 by de- +fault. We use one machine with 4 V100 GPUs and vary +the tensor model-parallel size and the pipeline model- +parallel size across the following three parallelism degrees: +{(1, 4), (2, 2), (4, 1)}, where the first number of the tuple +represents the tensor model-parallel degree and the second +number of the tuple stands for the pipeline model-parallel +degree. To investigate the impact of hyper-parameters, we +conduct experiments that vary the batch size from {8, 32}, +and sequence length from {128, 512} on fine-tuning. +For pre-training, we use micro-batch size 128, global batch +size 1024, and sequence length 128. To study the impact of +the distributed settings, we use the following three different +parallelism degrees: {(2, 8), (4, 4), (8, 2)}, where the first + +Does compressing activations help model parallel training? +g +g +C +C +C +C +Machine 1 +Machine 2 +C +Transformer Layer +Transformer Layer + Activation +Transformer Layer +Machine 1,2 +Machine 3,4 +DC +DC +DC +DC +Micro-batch +C +DC +Transformer Layer +Transformer Layer +Transformer Layer + Activation +DC +Figure 3. Illustration of compression on a 6-Layer Transformer model with 4 machines. Machine 1 and Machine 2 maintain the first three +layers according to the TP strategy (pipeline stage 1). g stands for an all-reduce operation in the forward pass. A compression method C +is used to reduce the message size for the all-reduce operation to reduce TP communication time. Correspondingly, a de-compression +method DC is used after the communication. For instance, if AE are used, then C is an encoder, and DC is a decoder. Machine 3 and +Machine 4 are responsible for the last three layers (pipeline stage 2). A compression method is used before Machine 1 sends the activation +to Machine 3, and before Machine 2 sends the activation to Machine 4 to reduce PP communication time. The goal of this paper is to +study the effect of different pairs of C and DC. +number of the tuple represents the tensor model-parallel +degree and the second number of the tuple represents the +pipeline model-parallel degree. +We also evaluate compression algorithms with different +parameters. For AE, we use different dimension after com- +pression from {50, 100}. For Top-K and Random-K algo- +rithms, we use two comparable settings: (1) Keep the same +compression ratio as AE (i.e., we compress the activation +around 10 and 20 times.) (2) Keep the same communica- +tion cost as AE. Finally, we also tune the parameters for +quantization and compress the activation to {2, 4, 8} bits. +By default, we perform experiments on BERTLarge model +with 24 layers and compress the activation for the last 12 +layers. For instance, when the pipeline model-parallel de- +gree is 2 and tensor model-parallel degree is 2, we compress +the activation between two pipeline stages and the communi- +cation cost over tensor parallelism in the last 12 layers. We +also vary the number of layers compressed in Section 4.5. +4.2 +Throughput Benefits for Fine-Tuning +Takeaway 1 Using non-learning-based compression tech- +niques to compress activations only slightly improves system +throughput (by 1% or less) due to the large overhead of these +methods. However, we see end-to-end speedups of up to +Notation +Description +A1 +AE with encoder output dimension 50 +A2 +AE with encoder output dimension 100 +T1/R1 +Top/Rand-K: same comm. cost as A1 +T2/R2 +Top/Rand-K: same comm. cost as A2 +T3/R3 +Top/Rand-K: same comp. ratio as A1 +T4/R4 +Top/Rand-K: same comp. ratio as A2 +Q1 +Quantization: reduce the precision to 2 bits +Q2 +Quantization: reduce the precision to 4 bits +TP +Tensor model-parallelism degree +PP +Pipeline model-parallelism degree +Table 1. Notation Table. For ease of notation, we use TP/PP to +denote the degree of tensor/pipeline model parallelism. ‘comm’ +and ‘comp’ are short for ‘communication’ and ‘compression’. +17.8% when using learning-based compression methods on +a machine without NVLink. +When running fine-tune experiments on a p3.8xlarge in- +stance on Amazon EC2, we cannot improve system through- +put by using non-learning-based compression algorithms +(Table 2). Comparing Tables 2 and 3, we can see that the net- + +Does compressing activations help model parallel training? +Distributed Setting +w/o +A1 +A2 +T1 +T2 +T3 +T4 +TP=1, PP=4 +591.96 +591.36 +591.47 +594.81 +595.53 +599.65 +605.05 +TP=2, PP=2 +440.71 +437.98 +444.02 +465.73 +473.64 +493.16 +528.93 +TP=4, PP=1 +261.48 +270.22 +275.54 +314.37 +323.90 +356.57 +409.23 +Distributed Setting +w/o +R1 +R2 +R3 +R4 +Q1 +Q2 +TP=1, PP=4 +591.96 +749.56 +1,008.64 +1,824.36 +5,572.87 +595.29 +595.45 +TP=2, PP=2 +440.71 +3,377.59 +6,616.30 +17,117.01 +71,058.64 +489.27 +486.54 +TP=4, PP=1 +261.48 +3,254.01 +6,561.22 +16,990.88 +65,121.79 +347.68 +350.50 +Table 2. The average iteration time (ms) for fine-tuning with various compression techniques by varying the distributed setting. The +results are collected from the AWS p3.8xlarge machine with NVLink by using batch size 32, and sequence length 512. The best setting is +bolded in the table. And the settings which see benefits compared with the baseline, are underlined. +With NVLink +w/o +A1 +A2 +TP=1, PP=4 +591.96 +591.36 +591.47 +TP=2, PP=2 +440.71 +437.98 +444.02 +TP=4, PP=1 +261.48 +270.22 +275.54 +Without NVLink +w/o +A1 +A2 +TP=1, PP=4 +633.17 +620.10 +620.44 +TP=2, PP=2 +646.14 +586.65 +595.25 +TP=4, PP=1 +736.01 +624.62 +636.15 +Table 3. The +average +iteration +time +(ms) +for +fine-tuning +with/without NVLink. We compare time without compression +and with AE on different distributed settings, with batch size 32, +and sequence length 512. The best setting on each machine is +bolded. And the settings, under which we can gain benefits com- +pared with the baseline, are underlined. +work bandwidth across the GPUs can affect the performance +benefits from compression. In other words, we can improve +system throughput by at most 17.8% when compressing +activation for fine-tuning tasks on a 4-GPU machine without +NVLink. That’s because, without NVLink, the communica- +tion time for model parallelism is much longer. Thus, while +the message encoding and decoding time remain unchanged, +compression methods can provide more throughput benefits +across lower bandwidth links. +Furthermore, from Tables 2 and 3, we observe that AE out- +performs other compression methods. In Table 4, we break- +down the time taken by each algorithm and find that Top-K, +Random-K and quantization have large encoding/decoding +overheads and thus cannot provide end-to-end throughput +improvements. Although AE slightly increases the time +taken by the backward step, the ∼ 2× reduction in commu- +nication time and the limited encoding/decoding overhead +lead to better overall throughput. +4.3 +Effect of Compression on Model Accuracy while +Fine-tuning +Takeaway 2 Among all evaluated compression algorithms, +only AE and quantization preserve fine-tuning accuracy. +From Table 5, we can see that, when using AE and quan- +tization algorithm for compression, the accuracy loss is +within 3% except for CoLA and RTE. In Figure 2, we have +shown that the activation for models is not low-rank. There- +fore, sparsification-based compression algorithms (Top- +K/Random-K) lose important information and do not pre- +serve model accuracy. Given that there is significant accu- +racy difference for CoLA and RTE, we study the impact +of varying the number and range of layers compressed for +these two datasets in Section 4.5. +4.4 +Throughput Benefits for Pre-training +Takeaway 3 Only AE and Top-K algorithms improve +throughput when performing distributed pre-training. +First, we recap the experimental environment here. For pre- +training, we use 4 p3.8xlarge instances on Amazon EC2 +and each instance has 4 GPUs with NVLink. From Table 6, +we can see that using Top-K and AE can speed up pre- +training by 7% and 16% respectively. Among the three +distributed settings, TP=4, PP=4 is the best setting for +pre-training. That is because the communication cost of +tensor parallelism is larger than that of pipeline parallelism +and with TP=4, tensor parallel communication happens over +faster NVLinks. +Takeaway 4 Compressing activation for models can im- +prove throughput for pre-training by 16%. +From Table 7, we notice that using AE and Top-K can +reduce the waiting time and pipeline communication time +of pre-training. This is because the inter-node bandwidth +(10Gbps) is smaller than the intra-node bandwidth (40GB/s + +Does compressing activations help model parallel training? +Compression +Algorithm +Forward +Backward +Optimizer +Waiting & +Pipeline Comm. +Total Time +Tensor Enc. +Tensor Dec. +Tensor +Comm. +w/o +276.34 +354.16 +5.80 +9.83 +646.14 +\ +\ +150.72 +A1 +213.83 +362.61 +6.16 +4.06 +586.65 +2.16 +3.12 +80.88 +A2 +219.01 +366.51 +5.67 +4.07 +595.25 +3.12 +4.56 +84.48 +T1 +298.93 +355.71 +6.79 +4.38 +665.81 +70.08 +13.68 +85.20 +T2 +305.47 +355.51 +6.36 +3.91 +671.24 +70.32 +16.80 +87.84 +T3 +331.70 +356.80 +5.78 +5.00 +699.27 +72.24 +27.36 +100.80 +T4 +376.72 +359.19 +5.89 +6.60 +748.41 +74.88 +45.36 +124.56 +R1 +2,408.68 +357.02 +6.10 +7.68 +2,779.49 +2,040.24 +15.84 +104.16 +R2 +4,696.99 +356.33 +6.28 +6.20 +5,065.80 +4,244.64 +19.44 +135.84 +R3 +12,603.79 +362.13 +6.81 +25.28 +12,998.01 +11,499.12 +29.76 +139.92 +R4 +46,968.21 +365.36 +7.61 +22.81 +47,363.98 +44,038.56 +47.52 +567.36 +Q1 +274.03 +354.56 +5.88 +7.98 +642.46 +20.64 +32.16 +91.68 +Q2 +282.64 +354.55 +5.58 +7.58 +650.36 +19.92 +30.24 +104.64 +Table 4. We breakdown the average iteration time (ms) for fine-tuning with various compression techniques when using TP=2 and PP=2, +batch size 32, and sequence length 512. The results are collected from the local machine without NVLink. The total time (ms) is divided +into following parts: forward step, backward step, optimizer, and waiting & pipeline communication. The last three columns further +breakdown the tensor encoder/decoder and communication times which are considered part of the forward step. +Compression +Algorithm +MNLI-(m/mm) +QQP +SST-2 +MRPC +CoLA +QNLI +RTE +STS-B +Avg. +w/o +88.07/88.70 +92.02 +95.07 +88.46 +62.22 +93.39 +82.67 +89.16 +86.64 +A1 +85.42/85.43 +91.07 +92.09 +86.14 +54.18 +91.31 +70.04 +87.61 +82.59 +A2 +85.53/85.65 +91.24 +93.23 +85.86 +55.93 +91.01 +65.34 +87.76 +82.40 +T1 +32.05/32.18 +74.31 +83.60 +70.78 +0.00 +58.37 +51.99 +0.00 +44.81 +T2 +44.12/45.67 +39.68 +90.83 +78.09 +0.00 +84.42 +49.82 +62.70 +55.04 +T3 +36.12/36.08 +74.75 +90.25 +81.51 +0.00 +85.41 +54.15 +0.00 +50.92 +T4 +83.85/84.41 +56.39 +93.69 +83.65 +0.00 +90.54 +59.21 +86.02 +70.86 +Q1 +87.25/87.81 +91.71 +93.46 +87.01 +55.99 +61.38 +67.51 +88.02 +80.02 +Q2 +87.85/88.47 +91.93 +93.23 +87.42 +57.67 +93.01 +78.34 +87.43 +85.04 +Table 5. Fine-tuning results over GLUE dataset under the setting that the tensor model-parallel size is 2 and pipeline model-parallel size is +2. F1 scores are reported for QQP and MRPC, Matthews correlation coefficients are reported for CoLA, and Spearman correlations are +reported for STS-B, and accuracy scores are reported for the other tasks. +with NVLink), so compression is effective at reducing the +communication time between two pipeline stages. From +Table 9, we can observe that, by using A2 to compress +the activation over the last 12 layers, we can reduce the +communication cost between two pipeline stages effectively. +Takeaway 5 Among all evaluated methods, AE is the +best strategy to compress activation over pre-training. It +achieves higher pre-training throughput and preserves the +model’s accuracy. +From Table 8, compared with the baseline (without com- +pression), we can observe that using AE is able to keep +the accuracy when compared to the uncompressed model. +In addition, we observe that we can use the AE at the pre- +training phase and remove it during the fine-tuning phase. +In other words, we only need to load the parameter of the +BERTLarge model to do fine-tuning, and the parameters of +the AE can be ignored. Furthermore, Table 8 shows that pre- +trained models suffer significant accuracy loss when using +Top-K for compression. Finally, quantization can preserve +the model’s accuracy, but we cannot achieve end-to-end +speedup by using quantization as strategy to compress ac- + +Does compressing activations help model parallel training? +Distributed Setting +w/o +A1 +A2 +T1 +T2 +T3 +T4 +TP=2, PP=8 +1,625.16 +1,550.18 +1,579.70 +1,508.34 +1,503.54 +1,593.37 +1,682.87 +TP=4, PP=4 +1,422.40 +1,242.97 +1,223.20 +1,360.37 +1,352.61 +1,410.47 +1,721.87 +TP=8, PP=2 +15,642.30 +14,577.29 +14,073.45 +14,308.12 +14,543.81 +18,919.92 +27,152.07 +Distributed Setting +w/o +R1 +R2 +R3 +R4 +Q1 +Q2 +TP=2, PP=8 +1,625.16 +10,308.03 +20,814.20 +55,925.28 +>100,000 +1,759.27 +1,752.24 +TP=4, PP=4 +1,422.40 +15,433.12 +31,565.19 +87,421.46 +>100,000 +2,435.03 +2,594.94 +TP=8, PP=2 +15,642.30 +32,522.47 +61,049.87 +>100,000 +>100,000 +16,414.57 +16,517.44 +Table 6. The average iteration time (ms) for pre-training with various compression techniques by varying the distributed setting. The +results are collected from 4 AWS p3.8xlarge machines with NVLink by using micro-batch size 128, global batch size 1024, and sequence +length 128. The best setting is bolded in the table. And the settings, under which we can gain benefits compared with the baseline, are +underlined. +Compression +Algorithm +Forward +Backward +Optimizer +Waiting & +Pipeline Comm. +Total Time +Tensor Enc. +Tensor Dec. +Tensor +Comm. +w/o +467.73 +419.26 +7.42 +527.99 +1,422.40 +\ +\ +91.08 +A1 +546.95 +455.26 +7.29 +233.47 +1,242.97 +8.64 +16.20 +32.76 +A2 +459.26 +467.51 +9.64 +286.78 +1,223.20 +12.96 +20.52 +43.56 +T1 +712.22 +423.91 +7.21 +217.03 +1,360.37 +73.44 +140.4 +80.28 +T2 +671.19 +424.27 +7.35 +249.80 +1,352.61 +81.00 +170.64 +81.36 +T3 +813.03 +433.42 +7.35 +156.67 +1,410.47 +108.00 +268.92 +115.92 +T4 +1,068.38 +444.26 +6.75 +202.48 +1,721.87 +153.36 +427.68 +151.56 +R1 +14,199.56 +421.40 +4.23 +807.93 +15,433.12 +13,185.72 +181.44 +193.68 +R2 +29,344.85 +427.18 +3.91 +1,789.25 +31,565.19 +27,975.24 +181.44 +187.20 +R3 +78,906.91 +444.88 +6.08 +3,707.37 +83,065.23 +73,847.16 +279.72 +649.44 +Q1 +803.63 +417.33 +8.61 +1,205.46 +2,435.03 +90.72 +304.56 +193.68 +Q2 +805.33 +417.74 +7.55 +1,364.32 +2,594.94 +85.32 +271.08 +111.60 +Table 7. We breakdown the average iteration time (ms) for pre-training with various compression techniques when using tensor model- +parallel size 4, pipeline model-parallel size 4, micro batch size 128, global batch size 1024, and sequence length 128. The results are +collected from 4 AWS p3.8xlarge machines with NVLink. +Compression +Algorithm +MNLI-(m/mm) +QQP +SST-2 +MRPC +CoLA +QNLI +RTE +STS-B +Avg. +w/o +84.87/84.79 +91.25 +92.43 +86.84 +56.36 +92.26 +70.40 +86.83 +82.89 +A2 +83.77/84.32 +91.14 +91.63 +86.55 +58.61 +91.96 +71.48 +87.16 +82.96 +T2 +61.06/60.93 +80.74 +80.16 +63.83 +10.01 +59.55 +47.29 +0.37 +51.55 +Q2 +84.47/85.32 +91.36 +93.23 +85.10 +58.84 +91.69 +71.84 +86.39 +83.14 +Table 8. Fine-tuning results over GLUE dataset by using the checkpoint obtained by pre-training. F1 scores are reported for QQP and +MRPC, Matthews correlation coefficient is reported for CoLA, and Spearman correlations are reported for STS-B, and accuracy scores +are reported for the other tasks. +tivation over pre-training. In conclusion, it is not a good +choice to compress the activation by using quantization or +Top-K. +4.5 +Varying Compression Layers and Location +Takeaway 6 When the number of compressed layers in- +creases, the model accuracy decreases. +From Figure 4(a), we can observe that the accuracy for RTE + +Does compressing activations help model parallel training? +Pipeline Stages +Comm. (w/o) +Comm. (A2) +0 ↔ 1 +77.82 +76.13 +1 ↔ 2 +88.69 +13.19 +2 ↔ 3 +97.67 +14.09 +Table 9. The average communication time (ms) per iteration be- +tween two pipeline stages. The first column indicates the pipeline +stage. And the second column shows the communication time +per iteration without compression. Moreover, the third column +presents the communication time with A2. We only compress +the activation in the last 12 layers and thus the time for the first +pipeline stage is unchanged. +w/o +6 +8 +10 +12 +14 +16 +18 +Number of Layers Compressed +0 +20 +40 +60 +80 +100 +Metrics (%) +CoLA +RTE +(a) Vary Number of Layers Compressed +1-12 +4-15 +7-18 +10-21 +13-24 +w/o +Compression Location +0 +20 +40 +60 +80 +100 +Metrics (%) +CoLA +RTE +(b) Vary Compression Location +Figure 4. Fine-tuning results over CoLA and RTE datasets by vary- +ing the compression location and number of layers compressed. +The above figure shows that model performance vs the number +of layers compressed. The below figure shows that model per- +formance versus the compression location. We use tensor model- +parallel degree 2, pipeline model-parallel degree 2, batch size 32, +and sequence length 512. +and the matthews correlation coefficient for CoLA decreases +as we increase the number of layers compressed. This is +because as we increase number of layers compressed, we +lose more information in the activations leading to a loss in +accuracy. From Figure 4(a), we observe that compressing +activations of the last 8 layers is the best strategy to keep +the accuracy loss within 3% for both datasets. +Takeaway 7 Compressing the activation for the initial lay- +ers harms the accuracy of the model. +We keep the number of layers compressed constant and +vary the location where we apply compression (Figure 4(b)). +The results indicate that compressing activations of the first +few layers of the model significantly harms the model’s +accuracy. This is because compressing activations generates +error and the error in the early layers can be accumulated +and propagated to later layers. +4.6 +Impact of Model Hyper-parameters +Takeaway 8 Using a smaller batch size or sequence length +for fine-tuning negates the throughput benefits from com- +pression because of the smaller communication cost. +We vary the batch size from {8, 32} and sequence length +from {128, 512}, and report the results in Table 11-14. We +provide more detailed experimental results in Appendix A. +We notice that when the communication cost over model par- +allelism is small, the overhead of the compression methods +can become the bottleneck. Therefore, we cannot improve +system throughput when using compression algorithms with +batch size 8 and sequence length 128. +4.7 +Performance Analysis +In this section, we develop an analytical cost model to an- +swer the question: +What will happen if we scale up the model size and the +cluster size? +Given that prior works (Li et al., 2022) have analyzed the +complexity of various model parallelism strategies, we only +consider a fixed strategy of using tensor model parallelism +here. Concretely, we use tensor model parallelism in the +same node, and pipeline model parallelism across the node, +a suggested strategy according to (Narayanan et al., 2021). +In particular, we build the performance analysis for real- +world settings similar to (Narayanan et al., 2019) in two +steps. First, we develop our own model on a single-node +scenario, and we scale up the model size on a single node. +Second, we increase the cluster size and, according to the +model-parallelism strategy we choose, assign additional +GPUs to pipeline parallelism, and use off-the-shelf pipeline +parallelism cost models to predict the performance (Li et al., +2022; Zheng et al., 2022). +Denote the vocabulary size as V , hidden size as h, sequence +length as s, and batch size as B. From (Narayanan et al., +2021), we know that the number of floating points opera- +tions (FLOPs) and all-reduce message size in a Transformer +layer is 96Bsh2 + 16Bs2h, and Bsh respectively. +If we do not use compression methods, the total time of a +Transformer layer can be modeled as a sum of the all-reduce +communication step and the computation time step. We note + +Does compressing activations help model parallel training? +2500 +5000 +7500 +10000 12500 +Hidden size +0 +50 +100 +150 +200 +Run-time (ms) +bs16(pred) +bs32(pred) +bs64(pred) +bs128(pred) +bs16 +bs32 +bs64 +bs128 +(a) Tcomp +2500 +5000 +7500 +10000 +12500 +Hidden size +0 +20 +40 +60 +Run-time (ms) +bs16(pred) +bs32(pred) +bs64(pred) +bs128(pred) +bs16 +bs32 +bs64 +bs128 +(b) Tcomm +2500 +5000 +7500 +10000 +12500 +Hidden size +0 +1 +2 +3 +4 +5 +Run-time (ms) +bs16(pred) +bs32(pred) +bs64(pred) +bs128(pred) +bs16 +bs32 +bs64 +bs128 +(c) Toverhead +2500 +5000 +7500 +10000 12500 +Hidden Size +1 +2 +3 +4 +5 +Speedup +bs16(pred) +bs32(pred) +bs64(pred) +bs128(pred) +bs16 +bs32 +bs64 +bs128 +(d) Speedup +Figure 5. We vary the batch size and the hidden size to show that our prediction model is accurate compared with the real experimental +results. The model we use here has only one transformer layer and the tensor model-parallel size is 4. In specific, Figure (a) shows the +real and predicted computation time with the increase of the hidden size. Figure (b) presents the real and predicted communication time +between tensor parallelism by varying the hidden size. As for the Figure (c), it presents the computation time of AE with the increase of +hidden size. In the end, Figure (d) show the total speedup when we use AE to compress activations over tensor parallelism. +that these two steps can hardly overlap because , the reason +behind it is that the all-reduce communication depends on +the previous computational results: +T = Tcomp(96Bsh2 + 16Bs2h) + Tcomm(Bsh) +(1) +Modeling Tcomp +We model Tcomp as a linear function of +FLOPs with the coefficient α that corresponds to the peak +performance of the GPU. In particular, we estimate α using +ground truth wall clock time of the largest hidden size we +can fit, where the GPU is more likely to be of the peak +utilization (Williams et al., 2009). During experiments, we +found that fitting α using time of smaller hidden sizes can +result in a 30x higher prediction time when we scale up the +hidden size because of low GPU utilization. Our prediction +versus the ground truth time is plotted in Figure 5(a). +Modeling Tcomm +we model Tcomm as a piece-wise func- +tion of the message size (Agarwal et al., 2022). Formally, +Tcomm(Bsh) = +� +c +if Bsh < d +βBsh +if Bsh ≥ d +If the message size is smaller than a threshold d, then +Tcomm(Bsh) is a constant c because the worker needs to +launch one communication round (Li et al., 2020). Other- +wise, the number of communication rounds is proportional +to the message size. The fitting result is in Figure 5(b). +Using AE as the compression method and a fixed encoder +dimension e (we set e to 100 in this section), the total time +of a single Transformer layer is: +TAE = Tcomp(96Bsh2 + 16Bs2h) + Tcomm(Bse) ++ Toverhead +Compared with the setting without compression, the compu- +tation time remains unchanged. In addition, Tcomm(Bse) +is roughly equal to c because Bse is usually smaller than +the threshold d. In our experiments, the threshold d = +16 × 128 × 100 = 409600 and c ≈ 0.2. +Modeling Toverhead +In AE, Toverhead is the encoder and +decoder computation time. It is a batched matrix multiplica- +tion with input dimension B × s × h and h × e. Assuming +e is kept constant, it can be modeled as Toverhead = γBsh. +The fitting result is shown in 5(c). +Since each Transformer layer has identical configurations in +popular Transformer models (Devlin et al., 2018; Radford +et al., 2018), the overall speedup ratio is the same as we vary +the number of layers. Thus, we can estimate the speedup of +different hidden sizes of any number of Transformer layers +using +T +TAE . We provide the fitting result in Figure 5(d). +Understanding the trend +We consider the asymptotic +behavior of large hidden size h: +T +TAE +≈ +α(96Bsh2 + 16Bs2h) + βBsh +α(96Bsh2 + 16Bs2h) + γBsh + c +(2) +Thus, we can see that as hidden layer size increases, the +benefits from compression diminish. +Scaling up the cluster size +Next we analyze the speedup +when scaling up the cluster size by combining the pipeline +parallelism cost model developed in (Li et al., 2022; Zheng +et al., 2022). Formally, the running time is modeled as a +sum of per-micro-batch pipeline communication time, per- +micro-batch of non-straggler pipeline execution time, and +the per-mini-batch straggler pipeline execution time. To use +the cost model, we denote the micro-batch size as m, the +number of nodes n, the number of layers L, the pipeline +communication time p or pAE. +We use the default pipeline layer assignment strategy +in (Shoeybi et al., 2019), which balances the number of +transformer layers. Thus, every stage takes the same time in +our scenario: L +nT or L +nTAE. We use the pipeline communi- +cation model in (Jia et al., 2019; Li et al., 2022), p = Bsh +w , +pAE = Bse +w , where w is the bandwidth. Thus the overall +speedup can be written as: +( m−1 +n ++ 1) × LT + (n − 1) × Bsh +w +( m−1 +n ++ 1) × LTAE + (n − 1) × Bse +w +(3) + +Does compressing activations help model parallel training? +From the Table 10, we see that we can maintain a ∼1.5x +speedup as we scale the hidden size to 25600. This shows +that if we increase the number of nodes when we increase in +hidden size, AE compression retains its benefits. However, +it is possible to avoid the diminishing speedup by properly +scaling up the number of nodes n, where the speedup will +asymptotically converge to h +e . +In summary, compression in model parallelism has dimin- +ishing returns if we only scale up the model on a fixed +cluster. To gain benefits from compression methods, one +needs to also properly manage other parameters in the +cost model, e.g. also scaling up the number of nodes and +use the pipeline parallelism. +5 +RELATED WORK +In this section, we first introduce work related to the de- +velopment of large Transformer models. Then, we discuss +strategies to train these models at scale. In the end, we +discuss prior work that accelerates distributed ML models +training by using compression techniques. +Transformer Models. Transformer models were first intro- +duced by Vaswani et al. (2017) in the machine translation +context. It has been shown to be effective in various other +language understanding tasks such as text generation, text +classification and question answering (Devlin et al., 2018; +Radford et al., 2018; Wang et al., 2018a; Rajpurkar et al., +2016). Recent research has also successfully applied Trans- +former models to images (Dosovitskiy et al., 2020; Touvron +et al., 2021), audio (Gong et al., 2021) and beyond (Sharir +et al., 2021). An N-layers transformer model is composed +of three major components: (1) An embedding layer that +maps an input token to a hidden state, (2) A stack of N +transformer layers, and (3) a prediction layer that maps the +hidden state proceeded by transformer layers to the task +output. A transformer layer is composed of an attention +module (Bahdanau et al., 2014) and several matrix multipli- +cations. Several optimizations have been proposed to speed +up Transformer model training such as carefully managing +the I/O (Dao et al., 2022) and reducing the complexity of the +attention module (Wang et al., 2020). In this work, we speed +up the Transformer model training in the distributed setting, +where we reduce the communication between workers. +Training Large Transformer models. Several parallelism +strategies have been proposed to train Transformer mod- +els. Megatron (Shoeybi et al., 2019) proposes tensor model +parallelism, which parallelizes the computation in attention +layers and in the following matrix multiplications. Deep- +Speed (Rasley et al., 2020) uses a specialized form of +pipeline parallelism (Huang et al., 2019; Narayanan et al., +2019) that treats a transformer layer as the smallest unit +in pipeline stages. It further combines the tensor model +parallelism developed in Megatron and data parallelism to +train Transformer models at the scale of trillion parame- +ters. (Li et al., 2022) considers a more sophisticated model +parallelism strategy space for Transformer models and uses +a cost model to automatically search for the optimal one. +Our work is orthogonal to the direction of developing new +parallel training strategies. In this work, we study how to +compress communication on existing parallel strategies. +Distributed training with Compression. Distributed ML +model training requires frequent and heavy synchronization +between workers. Several directions have been proposed +to reduce the communication bottleneck by compressing +the message size. One direction is developed on the data +parallelism setting, where workers communicate model gra- +dients (Wang et al., 2021; Agarwal et al., 2022) during +backward propagation. Common techniques to reduce the +gradient communication include low-rank updates (Wang +et al., 2018b), sparsification (Lin et al., 2017), and quanti- +zation (Seide et al., 2014; Bernstein et al., 2018; Dettmers, +2015). A more recent direction find that the activation pro- +duced during the forward propagation in neural networks is +large, and thus compressing them is beneficial (Wang et al., +2022). In particular, they use quantization to compress the +activation volume between pipeline parallelism workers. +However, they focus on the geo-distributed setting where +the network bandwidth is very low. In this paper, we study +the effect of a rich set of popular compression techniques +on tensor and pipeline parallelism, and in a typical cloud +computing setting. +6 +CONCLUSION +In this work, we studied the impact of compressing acti- +vations for models trained using model parallelism. We +implemented and integrated several popular compression +algorithms into an existing distributed training framework +(Megatron-LM) and evaluated their performance in terms +of throughput and accuracy under various settings. Our re- +sults show that learning-based compression algorithms are +the most effective approach for compressing activations in +model parallelism. We also developed a performance model +to analyze the speedup when scaling up the model. Our ex- +periments provide valuable insights for the development of +improved activation compression algorithms in the future. +Acknowledgments +Shivaram Venkataraman is supported by the Office of the +Vice Chancellor for Research and Graduate Education at +UW-Madison with funding from the Wisconsin Alumni +Research Foundation. +Eric Xing is supported by NSF +IIS1563887, NSF CCF1629559, NSF IIS1617583, NGA +HM04762010002, NSF IIS1955532, NSF CNS2008248, +NSF IIS2123952, and NSF BCS2040381. + +Does compressing activations help model parallel training? +hidden size +number of layers +number of nodes +batch size +speedup +6144 +40 +1 +1024 +1.91× +8192 +48 +2 +1536 +1.75× +10240 +60 +4 +1792 +1.63× +12288 +80 +8 +2304 +1.55× +16384 +96 +16 +2176 +1.46× +20480 +105 +35 +2528 +1.46× +25600 +128 +64 +3072 +1.47× +Table 10. Weak-scaling speedup for the Transformer models. 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In Proceedings of the +IEEE international conference on computer vision, pp. +19–27, 2015. + +Does compressing activations help model parallel training? +A +MORE EXPERIMENTAL RESULTS +We provide more experimental results in this section. +Distributed Setting +w/o +A1 +A2 +T1 +T2 +T3 +T4 +TP=1, PP=4 +151.82 +154.62 +155.03 +155.78 +155.12 +156.84 +158.58 +TP=2, PP=2 +145.58 +157.49 +163.63 +175.67 +177.39 +186.71 +178.91 +TP=4, PP=1 +136.66 +155.43 +145.97 +170.04 +176.88 +186.06 +190.01 +Distributed Setting +R1 +R2 +R3 +R4 +Q1 +Q2 +Q3 +TP=1, PP=4 +206.89 +273.49 +449.70 +1,292.15 +154.30 +153.65 +152.33 +TP=2, PP=2 +844.66 +1,589.66 +3,915.32 +15,732.57 +178.09 +175.23 +172.93 +TP=4, PP=1 +820.37 +1,588.59 +3,915.52 +15,469.87 +188.10 +168.90 +167.90 +Table 11. The total time (ms) for fine-tuning with various compression techniques by varying the distributed setting. The results are +collected from the AWS p3.8xlarge machine with NVLink by using batch size 32, and sequence length 128. +Distributed Setting +w/o +A1 +A2 +T1 +T2 +T3 +T4 +TP=1, PP=4 +106.04 +113.67 +106.35 +109.58 +109.10 +109.18 +110.57 +TP=2, PP=2 +121.26 +142.41 +140.05 +152.91 +154.60 +162.00 +157.12 +TP=4, PP=1 +122.22 +142.33 +139.47 +171.24 +165.77 +172.69 +170.61 +Distributed Setting +R1 +R2 +R3 +R4 +Q1 +Q2 +Q3 +TP=1, PP=4 +124.39 +137.51 +187.59 +333.61 +108.18 +109.56 +109.49 +TP=2, PP=2 +314.51 +507.00 +998.51 +3,197.42 +163.18 +155.48 +150.31 +TP=4, PP=1 +329.33 +513.89 +1,007.65 +3,406.20 +171.06 +163.96 +152.82 +Table 12. The total time (ms) for fine-tuning with various compression techniques by varying the distributed setting. The results are +collected from the AWS p3.8xlarge machine with NVLink by using batch size 8, and sequence length 128. +Distributed Setting +w/o +A1 +A2 +T1 +T2 +T3 +T4 +TP=1, PP=4 +154.82 +152.50 +153.47 +155.56 +156.01 +156.81 +158.37 +TP=2, PP=2 +184.48 +175.29 +180.35 +206.56 +204.48 +207.66 +214.30 +TP=4, PP=1 +212.76 +201.39 +200.31 +234.16 +240.42 +242.62 +261.39 +Distributed Setting +R1 +R2 +R3 +R4 +Q1 +Q2 +Q3 +TP=1, PP=4 +185.83 +231.78 +368.95 +963.62 +155.33 +154.85 +154.82 +TP=2, PP=2 +684.28 +1,228.36 +2,900.86 +10,499.14 +188.82 +189.14 +194.25 +TP=4, PP=1 +722.87 +1,275.57 +2,973.04 +10,891.70 +225.42 +230.69 +242.42 +Table 13. The total time (ms) for fine-tuning with various compression techniques by varying the distributed setting. The results are +collected from the local machine without NVLink by using batch size 32, and sequence length 128. +Distributed Setting +w/o +A1 +A2 +T1 +T2 +T3 +T4 +TP=1, PP=4 +73.19 +72.94 +72.58 +75.98 +74.15 +73.62 +74.86 +TP=2, PP=2 +100.86 +107.73 +100.54 +113.59 +117.36 +114.86 +112.11 +TP=4, PP=1 +100.73 +107.90 +115.18 +129.31 +124.94 +136.18 +133.91 +Distributed Setting +R1 +R2 +R3 +R4 +Q1 +Q2 +Q3 +TP=1, PP=4 +82.45 +94.84 +123.78 +239.81 +73.33 +74.41 +71.80 +TP=2, PP=2 +235.02 +366.59 +769.47 +2,183.39 +111.61 +106.75 +101.25 +TP=4, PP=1 +238.28 +368.45 +733.03 +2,509.73 +120.14 +114.73 +118.98 +Table 14. The total time (ms) for fine-tuning with various compression techniques by varying the distributed setting. The results are +collected from the local machine without NVLink by using batch size 8, and sequence length 128. + +Does compressing activations help model parallel training? +Compression +Algorithm +MNLI-(m/mm) +QQP +SST-2 +MRPC +CoLA +QNLI +RTE +STS-B +w/o +87.87/88.02 +91.96 +95.18 +87.71 +59.40 +92.99 +76.90 +88.43 +A1 +85.30/85.33 +91.28 +92.32 +84.58 +55.18 +90.87 +59.93 +87.92 +A2 +85.25/85.19 +91.41 +93.23 +86.72 +57.02 +90.92 +64.26 +87.74 +T1 +34.38/34.01 +72.29 +49.54 +70.38 +36.64 +59.89 +53.43 +70.81 +T2 +40.10/38.97 +58.91 +79.24 +66.49 +0.00 +80.40 +45.49 +11.32 +T3 +68.76/69.23 +64.58 +91.40 +80.93 +0.00 +67.34 +66.43 +69.24 +T4 +84.24/85.23 +89.17 +92.09 +81.68 +51.54 +91.71 +63.54 +84.80 +Q1 +86.85/87.58 +91.50 +93.58 +86.96 +59.20 +92.24 +59.57 +86.89 +Q2 +87.46/88.02 +91.82 +94.95 +87.48 +57.02 +93.36 +68.95 +87.84 +Table 15. Fintune results over GLUE dataset under the setting using tensor parallelism size 2, pipeline parallelism size 2, batch size +32, and sequence length 128. F1 scores are reported for QQP and MRPC, Matthews correlation coefficient is reported for CoLA, and +Spearman correlations are reported for STS-B, and accuracy scores are reported for the other tasks. +Compression +Algorithm +MNLI-(m/mm) +QQP +SST-2 +MRPC +CoLA +QNLI +RTE +STS-B +w/o +86.23/86.07 +91.22 +91.74 +88.17 +59.02 +92.09 +78.70 +88.40 +A1 +82.49/82.41 +89.93 +91.85 +82.43 +43.56 +89.84 +47.29 +87.03 +A2 +82.18/82.23 +90.45 +90.52 +83.54 +0.00 +89.02 +62.82 +87.66 +T1 +36.69/38.13 +66.85 +55.32 +68.93 +0.00 +59.13 +52.71 +1.97 +T2 +43.92/43.66 +73.63 +51.26 +62.26 +0.00 +60.13 +49.82 +0.00 +T3 +49.07/47.96 +72.02 +83.57 +69.33 +12.04 +83.60 +55.60 +84.96 +T4 +83.99/84.37 +35.78 +68.30 +83.54 +47.33 +60.52 +64.62 +86.72 +Q1 +84.91/85.18 +90.54 +92.43 +85.91 +53.25 +60.68 +57.04 +87.91 +Q2 +85.66/86.09 +90.99 +91.74 +86.84 +53.92 +91.31 +75.81 +88.19 +Table 16. Fintune results over GLUE dataset under the setting using tensor parallelism size 2, pipeline parallelism size 2, batch size 8, and +sequence length 128. F1 scores are reported for QQP and MRPC, Matthews correlation coefficient is reported for CoLA, and Spearman +correlations are reported for STS-B, and accuracy scores are reported for the other tasks. + diff --git a/B9E0T4oBgHgl3EQfyAKb/content/tmp_files/load_file.txt b/B9E0T4oBgHgl3EQfyAKb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3fe01c96e08a59970ec213a465a7f78eb604c9a0 --- /dev/null +++ b/B9E0T4oBgHgl3EQfyAKb/content/tmp_files/load_file.txt @@ -0,0 +1,1880 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf,len=1879 +page_content='DOES COMPRESSING ACTIVATIONS HELP MODEL PARALLEL TRAINING?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Song Bian * 1 Dacheng Li * 2 Hongyi Wang 2 Eric P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Xing 2 3 4 Shivaram Venkataraman 1 ABSTRACT Large-scale Transformer models are known for their exceptional performance in a range of tasks, but training them can be difficult due to the requirement for communication-intensive model parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' One way to improve training speed is to compress the message size in communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Previous approaches have primarily focused on compressing gradients in a data parallelism setting, but compression in a model-parallel setting is an understudied area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' We have discovered that model parallelism has fundamentally different characteristics than data parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' In this work, we present the first empirical study on the effectiveness of compression methods for model parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' We implement and evaluate three common classes of compression algorithms - pruning-based, learning-based, and quantization-based - using a popular Transformer training framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' We evaluate these methods across more than 160 settings and 8 popular datasets, taking into account different hyperparameters, hardware, and both fine-tuning and pre-training stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' We also provide analysis when the model is scaled up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Finally, we provide insights for future development of model parallelism compression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' 1 INTRODUCTION Transformer models have become the dominant model for many machine learning tasks (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Sharir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' However, state-of-the-art Transformer models have a large number of parameters, making it difficult for a single GPU to hold the entire model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' As a result, training large Transformer models often requires partitioning the model parameters among multiple GPUs, a technique known as model paral- lelism (Shoeybi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Rasley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Model parallelism strategies often introduce significant commu- nication overhead, as demonstrated in Figure 1 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' For instance, the most commonly used tensor model parallelism strategy requires two all-reduce operations over a large tensor in each Transformer encoder block per iter- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' This can greatly increase the overall computational cost of training the model (Shoeybi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' To address the issue of high communication overhead in model parallelism, one approach is to compress the mes- sages communicated among GPUs, such as activation val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' In the data-parallel setting, several prior works have explored compressing gradients to reduce the communica- tion cost of training (Seide et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Bernstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Dettmers, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=', 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content=' Equal contribution 1Department of Computer Science, Uni- versity of Wisconsin-Madison 2Machine Learning Department, Carnegie Mellon University 3MBZUAI 4Petuum Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E0T4oBgHgl3EQfyAKb/content/2301.02654v1.pdf'} +page_content='. Correspon- dence to: Song Bian 0,αk̸=j +i += 0 +� +. +(21) +These local Sobol’ indices are used in the DAL-PCE to determine the cut direction (see Section 3.6). Likewise, +global Sobol’ indices can be obtained easily from weighted summation of local contributions to partial variances +normalized by σ2 +Y as follows: +SX j = +�n� +i=1 Wi +� +αααi∈A +X j +i +β2 +αααi +σ2 +Y +. +(22) +Similarly, global LOO-CV, Q2, of a QoI can be approximated by the weighted summation of the local contributions +as +Q2 = +n� +� +i=1 +WiQ2 +�i, +(23) +where Q2 +�i are obtained from each local PCE using Eq. (9). +These estimates are used throughout the proposed DAL-PCE, as described in detail next. +9 + +3.6. Numerical Algorithm +Based on the presented theoretical background, we now present the numerical algorithm for the domain +adaptive localized PCE. As mentioned above, the whole process can be divided to two iterative tasks: (i) decom- +position of the input random space and (ii) construction of localized PCEs. Both of these tasks are described in +the following paragraphs with specific reference to the steps in Algorithm 1. +Algorithm 1 DAL-PCE: Active Domain Decomposition and Construction of Localized PCEs +Input: maximum local polynomial order p, number of screening global candidates nc,g, number of local +candidates nc,l, number of iterations niter +1: set the minimum number of realizations for local PCE construction nsim ∈ 〈P,2P〉 +2: generate a rich pool of nc,g screening candidates +3: generate the initial ED (size nsim) and construct the initial global PCE +4: for 1 to niter do +5: +identify the sub-domain �i with the highest Θi based on screening candidates +6: +ni ← number of ED samples existing in �i +7: +if ni ≥ nsim then +8: +the identified sub-domain �i becomes a parent �i +9: +identify the direction of the highest first-order Sobol’ index S�i of the parent �i +10: +restrict coordinates of �i → �i and create �⋆ +i +11: +ni ← number of ED samples existing in �i +12: +end if +13: +generate nc,l local candidates in �i +14: +while ni < nsim do +15: +extend size of local ED ni using the local Θc criterion +16: +end while +17: +reconstruct local PCEs in the �i +18: end for +Output: list of subdomains and corresponding PCEs +The first task identifies the important sub-domain �i that should be divided and over which low-order local +PCE should be constructed. The sub-domain �i is specifically identified using the Θi criterion from Eq. (17), +which again incorporates three important characteristics for accurate surrogate modeling – the size of the sub- +domain Wi, the accuracy of the existing local PCE measured by Q2 +�i, and the original Θc criterion measuring the +variance contribution in �i. While Wi and Q2 +�i are computed for the whole sub-domain, Θc is computed at specific +realizations of input random vector. Therefore, it is necessary to cover the sub-domains by a sufficiently large +number of screening candidates, such that the total global number of screening candidates is given by nc,g. Based +on numerical experiments, we recommend nc,g ≥ 1000 M to ensure that each sub-domain contains a sufficient +number of screening candidates. Note that the screening candidates are used only to identify �i [step 5]. They +are not used for the ED, and thus even high nc,g does not bring any additional computational demand. +Once �i is identified, it is necessary to check whether there are enough samples to construct a PCE inside the +sub-domain. We start with finding out how many points belong to the selected domain �i [step 6]. If the number +of samples in the identified sub-domain, ni, is greater than (or equal to) nsim [step 7], a local PCE already exists +for �i. The subdomain is then assigned as a parent �i for division [step 8] and the first-order Sobol’ indices +are estimated by Eq. (22) [step 9]. This identified parent �i is divided in the direction of the highest first-order +Sobol’ index S +X j +�i . The new restricted coordinates of refinement-child �i are identified and the inheriting-child �⋆ +i +is created [step 10]. Further, the number of ED samples ni in the refinement-child �i is determined [step 11]. On +the other hand, if the identified sub-domain �i does not contain enough samples (i.e. ni < nsim), the inherited +PCE from the previous iteration is not sufficiently local (it was trained over a domain that has since been divided) +and it is necessary to add new samples to �i before constructing a new local PCE. +The second task of the proposed algorithm is sequential sampling and adaptive PCE construction in sub- +domain �i. Recall that this domain may be either +10 + +(i) a refinement-child that was just divided but does not contain a sufficient number of points (ni < nsim) or, +(ii) an inheriting-child that now does not contain at least nsim ED samples. +Next, a set of local candidates is generated in region �i [step 13]. To ensure sufficient assessment of the coverage +of the domain, the number of local candidates is empirically recommended as nc,l ∈ 〈3P,5P〉 [1]. From these +candidates, the standard Θc criterion in Eq. (12) is used to iteratively select the best candidates until there are +nsim samples in �i [step 14-16]. This sequential extension of the sample in �i is adaptive in the sense that the +pairwise distances in Eq. (12) between candidates and existing ED points are updated after the addition of each +new point. However, because ni < nsim the local variance densities are estimated from the previously existing +PCE, which cannot be updated until a sufficient number of samples are available in �i. +The last step of each iteration is to construct the local PCE using scaled Legendre polynomials as basis func- +tions (see Eq. (18)) [step 17]. Any non-intrusive technique can be used to estimate the coefficients βββ; we use LARS +and OLS for an adaptive construction of the local PCEs in this paper. At the end of the iteration, all sub-domains +are re-numbered and a list of sub-domains with corresponding PCEs can be exported or the next iteration can be +started. +3.7. Adaptivity in PCE Construction and Domain Decomposition +Adaptivity is central to the proposed DAL-PCE. In the proposed algorithm, there are two types of adaptivity +employed: +(i) adaptivity in PCE construction (selection of the optimal set of basis functions), and +(ii) adaptivity in domain decomposition +Since the PCE can be constructed by any regression technique in each sub-domain, PCE adaptivity is incorporated +by sparse solvers and best model selection algorithms, e.g. Least Angle Regression [32], orthogonal matching +pursuit [33] or Bayesian compressive sensing [34]. Although sparse solvers are often used for PCE with high p, +this adaptivity is also important for reducing the number of basis functions (and thus the minimum number of ED +samples) for high-dimensional examples or, in our case, for very low-size ED in each �i approximated by low-p +local PCE. +The second type of adaptivity is the proposed adaptivity in the domain decomposition. At any point in the +iterative process, the existing ED samples can be used to construct local PCEs or a single global PCE. The DAL- +PCE is not guaranteed to provide a better approximation than the global PCE. This can be measured via Q2, +specifically by computing Q2 +local from Eq. (23) and Q2 +global from a single global PCE according to Eq. (9). If +Q2 +local > Q2 +global at a given iteration, the domain decomposition is deemed to be poor and the whole decomposition +process is re-started. That is, the complete geometrical decomposition is forgotten and all existing ED points +are taken as an initial ED for a brand new run of the algorithm. This is illustrated in Fig. 3 which shows the +decomposition (top) and the associated error (bottom) right before the restart a) at Nsim = 181, b) the new +decomposition and error right after the restart, and c) the final decomposition/error which shows significant +improvement over the global PCE. These histories show the standard R2 error defined in Eq. (24). It is not +necessary to check this criterion at every iteration, but it is suggested to check it periodically, every nr steps, to +ensure adequate local refinement. +3.8. Stopping Criteria +The proposed DAL-PCE algorithm can be fully automated by adding an adequate stopping criterion. A simple +but practical stopping criterion is based on computational budget, i.e. once the total number of model evaluations +Nsim or number of iterations niter have reached a critical level/budget. One may also use a stopping criterion +based on decomposition pattern, e.g. the smallest or the largest volumes of any subdomain, to ensure a desired +resolution. Valuable stopping criterion can be also obtained directly from Q2, corresponding to a target/threshold +level of achieved accuracy. Regardless of the selected stopping criteria, it can easily be applied before step 5 of +the proposed algorithm (start of each iteration). +11 + +Figure 3: Illustration of domain decomposition restart. a) decomposition and error evolution prior to restart, b) rebuilt decomposition and +error drop right after the restart, c) final decomposition and error showing that the restart unlocks a dramatic decrease in approximation +error. +4. Numerical Experiments +The proposed DAL-PCE is presented on four numerical examples of increasing complexity and which illus- +trated different aspects of the approach. The obtained results are compared (a) to the standard global PCE +approach with adaptive maximum order p ∈ [5,25] and (b) to SSE [2], as current state-of-the-art non-intrusive +surrogate modeling technique based on the domain decomposition. The PCE is constructed using the UQPy pack- +age [36] and the original implementation of SSE is used from the UQLab package [37]. To compare methods, +the relative mean squared errors ε are calculated for all three approximations ˜f on a validation set containing +a large pool of 106 integration points generated by crude Monte Carlo according to: +ε(XXX) := +� +�� +f (XXX) − ˜f (XXX) +�2� +� +� +f (XXX) +� +, +(24) +where �[] and �[] are the mean value and variance operators, respectively. +To show representative results of the proposed DAL-PCE algorithm, the calculations were repeated 100 times, +and the same settings of the algorithm for all examples were selected as follows: maximum local polynomial +degree p = 2, number of global candidates nc,g = 1000 M, number of local candidates nc,l = 5P, minimum +number of samples for local PCE construction nsim = 1.5P, minimum number of iterations before checking for +restart nr = 20, and βββ are obtained by LARS and OLS algorithm. Minimum number of samples in sub-domains +required to justify an expansions for SSE was set identically to DAL-PCE and polynomial order is adaptively +selected in the range p ∈ [2,6]. Since the SSE is not a sequential approach, the presented results were obtained +for 10 discrete sample sets of increasing size to compare convergence of the method. Note that all samples and +candidates are generated by LHS for all compared approaches, though it was shown [1] that for the variance- +based sequential sampling, it is significantly better to use advanced techniques such as Coherence D-optimal +sampling [41]. +12 + +4.1. One-dimensional Toy Example +The first example involves a simple 1D function [2] that is extremely difficult to approximate with PCE due +to the third, highly nonlinear “exp” term: +f (X) = −X + 0.1sin(30X) + exp(−(50(X − 0.65))2), +X ∼ U[0,1]. +(25) +The poor performance of a single global PCE learned from 200 samples is depicted by the blue line in Fig. 4c +where it is clear that a single global PCE is not able to accurately approximate the function even for a high +number of samples and high maximum polynomial order p ∈ [5,25]. This function was originally developed to +demonstrate the efficiency of SSE based on domain decomposition and thus it is a natural choice for comparison +of the proposed DAL-PCE and SSE. +Fig. 4a-b show a typical realization of the DAL-PCE where the algorithm sequentially decomposes the domain +and adds additional samples to the ED. Specifically shown are the 4th and 11th iterations. The boundaries of +sub-domains are represented by blue vertical lines and red dots show the positions of samples in the ED. Once +the algorithm discovers the highly nonlinear region (the steep peak caused by exp), it progressively refines this +region and adds more samples there as a result of the high variance density. Of course, these figures show only +one realization of the algorithm and the decomposition is dependent on the initial ED. Therefore, it is necessary +to repeat the algorithm many times with random initial ED to assess convergence. Fig. 4d shows convergence +Figure 4: (a), (b) The adapted domain and ED before (iteration 4) and after (iteration 11) exploration and discovery of the exponential part +of the mathematical model. (c) Final surrogate models from global PCE and DAL-PCE. (d) Convergence plot comparing the mean square +error for global PCE SSE, and DAL-PCE. The convergence plots for Global PCE and DAL-PCE show continuous mean value ±σ intervals +from 100 repeated trials, while those for SSE are plotted for several discrete ED sizes. +of the error ε from 100 repeated trials. The single global PCE is unable to accurately approximate the original +function even when using high p and thus the ε does not converge, as expected. Both methods based on domain +decomposition (DAL-PCE and SSE) achieve great accuracy already for 200 samples. However, the DAL-PCE +consistently has 1–2 orders of magnitude higher accuracy than SSE for the given number of samples. Moreover, +increase in variance of ε is, in general, slower in DAL-PCE than in SSE. Fast increment in variance of SSE can +be seen also in the original paper [2]. Finally, we again observe that convergence is continuous with DAL-PCE, +where convergence can only be assessed at discrete sample sizes with SSE through a new analysis. All of these +13 + +Figure 5: Results for the 2-dimensional Singularity function: a) original mathematical model, b) approximation via DAL-PCE (background +color), current domain division and the corresponding ED, c) local LOO-CV Q2 +�i and Θi value for each sub-domain, d) convergence plots for +DAL-PCE, Global PCE, and SSE showing the mean value and ±σ interval. Convergence plots for SSE show the mean ±σ at discrete sample +sizes. +advantages of the DAL-PCE can be attributed to the active learning, which both explores the space and exploits +the behavior of the function to decompose the domain and add samples. Although active learning might lead to +lower accuracy (higher ε) initially (for small nsim = 10–20) as it is dominated by exploration, it rapidly improves +once it identifies important features and begins to favor exploitation. +4.2. Two-dimensional Singularity +The second example involves a 2D function with mirrored quarter-circle arc line singularities [1]. The form +of the function is give by: +f (XXX) = +1 +|0.3 − X 2 +1 − X 2 +2| + δ − +1 +|0.3 − (1 − X1)2 − (1 − X2)2| + δ, +XXX ∼ U[0,1]2, +(26) +where the strength of the singularities is controlled by the parameter δ, which we set as δ = 0.1. The singularities +in this example represent a challenging task for a global PCE even with high order, due to the well-known Gibbs +phenomenon [49]. It is thus beneficial to identify the location of the singularity, locally decompose the domain, +and construct low-order local PCEs. +Fig. 5 illustrates the decomposition and DAL-PCE approximation at a given stage of the computation. Panel +a) visualizes the true values of the function via a background color. The same coloring scheme is used in panel b) +for the pointwise information available in the current ED (small circles) and for the function approximation via +DAL-PCE by the background color. Panels b) and c) show also the final domain decomposition. The symmetry +14 + +Figure 6: Results for the 2-dimensional discontinuiy function: a) original mathematical model, b) approximation via DAL-PCE and ED, c) +local LOO-CV Q2 +�i and Θi value for each sub-domain, d) convergence plots for DAL-PCE, Global PCE, and SEE showing the mean value and +±σ interval. Convergence plots for SSE show the mean ±σ at discrete sample sizes. +in the decomposition documents the great convergence of the DAL-PCE thanks to an adaptive decomposition +described in the previous section. Plot c) shows the local Q2 +�i error in each individual sub-domain (darker color +corresponds to higher local error). These local errors clearly show localization of the prediction error to very +small areas near singularities, which are continually being refined. The color of the small solid squares in the +center of each sub-domains shows the Θi value for that sub-domain. +Finally, the convergence plot in Fig. 5d) shows that both DAL-PCE and SSE outperform the global PCE, as +expected. The SSE performs comparable to or slightly better than DAL-PCE for small Nsim, but the DAL-PCE +begins to outperform SSE as Nsim grows thanks to the active learning approach that targets samples in the vicinity +of the singularities. Note that the error converges for both SSE and DAL-PCE as we approach 1000 samples and +does not seem to substantially reduce after this. This is due to the fundamental limitation of trying to approximate +this singularity, even locally, with low-order polynomials. +4.3. M-dimensional Discontinuity +The third example investigates the role of dimensionality on the performance of the proposed DAL-PCE. The +following discontinuous function is defined for an arbitrary number of input random variables M [26]: +f (XXX) = +� +sin(X1π)sin(X2π) +if x1 ≤ 0.5 and x2 ≤ 0.5 +�M +i=3 Xi +otherwise +, +XXX ∼ U[0,1]M. +(27) +This function has a discontinuity in the first two input random variables, which can be seen in Fig. 6a. A single +global PCE cannot accurately approximate the function because of the discontinuity, although the function f (XXX) +15 + +Figure 7: Convergence plots for the M-dimensional function: a) 3-dimensional version, b) 5-dimensional version, c) 6-dimensional version, +and d) 8-dimensional version. Convergence plots for the DAL-PCE and global PCE show the mean value ±σ interval. Convergence plots +for SSE also show the mean ±σ, but at discrete sample sizes. +can be easily approximated by two separate PCEs in the two regions for which the definitions differ. But, this +requires a priori knowledge of the discontinuity location. Since the location of the discontinuity is assumed to be +unknown, this function is a good example for domain adaptation using DAL-PCE. +The detailed results for a 2D version of this problem are depicted in Fig. 6 in identical form as in the previous +example. Note that the local Q2 +i errors Fig. 6c show perfect accuracy in the part of the input random space where +f (XXX) = 0 and thus the associated sub-domains are not preferred for further decomposition. The convergence +plot in Fig. 6d confirms that a single global PCE is not able to create an accurate approximation and adding +more points to ED does not lead to significant improvements in the approximation. The mean values of errors +ε associated to the proposed DAL-PCE approach are significantly lower in comparison to SSE (1–2 orders of +magnitude) similarly as in the first example, though the convergence trend is similar for both methods. SSE, +however, uses a random splitting routine. This can lead to very high variance of results, since the accuracy is +highly dependent on the pattern of the decomposed input random space. This clearly shows the advantage of an +active learning approach. +The influence of dimensionality M on convergence of the DAL-PCE, SSE, and global PCE is studied in Fig. 7 +for a) 3, b) 5, c) 6, and d) 8 input random variables. As the domain dimension increases, the linear part of the +function f (XXX) occupies an increasing proportion of the domain while the discontinuity remain low-dimensional. +The proposed DAL-PCE greatly improves the convergence because it is able to identify an ideal decomposition +and local samples to resolve the discontinuity. For low-dimensions (M = 2,3), SSE error ε shows a decreasing +trend that is better than global PCE but has an extremely high variance. This is caused by a lack of control in +sample placement. The domain decomposition in SSE is a product of sample location and without active learning +to guide sample placement, SSE will sometimes produce a very good decomposition and sometimes a very poor +decomposition. Meanwhile, the proposed DAL-PCE errors have comparably low variance for low-dimensions +and consistently have accuracy comparable to, or better than, the best SSE realizations. +As the dimension, M, increases the DAL-PCE is able to maintain a very high level of accuracy, while the +accuracy degrades completely for the SSE such that it is comparable to the global PCE. The DAL-PCE is able +to maintain its low error because the discontinuity remains low-dimensional and the active learning process is +able to target this region for domain refinement and sampling. This means that the DAL-PCE remains largely +independent of the problem dimension, and instead depends predominantly on the intrinsic dimension of the +16 + +Figure 8: Convergence plots for the modified M-dimensional function: a) 3-dimensional version, b) 5-dimensional version, c) 6-dimensional +version, and d) 8-dimensional version. Convergence plots for the DAL-PCE and global PCE show the mean value ±σ interval. Convergence +plots for SSE also show the mean ±σ, but at discrete sample sizes. +discontinuous/nonlinear features of the model. The performance of SSE, on the other hand, degrades with +dimension because its domain decomposition depends only on a set of a priori specified points that are not selected +in a way that is aware of the important features of the model. Consequently, as the dimension increases the +algorithm becomes less likely to refine the domain appropriately around an embedded low-dimensional feature. +We remark that this desirable scalable convergence trend of the DAL-PCE is not likely a universal property, as +the trend may break down in problems where the intrinsic dimension of the discontinuity/nonlinearity is high or +where the discontinuity occupies a very small proportion of the domain – in which case exploration of the space +to find the important feature may take a very large number of samples. +In the present example, the discontinuity in the function given in Eq. (27) lies at x1 = 0.5 and x2 = 0.5, which +corresponds to the exact location where the domain will be split for both SSE and during the early iterations of +the DAL-PCE. One might argue that this presents an unreasonable advantage for the proposed algorithm. We +therefore modified the function such that the discontinuity lies at x1 = 0.61 and x2 = 0.61. Fig. 8 shows the +convergence for the DAL-PCE and SSE for this modified function with varying dimension, M. The absolute errors +ε exhibit slower decrease, especially for dimensions M = 3 and M = 5. However, the proposed active learning +still leads to superior results (especially for higher dimensions as in the previous case). Note that there are visible +spikes in the DAL-PCE convergence graph for the 3-dimensional example. Although the results were statistically +processed, these spikes are caused by the restart adaptivity occurring at the same Nsim in each replication. In +this case, the optimal decomposition pattern is very complicated and therefore the algorithm activates the restart +adaptivity frequently (after multiples of nr steps), until it finds a suitable pattern to continue convergence. SSE +in the 3- and 5-dimensional cases has higher mean error and significantly lower variance in comparison to the +previous example. This is caused by the fact that the modified discontinuity location no longer lies along the +boundary of the domain decomposition. In the previous example, some SSE realizations achieved near-perfect +accuracy because the domain was coincidentally divided along the discontinuity. +This phenomenon is investigated more closely in Fig. 9, which compares number of outliers in both versions +of 3D examples. In addition to the mean ±σ seen previously, the figure also shows standard boxplots for SSE +(median along with lower and upper quartiles) and the corresponding number of “extreme” realizations produc- +ing very high accuracy (top axis) for a) the original position of discontinuity; and b) discontinuity at x1 = 0.61 +17 + +Figure 9: Convergence plots for DAL-PCE and SSE with additional boxplots for SSE showing the median, lower and upper quartiles and +outliers for: a) the 3D example with discontinuity at x1 = 0.5 and x2 = 0.5, b) the 3D example with discontinuity at x1 = 0.61 and x2 = 0.61. +and x2 = 0.61. As can be seen, in panel a) there are many outliers producing ε < −7, which effectively decreases +µ relative to the median while also significantly increasing the variance. In contrast DAL-PCE has no outliers and +it leads to very consistent results. In panel b), there are no outliers for either SSE or DAL-PCE and the results +are thus consistent with low variance for both methods. +4.4. Asymmetric shallow von Mises truss +In this section, we demonstrate the relevance of the proposed method for a representative engineering exam- +ple exhibiting discontinuous response. Consider the shallow two-bar planar truss subjected to a vertical load at +its top joint, as presented in [50] and illustrated in Fig. 10a. +The truss is formed by two prismatic bars made of a hard wood (density 800 kg/m3, modulus of elasticity +E = 12 GPa). There are two variables in the studied von Mises truss: (i) the loading vertical force F, and (ii) +a half sine-wave imperfection of the left bar having magnitude δ, see the sketch in Fig. 10a. The load is applied +dynamically as a step function at time zero for an unlimited duration. The structure is modeled, as illustrated +in Fig. 10b. In particular, the mass of the bar is concentrated in 21 mass points, including the supports and +the loading point. These mass points are connected via 10 + 10 translational springs representing the normal +stiffness of the true bars. The pairs of the axial members are connected via rotational spring having zero moment +for a zero angle between adjacent bars. The only exceptions are the loading ans support points where there +are no rotational springs attached (hinges). The damping is associated with the mass points via linear viscous +damping coefficient set to 11 N · s/(kg · m) approximating the relative damping of about 3%. Explicit dynamics +solver FyDiK [51, 52] was used to solve the equations of equilibrium at the mass points. The numerical solution +lasts to up to two seconds, which is the time needed for almost complete stabilization of the solution (kinetic +energy drops below a negligible threshold). +Since the structure is very shallow, sudden application of the vertical force can cause snap-through buckling, +wherein the loading point drops down between the supports and the members switch from a state of compression +to tensile stresses in the final stable state. We specifically study the horizontal coordinate yF of the loading point +after the dynamic response stabilizes to the final deformed shape. The force F ∈ (31.6,772.6) kN and initial +imperfection δ ∈ (−0.4,0.4) m are treated as uniform random variables mapped to the unit square such that +the model input X ∼ U[0,1]2. Because of the potential snap-through buckling, the solution is discontinuous as +illustrated in Fig. 10c. On each side of the discontinuity, the solution yF is smooth and slowly-varying having +values near +1 m and -1 m, respectively. Note that the output is not symmetric with respect to δ = 0 because the +dynamical response evolves differently for concave and convex initial displacements. +The sharp boundary between the buckled and unbuckled regions, shown in Fig. 11a cause global PCE to +produce poor approximations that are vulnerable to the Gibbs phenomenon, similar to the example in subsection +18 + +b) +a) +c) +Figure 10: Asymmetric shallow von Mises truss. a) Initial geometry with two random variables F and δ; b) illustrative sketch of the discrete +dynamical model and the meaning of output variable yF, c) illustration of the discontinuous response function of the two input variables. +Figure 11: Results for the von Misses truss example: a) original mathematical model (numerical solution), b) approximation via DAL-PCE +and ED, c) local LOO-CV Q2 +�i and Θi value for each sub-domain, d) convergence plots for DAL-PCE, Global PCE, and SSE showing the +mean value and ±σ interval; convergence plots for SSE show the mean ±σ at discrete sample sizes. +4.2. This is shown by the convergence plots in Fig. 11d comparing global PCE, DAL-PCE, and SSE. Clearly, the +complexity of this example and the complicated shape of the discontinuity limits the accuracy of all the surrogate +models. The proposed DAL-PCE achieves low accuracy for small sample sizes because the corresponding small +number of sub-domains and low-order PCEs are unable to sufficiently approximate the boundary. Therefore, the +global PCE and SSE (with a low number of embedding levels) are initially better. With increasing number of +samples, the proposed DAL-PCE approach leads to superior results because the active learning is able to resolve +the discontinuity as illustrated in Fig. 11b, which shows the domain decomposition and approximation after +2000 samples. Fig. 11c shows the corresponding LOO-CV errors for each subdomain, demonstrating the errors +are confined to small, localized regions near the boundary. +19 + +5. Discussion & Future Work +The proposed DAL-PCE approach is a general methodology for the decomposition of the input random space +and construction of localized PCEs using active learning. The proposed active learning is based on a novel Θ +criterion that optimally balances global exploration with local exploitation of the model. Although this paper +presents one specific learning algorithm, the methodology is general and amenable to modifications to reflect the +specific user’s needs. The whole process can be divided into two tasks: A) decomposition of the input random +space and B) construction of localized PCEs; and both can be easily modified as discussed further: +A) The most important sub-domain �i is identified by extended Θ according to Eq. (17) evaluated for a large +number of global candidates. In this paper, we use standard LHS for candidate generation, but it may +be beneficial to use different sampling methods that produce more uniform coverage of the whole input +random space (see e.g. [53, 54, 45]). Although it is generally possible to generate a large number of +candidates, it might be challenging to uniformly cover the entire input random space, especially in high +dimensions. Thus, one can use any sampling technique suitable for a specific example, e.g. [55]. +Once the �i is identified via Eq. (17), it is either divided (providing it contains enough ED points) or +the sample is extended inside it, to achieve a better PCE approximation. The simplest division occurs by +splitting the volume into two parts of identical hypervolume in the direction of the highest first-order Sobol’ +index. However, the algorithm can accommodate various different approaches. For example, it is possible +to divide the �i into a higher number of sub-domains, not just two. Moreover, instead of splitting the +domain into parts of equal hypervolume, other criteria can be used. For example, the cutting plane can be +positioned so to split the domain variance into equal parts. +B) The user can choose to employ any existing method to construct the non-intrusive PCEs, including various +sparse solvers or adaptive algorithms, which may be preferable for certain applications [12]. For example, +we use LARS with OLS. However, it is generally more efficient to use active learning based on the Θ criterion +for PCE as shown in [1], which employs variance-based sequential sampling. This improvement can be +integrated within the DAL-PCE to make local PCE more efficient in each subdomain, and thereby improving +the overall convergence. The can be compounded by the use of advanced sampling techniques within the +subdomains such as Coherence D-optimal sampling [40, 41]. +As seen from the previous paragraphs, the whole algorithm can be adapted for specific needs reflecting the +characteristics of a given mathematical model, such as dimensionality, sparsity, non-linearity etc., by simply ex- +changing components of the proposed algorithm for suitable existing (or new) techniques. Note that even after +the modification, the whole methodology based on Θ criterion is still valid and can be used for uncertainty +quantification and surrogate modelling as described in this paper. Moreover, in comparison to SSE, the DAL- +PCE sequentially adds points and divides the sub-domains one-by-one based on information obtained from the +previous iteration. +Another significant advantage of the DAL-PCE is that it provides estimates of the local errors, Q�i, associ- +ated with each sub-domain. Since localized PCEs are constructed independently, local errors estimate the local +accuracy of the surrogate model directly, and can be assembled to provide global error measures. Naturally, local +accuracy is very important information that can be used for further probabilistic analysis and active learning. +Although this paper does not propose any specific approach for further processing of this information, it could +serve as a main ingredient for various active learning algorithms. For example, it could be directly used to predict +uncertainty in industrial applications and possibly extend the ED in a sub-domain of interest. +Finally, an important topic of further research is to study the behavior of the proposed criterion in higher +dimensions. In particular, the geometrical terms l M +c,s and �i likely cause poor convergence in high dimensions. +Although some preliminary results focused on investigating of l M +c,s in high dimensions was previously performed in +the paper [1] proposing the original Θ criterion, it is still necessary to perform an extensive study of its behavior +as well as investigating the influence of �i, which may need to be reformulated for high dimensions. +20 + +6. Conclusion +The paper presented a novel approach, domain adaptively localzed PCE, for the adaptive sequential con- +struction of localized PCEs based on active learning and decomposition of the input random space. It combines +adaptive sequential sampling based on the recently proposed Θ criterion to maintain the balance between ex- +ploration of the input random space and exploitation of the current characteristics of the PCE together with the +adaptive sequential decomposition of the input random space creating sub-domains approximated by local sur- +rogate models. The methodology offers a general technique that can be easily adapted or modified for specific +functions extending its applicability. The performance of the proposed methodology was validated on several nu- +merical examples of increasing complexity investigating different aspects of the algorithm and leading to superior +results in comparison to a single global PCE and the recently proposed SSE. +Acknowledgments +The first author acknowledge financial support provided by the Czech Science Foundation under project num- +ber 22-00774S. 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Mašek, Distance-based optimal sampling in a hypercube: Energy potentials for high-dimensional and low-saturation +designs, Advances in Engineering Software 149 (2020) 102880. doi:10.1016/j.advengsoft.2020.102880. +23 + diff --git a/B9FRT4oBgHgl3EQfvjiF/content/tmp_files/load_file.txt b/B9FRT4oBgHgl3EQfvjiF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b2303f2b20f98a2cdbb73c1bd251bece2bf70f2 --- /dev/null +++ b/B9FRT4oBgHgl3EQfvjiF/content/tmp_files/load_file.txt @@ -0,0 +1,1043 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf,len=1042 +page_content='Highlights Active Learning-based Domain Adaptive Localized Polynomial Chaos Expansion Lukáš Novák, Michael D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Shields, Václav Sadílek, Miroslav Voˇrechovský Effective construction of a general purpose surrogate model based on polynomial chaos expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Novel method for sequential decomposition of the input random space and construction of local approxi- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Sequential domain decomposition and sample size extension based on an active learning methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Active learning is represented by variance-based Θ criterion developed for polynomial chaos expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='13635v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='LG] 31 Jan 2023 Active Learning-based Domain Adaptive Localized Polynomial Chaos Expansion Lukáš Novák∗ Brno University of Technology, Brno, Czech Republic Michael D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Shields Johns Hopkins University, Baltimore, USA Václav Sadílek, Miroslav Voˇrechovský Brno University of Technology, Brno, Czech Republic Abstract The paper presents a novel methodology to build surrogate models of complicated functions by an active learning- based sequential decomposition of the input random space and construction of localized polynomial chaos expan- sions, referred to as domain adaptive localized polynomial chaos expansion (DAL-PCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The approach utilizes sequential decomposition of the input random space into smaller sub-domains approximated by low-order poly- nomial expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This allows approximation of functions with strong nonlinearties, discontinuities, and/or sin- gularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Decomposition of the input random space and local approximations alleviates the Gibbs phenomenon for these types of problems and confines error to a very small vicinity near the non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The global behavior of the surrogate model is therefore significantly better than existing methods as shown in numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The whole process is driven by an active learning routine that uses the recently proposed Θ criterion to assess local variance contributions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The proposed approach balances both exploitation of the surrogate model and exploration of the input random space and thus leads to efficient and accurate approximation of the original mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The numerical results show the superiority of the DAL-PCE in comparison to (i) a single global polynomial chaos expansion and (ii) the recently proposed stochastic spectral embedding (SSE) method [2] developed as an accurate surrogate model and which is based on a similar domain decomposition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This method represents general framework upon which further extensions and refinements can be based, and which can be combined with any technique for non-intrusive polynomial chaos expansion construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Keywords: Polynomial Chaos Expansion, Adaptive Sampling, Sequential Sampling, Local Approximations, Active Learning, Stochastic Spectral Embedding 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Introduction The Polynomial Chaos Expansion (PCE), originally proposed by Norbert Wiener [3] and further investigated in the context of engineering problems by many researchers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' [4, 5], is a preferred method for uncertainty quantification (UQ) and surrogate modeling in industrial applications [6, 7] thanks to its efficiency and powerful post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Once a PCE is available for a given problem, the constructed explicit function can be exploited ∗Corresponding author Email addresses: novak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='l@fce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='vutbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='cz (Lukáš Novák), michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='shields@jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='edu (Michael D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Shields), sadilek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='v@fce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='vutbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='cz (Václav Sadílek), vorechovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='m@vut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='cz (Miroslav Voˇrechovský) Preprint submitted to Computer Methods in Applied Mechanics and Engineering February 1, 2023 to directly estimate important properties of the original problem including its statistical moments, response prob- ability distribution or sensitivity indices (without additional sampling [8]), which brings significant efficiency for surrogate modeling, sensitivity analysis, uncertainty quantification and reliability analysis [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The PCE, in its non-intrusive form, offers a convenient way to perform probabilistic analysis of any black-box model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' finite element models representing complex physical systems in engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' There are generally two types of non-intrusive methods to calculate the deterministic PCE coefficients: spectral projection and lin- ear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The spectral projection approach utilizes the orthogonality of the multivariate polynomials and calculates the coefficients using inner products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The spectral projection leads to an explosion of computational complexity referred to as the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Therefore, the non-intrusive approach based on linear re- gression is often preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although it is typically less expensive than the spectral projection (the number of samples should be at least � (P ln(P)), where P is the number of terms in the PCE [10, 11]), it suffers from the curse of dimensionality as well, since the number of PCE terms grows rapidly with both dimension and maximum polynomial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Therefore, it becomes necessary to employ advanced adaptive techniques to construct sparse PCEs that yield efficient solutions for real-world physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Regression-based PCE can be significantly affected by the selected sampling scheme, as was recently shown in an extensive review paper [12] comparing several general statistical sampling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' However, PCE construction as a linear regression model is a very problem specific task and it can be highly beneficial to use methods that exploit information from the given mathematical model and sequentially update the surrogate model – referred to as active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Active learning is a common approach for surrogate-based reliability analysis, wherein an initial experimental design is iteratively updated based on the current estimate of the limit- state surface [13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Active learning for reliability analysis with PCE was used e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' in [16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' For general UQ studies, some recent studies have focused on general sequential sampling for PCE based on space- filling criteria or alphabetical optimality [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' However, it is beneficial to use both exploitation (leveraging model behavior) criteria and exploration (space filling) criteria to define an optimally balanced criterion [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Such sequential sampling for sparse Bayesian learning PCE combining both aspects – epistemic uncertainty of the statistical inference (exploration) together with quadratic loss function (local exploitation) – was recently proposed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' However, its application is limited to PCE built by sparse Bayesian learning only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The authors of this paper recently proposed a general active learning method based on sequential adaptive variance-based sampling [1], which is an efficient tool for accurate surrogate modeling that is sufficiently general for further extension [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although this approach leads to superior results in comparison to standard approaches without active learning, it is limited by the inherently smooth nature of the PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' More specifically, polynomial basis functions are not able to approximate functions with discontinuities or singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Moreover, it is nec- essary to use high-order polynomials to approximate functions with local non-linearities, even when the rest of the input random space could be easily approximated by a low-order PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This can lead to spurious oscillations in the approximation and over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' To overcome this limitation, we propose a method to construct localized PCEs based on the concept of divide-and-conquer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' decomposition of the input random space to sub-domains approximated by many low-order PCEs instead of a single high-order global PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although this concept is not entirely new in stochastic finite elements [24] and stochastic collocation [25, 26], there is no such approach for non-intrusive PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' However there are two primary techniques based on similar concepts as described in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Related Developments Stochastic Spectral Embedding (SSE) [2] is a general approximation technique based on a decomposition of the input random space and the construction of embedded local approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although it is generally possible to use any spectral approximation technique, it is beneficially coupled with PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' SSE is based on a novel idea of embedding – instead of constructing local approximations of the original mathematical model, local surrogates are constructed to approximate the residuals between the model and approximation from the previous level of the decomposed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although such an approach can lead to significant improvement in comparison to a single global approximation [2], it is not a sequential approach based on active learning and thus it does not iteratively reflect new information obtained from the previous steps of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Active learning is crucial in analysis of functions with discontinuity or singularity because it allows for the aforementioned exploration and exploitation necessary to find and resolve these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' For the sake of completeness, active learning for SSE has been 2 proposed for reliability analysis [27], but it does not lead to an accurate approximation over the entire input random space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Its accuracy is limited to regions around the limit surface, which are important for an estimation of failure probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The second related technique is Multi-element generalized Polynomial Chaos Expansion (ME-gPC) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' ME- gPC was developed as an extension of generalized PCE based on Wiener-Askey scheme [29] allowing analysis of models with arbitrary distribution of input random vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The ME-gPC method consists of three main parts: de- composition of the input random space, numerical construction of locally orthogonal polynomials and an adaptive procedure based on the decay rate of local error in estimated variance derived from local PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' ME-PCE applies an h-type mesh refinement procedure akin to mesh refinement in finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' By doing so, they introduce a structured grid of uniform points in each new element and solve for the PCE coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This can be cumbersome and does not afford the flexibility to adaptively select sparse and near-optimal training points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Moreover, we note that the ME-gPC was created mainly for uncertainty propagation in models with arbitrary input distributions, and thus in contrast to SSE, its objective is not necessarily to construct the best possible sur- rogate model using adaptive algorithms, but rather to minimize errors in response statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This is a subtle, but important difference that distinguishes its use as a predictive tool from that of a tool for statistical estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Contributions of this paper This paper describes a novel method, termed Domain Adaptive Localized PCE (DAL-PCE) that applies adap- tive sequential decomposition of the input random space and adaptive sequential sampling within the sub- domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Both of these features are based on recently a proposed criterion for variance-based sequential sta- tistical sampling, developed specifically for PCE in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In the context of previously described methods SSE and ME-gPC, the proposed novel approach can be though to lie between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Like SSE, it is developed specifically for the construction of accurate surrogate models, especially for functions with high non-linearity or disconti- nuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' But the decomposition of the input random space is rather similar to ME-gPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The uniqueness of our proposal lies in the combination of active learning, sequential sampling, sequential decomposition of the input space and regression-based PCE using sparse solvers such as Least Angle Regression (LARS) allowing adaptivity and learning in each iteration of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Polynomial Chaos Expansion Assume a probability space (Ω,F,P), where Ω is an event space, F is a σ-algebra on Ω and P is a probability measure on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' If the input variable of a mathematical model, Y = f (X), is a random variable X(ω),ω ∈ Ω, the model response Y (ω) is also a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Assuming that Y has a finite variance, PCE represents the output variable Y as a function of an another random variable ξ called the germ with a known distribution Y = f (X) = f PCE(ξ), (1) and represents the function f (X) via infinite polynomial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' A set of polynomials, orthogonal with respect to the distribution of the germ, are used as a basis of the Hilbert space L2 (Ω,F,P) of all real-valued random variables of finite variance, where P takes over the meaning of the probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The orthogonality condition is given by the inner product of L2 (Ω,F,P) defined for any two functions ψj and ψk for all j ̸= k with respect to the weight function pξ (probability density function of ξ) as: 〈ψj,ψk〉 = � ψj(ξ)ψk(ξ)pξ(ξ) dξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (2) This means that there are specific orthogonal polynomials associated with the corresponding distribution of the germ via its weighting function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' For example, Hermite polynomials orthogonal to the Gaussian measure are associated with normally distributed germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Orthogonal polynomials corresponding to other distributions can be chosen according to Wiener-Askey scheme [29] or constructed numerically [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' For further processing, it is beneficial to use normalized polynomials (orthonormal), where the inner product of ith and jth polynomials is equal to the Kronecker delta δjk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' δjk = 1 if and only if j = k, and δjk = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 3 In the case of XXX and ξ being vectors containing M independent random variables, the polynomial Ψ(ξ) is multivariate and it is built up as a tensor product of univariate orthonormal polynomials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Ψααα(ξ) = M � i=1 ψαi(ξi), (3) where ααα ∈ �M is a set of integers called the multi-index reflecting polynomial degrees associated to each ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The quantity of interest (QoI), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' the response of the mathematical model Y = f (XXX), can then be represented as [5] Y = f (XXX) = � ααα∈�M βαααΨααα(ξ), (4) where βααα are deterministic coefficients and Ψααα are multivariate orthonormal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Non-intrusive computation of PCE coefficients For practical computation, the PCE expressed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (4) must be truncated to a finite number of terms P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' One can generally choose any truncation rule (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' tensor product of polynomials up to the selected order p), but the most common truncation is achieved by retaining only terms whose total degree |ααα| is less than or equal to a given p, in which case the truncated set of PCE terms is then defined as AM,p = � ααα ∈ �M : |ααα| = M � i=1 αi ≤ p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (5) The cardinality of the truncated index set AM,p is given by card AM,p = (M + p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' M!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' ≡ P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (6) When the PCE is truncated to a finite number of terms, there is an error ϵ in the approximation such that Y = f (XXX) = � ααα∈A βαααΨααα(ξ) + ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' From a statistical point of view, PCE is a simple linear regression model with intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Therefore, it is possible to use ordinary least squares (OLS) regression to minimize the error ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Knowledge of vector βββ fully characterizes the approximation via PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' To solve for βββ, first it is necessary to create Nsim realizations of the input random vector XXX and the corresponding results of the original mathematical model Y, together called the experimental design (ED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Then, the vector of P deterministic coefficients βββ can be determined by OLS as βββ = (Ψ TΨ)−1 Ψ TY, (7) where Ψ is the data matrix Ψ = � Ψi j = Ψj(ξ(i)), i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=', Nsim, j = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=', P − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (8) A well-known problem, the curse of dimensionality, states that P is highly dependent on the number of input random variables M and the maximum total degree of polynomials p, which is clear from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Considering that estimation of βββ by regression requires at least � (P ln(P)) number of samples for stable solution [10, 11], the problem can become computationally highly demanding in case of a large or strongly non-linear stochastic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although one can use advanced model selection algorithms such as Least Angle Regression (LAR) [32, 4], orthogonal matching pursuit [33] or Bayesian compressive sensing [34] to find an optimal set of PCE terms, and thus reduce the number of samples needed to compute the unknown coefficients, the benefit of these techniques is significant only if the true coefficient vector is sparse or compressible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The sparse set of basis functions obtained by any adaptive algorithm is further denoted by A for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Approximation Error Estimation Once the PCE is constructed, it is crucial to estimate its accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Further, the PCE accuracy can be used to directly compare several PCEs to choose the best surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Ideally the ED should be divided into validation and training sets, but this might be extremely computationally demanding in engineering applications with complex numerical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Therefore in the field of uncertainty quantification (UQ) of engineering models, it is preferred to estimate the approximation error directly from the training set, without any additional sampling of the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' A common choice is the coefficient of determination R2, which is well-known from machine learning or statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' However, R2 may lead to over-fitting and thus advanced methods should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' One of the most widely-used methods is the leave-one-out cross-validation (LOO-CV) error Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The LOO-CV is based on residuals between the original surrogate model and the surrogate model built with the ED while excluding one realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This approach is repeated for all realizations in the ED and the average error is estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although the calculation of Q2 is typically highly time-consuming, it is possible to obtain results analytically from a single PCE as follows [35]: Q2 = 1 Nsim Nsim � i=1 � g � x (i)� − gPCE � x (i)� 1 − hi �2 σ2 Y,ED , (9) where σ2 Y,ED is the variance of the ED calculated using the original mathematical model and hi represents the ith diagonal term of matrix H = Ψ � Ψ TΨ �−1 Ψ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Statistical Moments Derived from PCE The form of PCE as a linear summation over orthonormal polynomials allows for powerful and efficient post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In particular, once a PCE approximation is created, it is possible to directly estimate statistical moments of the output from the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The first statistical moment (the mean value) is simply the first deterministic coefficient of the expansion µY = � Y 1� = β000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The second raw statistical moment, � Y 2� , can be estimated by � Y 2� = � �� ααα∈A βαααΨααα (ξ) �2 pξ (ξ) dξ = � ααα1∈A � ααα2∈A βααα1βααα2 � Ψααα1 (ξ)Ψααα2 (ξ) pξ (ξ) dξ (10) = � ααα∈A β2 ααα � Ψααα (ξ)2pξ (ξ) dξ = � ααα∈A β2 ααα 〈Ψααα,Ψααα〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Considering the orthonormality of the polynomials, it is possible to obtain the variance σ2 Y = � Y 2� − µ2 Y as the sum of all squared deterministic coefficients except the intercept (which represents the mean value), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' σ2 Y = � ααα∈A ααα̸=000 β2 ααα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (11) Note that the computation of higher statistical central moments, specifically skewness γY (3rd moment) and kurtosis κY (4th moment), are more complicated since they require triple and quad products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' These can be obtained analytically only for certain polynomial families, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' formulas for Hermite and Legendre polynomials (and their combination) can be found in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Active Learning-based Domain Adaptive Localized PCE (DAL-PCE) In this section, we propose a novel methodology to constructed localized PCEs designed for highly non-linear functions, termed Domain Adaptive Localized PCE (DAL-PCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Instead of increasing the maximum polynomial order p (p-adaptivity), which brings high computational requirements due to the curse of dimensionality, we 5 propose to decompose the input random space into several sub-domains approximated by low-order PCEs (h- adaptivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although this idea is not entirely new, we use this approach in combination with novel active learning methods to identify domains for refinement and for sequential sample selection and regression-based PCEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This allows us to use any sparse adaptive solver (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' LAR) and thus it can be easily implemented into the existing software packages [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In the following sections, we define the requisite components of the proposed method and provide an algorithm (Algorithm 1) for its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Variance-based Adaptive Sequential Sampling The decomposition of the input random space is a sequential process coupled with adaptive sampling assuring optimal coverage of the sub-domains of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The whole process thus consists of two steps: (i) identification of an important sub-domain, that is, a domain that is either large compared to other sub-domains or that is associated with a high local variance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' and (ii) identification of the best positions for additional samples extending the current ED in the selected sub-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Each of these steps must be based on a criterion that balances exploration of the input random space with exploitation of the surrogate model, which in our case is in the form of a PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The Θ-criterion for adaptive sequential sampling, which is driven by the output variance and its approximation via local variance using PCE [1], is employed for both steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' We will first discuss the process for adaptive sequential sampling within a specified sub-domain in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This will be followed by the process for refinement of the domain in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Consider a pool of candidate samples containing realizations of the random vector ξ generated by an arbitrary sampling technique, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=', Latin Hypercube Sampling (LHS) [38, 39] or Coherence sampling [40, 41, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' From this pool of candidates, we select the best sample using a method inspired by the sequential sampling proposed in [21] and based on Koksma-Hlawka inequality [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The Θ-criterion for PCE, which accounts for both variation of the function and discrepancy of the samples, was proposed as follows [1]: Θ(ξ(c)) ≡ Θc = � σ2 A(ξ(c)) · σ2 A(ξ(s)) ave variance density l M c,s vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' ≡ � σ2 c · σ2 s l M c,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (12) The criterion is a product of two terms – the exploitation term (denoted as “ave variance density”) and the exploration part (the distance term lc,s raised to the domain dimension) – which are multiplied to maintain an optimal balance between exploration and exploitation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The exploration aspect is maintained by accounting for the distance lc,s between a candidate ξ(c) and its nearest neighboring realization from the existing ED, ξ(s) as lc,s = � � � M � i=1 |ξ(c) i − ξ(s) i |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (13) If the criterion was reduced to this term only, sequential filling of the greatest empty regions would occur, con- verging to uniform space coverage in the spirit of the space-filling “miniMax criterion” [43, 44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The exploitation component is motivated by the desire to sample points in regions with the greatest contribu- tions to the total variance of the QoI σ2 Y , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' at points with the highest variance density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Once the PCE has been established at any given stage of the algorithm, the variance density is computationally cheap to evaluate for any location ξ as σ2 A(ξ) = � � ααα∈A ααα̸=000 βαααΨααα (ξ) �2pξ (ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (14) The local variance is therefore estimated directly using the basis functions and coefficients β of the PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' When considering a candidate “c”, an estimate of the variance contribution of the region between the candidate and its nearest neighbor “s” may be obtained by averaging the local variance densities between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Therefore, we can say that the candidate with the greatest Θc criterion is the one that represents the largest amount of total variance to be refined by its selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' A significant advantage of this method is the ability to add candidates into an existing ED one-by-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Thus, it can be employed at any moment of the PCE construction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Moreover, this learning function can be 6 combined with any sampling algorithm for the construction of the initial ED and candidates for extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The ideas behind the Θ criterion will now be used in the proposed domain decomposition and ED extension algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Decomposition of Input Random Space The core of the proposed approach is a sequential decomposition of the input random space � for the construc- tion of local approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This approach assumes that the original mathematical model can be approximated by piecewise low-order PCEs that are valid only in individual sub-domains of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Therefore, in the proposed ap- proach, the input random space is sequentially decomposed into n� smaller non-overlapping sub-domains �i ⊂ � that collectively fill the full input random space �, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' n� � i=1 �i = � such that �i ∩ �j = � ∀i, j (15) In each iteration of the algorithm, a single sub-domain �i (referred to as the parent) is identified for refinement and divided by a plane perpendicular to the direction of one selected input random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Specifically, �i is divided into a refinement-child �i, which is further processed, and an inheriting-child �⋆ i adopting the PCE from the parent as illustrated for a one-dimensional function in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In this case, we see that the space is divided into two subdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In the left (refinement child) a new PCE is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In the right (inheriting child), the original PCE is retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Such process assures an exhaustive decomposition into disjoint subsets i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' �i = �i⊕�⋆ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This sequential domain decomposition is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 2, which depicts the original input random space and the first four iterations of the decomposition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Figure 1: The first iteration of the algorithm: the original sub-domain is split and the new local PCE is constructed in �i (red background), while the second part in �⋆ i inherits the PCE approximation from the original domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In contrast to SSE [2], the selection of a single sub-domain for refinement in each iteration is based on an active learning approach, the details of which are provided in subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Importantly, actively integrating information from the original mathematical model leads to a significantly more effective decomposition of the space and thus assures accurate approximations, even for small-size EDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' On the other hand, the identified decomposition and the associated ED are directly connected to the given mathematical model and therefore might be inefficient for general statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The complete surrogate model is assembled from the n� local PCEs associated with all sub-domains �i as: Y ≈ n� � i=0 � αααi∈Ai βαααiΨαααi(ξ)��i(ξ), (16) where ��i(ξ) represents indicator function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' ��i(ξ) = 1 only if ξ ∈ �i and ��i(ξ) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In other words, to approximate the original model at any point, it suffices to determine the one relevant sub-domain and use the corresponding local PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Each such local PCE has its own set of basis functions Ai and corresponding co- efficients βαααi, which can be obtained by any model-selection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In this paper the OLS and LAR algorithms are employed, but generally any non-intrusive technique can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 7 Figure 2: The first four steps of the decomposition of a 3D space of input random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The thick black lines outline the parent domain selected for division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The red and green boxes inside it represent the two newly created refinement-child �i (red) and inheriting-child �⋆ i (green) sub-domains created by splitting the parent domain �i (bold boundaries), selected via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (17), by the cutting plane (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The cutting plane is perpendicular to the variable selected for splitting (blue arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Domain Selection via Modified Variance-based Criterion The selection process to identify the “best” subdomain for possible division is governed by extending the Θ-criterion from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (12) as follows: Θi = Wi · exp(Q2 i ) weight of subdomain � σ2 Ai(ξ(c)) · σ2 Ai(ξ(s)) l M c,s Θc in ith subdomain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (17) This extended criterion aims to identify sub-domains of the input random space associated with the maximum value of Θc, while simultaneously accounting for the size of each subdomain and the accuracy of the existing local PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The former is calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (12) calculated for a rich pool of screening global candidates, while the latter are measured by incorporating the volume of each sub-domain Wi and the LOO-CV error Q2 i , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The LOO-CV term, exp(Q2 i ), can be thought to artificially inflate the domain volume as a penalization for inaccurate approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' When the approximation is perfect (Q2 i = 0) the true volume of the sub-domain is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Meanwhile, a poor approximation with Q2 i = 1 leads to roughly 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='72 times increased volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The three terms featured in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (17) aim at different aspects affecting the accuracy of the final surrogate model: large sub-domains are preferred by Wi, sub-domains containing poor PCE approximation are promoted via exp(Q2 i ) and finally, Θc prefers sub-domains with high concentration of variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Note that Θc is calculated for a rich pool of screening candidates, and Wi and exp(Q2 i ) are calculated directly from the geometry of existing sub-domain and the local PCE model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The product of all three terms in the extended criterion therefore maintains the desired balance and assures the selection of the sub-domain, �i, that currently seems to be the most important for increasing the accuracy of the PCE surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Sub-domain � with the greatest Θi is selected and one of the operations described in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='6 is performed, depending on whether �i contains a critical number of ED points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Two scenarios can occur: �i contains a sufficient number of ED points (ni ≥ nsim) to ensure accuracy of a PCE on the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Therefore, it becomes a parent �i (bold boundaries in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 2) and is divided into two parts by a selected rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The child domain containing the decisive candidate with the greatest Θc becomes the refinement-child �i (see the red subdomains in steps 1 − 4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The remaining volume becomes an inheriting-child denoted �⋆ i (see the green subdomains in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 2), which retains the PCE from the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Division occurs by a cutting plane, oriented perpendicular to the selected direction (blue arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 2) and naturally, the coordinates of the cutting plane are restricted to the bounding box of the selected parent �i, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' If needed, the refinement-child domain �i is sequentially filled with additional ED points (according to Θc) to reach ni = nsim needed to construct a new PCE approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' �i does not contain a sufficient number of ED points (ni < nsim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The domain is not divided because the suggestion for division is based on insufficient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Instead, new ED points are sequentially added to �i, again using the Θc criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Note that this scenario practically arises when the selected domain was an inheriting-child in the previous iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In this case, the selected domain has inherited a PCE model that was constructed over a larger domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' When that domain was divided, it was left with an insufficient number of points from which to construct a new PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' PCE Basis Functions Without loss of generality, the proposed method operates on the M-dimensional unit hypercube with uniform distributions of input random variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' XXX ∼ U[0,1]M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In the case of a general joint probability distribution of XXX, it is always possible to transform input random vector to the unit hypercube by Rosenblatt transformation [46], Nataf transformation [47] or various methods based on copulas [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Standard normalized Legendre polynomials, orthonormal to the uniform distribution, can thus be used as basis functions for the PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' However, due to the decomposition of the input random space to smaller sub-domains, each with lower bound ai and upper bound bi, it is necessary to use univariate scaled orthonormal Legendre polynomials of nth order ˜ ψn(ξ) defined as follows: ˜ ψn(ξ) = ψn �2ξ − ai − bi bi − ai � , (18) where ψn represents standard orthonormal Legendre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Naturally, the transformation of the original input random vector to the unit hypercube might bring additional non-linearity, and thus one might prefer the direct construction of polynomials locally orthonormal to the given original probability measure as proposed in the Me-gPC [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' While certainly possible, this brings additional computational demands and thus it is not employed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Local and Global Statistical Estimates from DAL-PCE The significant advantage of PCE is that analytically post-processing of the expansion yields highly efficient estimates of statistical moments [30], sensitivity indices [8] and LOO-CV [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In the proposed DAL-PCE, since the original domain � is decomposed into a set of sub-domains (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (15)), standard analytical post-processing can be applied locally and global characteristics can be obtained by simple weighted summations that converge to the true values as n� increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Specifically, the global mean value and variance of a QoI are obtained from localized PCEs (denoted by subscript �i) as follows: µY = n� � i=1 Wiβ0i = n� � i=1 Wiµ�i, (19) σ2 Y = n� � i=1 Wi � αααi∈Ai αααi̸=000 β2 αααi = n� � i=1 Wiσ2 �i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (20) where the local mean µ�i and variance σ2 �i are obtained as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Local Sobol’ indices, S�i, of any order can be derived directly from localized PCEs and their first-order (main effect) estimates are given by S X j �i = 1 σ2 �i � αααi∈A X j i β2 αααi A X j i = � αααi ∈ Ai : αj i > 0,αk̸=j i = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (21) These local Sobol’ indices are used in the DAL-PCE to determine the cut direction (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Likewise, global Sobol’ indices can be obtained easily from weighted summation of local contributions to partial variances normalized by σ2 Y as follows: SX j = �n� i=1 Wi � αααi∈A X j i β2 αααi σ2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (22) Similarly, global LOO-CV, Q2, of a QoI can be approximated by the weighted summation of the local contributions as Q2 = n� � i=1 WiQ2 �i, (23) where Q2 �i are obtained from each local PCE using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' These estimates are used throughout the proposed DAL-PCE, as described in detail next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Numerical Algorithm Based on the presented theoretical background, we now present the numerical algorithm for the domain adaptive localized PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' As mentioned above, the whole process can be divided to two iterative tasks: (i) decom- position of the input random space and (ii) construction of localized PCEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Both of these tasks are described in the following paragraphs with specific reference to the steps in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Algorithm 1 DAL-PCE: Active Domain Decomposition and Construction of Localized PCEs Input: maximum local polynomial order p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' number of screening global candidates nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' number of local candidates nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' number of iterations niter 1: set the minimum number of realizations for local PCE construction nsim ∈ 〈P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='2P〉 2: generate a rich pool of nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g screening candidates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='3: generate the initial ED (size nsim) and construct the initial global PCE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='4: for 1 to niter do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='identify the sub-domain �i with the highest Θi based on screening candidates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='6: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='ni ← number of ED samples existing in �i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='if ni ≥ nsim then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='the identified sub-domain �i becomes a parent �i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='identify the direction of the highest first-order Sobol’ index S�i of the parent �i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='restrict coordinates of �i → �i and create �⋆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='ni ← number of ED samples existing in �i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='generate nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='l local candidates in �i 14: while ni < nsim do 15: extend size of local ED ni using the local Θc criterion 16: end while 17: reconstruct local PCEs in the �i 18: end for Output: list of subdomains and corresponding PCEs The first task identifies the important sub-domain �i that should be divided and over which low-order local PCE should be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The sub-domain �i is specifically identified using the Θi criterion from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (17), which again incorporates three important characteristics for accurate surrogate modeling – the size of the sub- domain Wi, the accuracy of the existing local PCE measured by Q2 �i, and the original Θc criterion measuring the variance contribution in �i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' While Wi and Q2 �i are computed for the whole sub-domain, Θc is computed at specific realizations of input random vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Therefore, it is necessary to cover the sub-domains by a sufficiently large number of screening candidates, such that the total global number of screening candidates is given by nc,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Based on numerical experiments, we recommend nc,g ≥ 1000 M to ensure that each sub-domain contains a sufficient number of screening candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Note that the screening candidates are used only to identify �i [step 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' They are not used for the ED, and thus even high nc,g does not bring any additional computational demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Once �i is identified, it is necessary to check whether there are enough samples to construct a PCE inside the sub-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' We start with finding out how many points belong to the selected domain �i [step 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' If the number of samples in the identified sub-domain, ni, is greater than (or equal to) nsim [step 7], a local PCE already exists for �i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The subdomain is then assigned as a parent �i for division [step 8] and the first-order Sobol’ indices are estimated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (22) [step 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This identified parent �i is divided in the direction of the highest first-order Sobol’ index S X j �i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The new restricted coordinates of refinement-child �i are identified and the inheriting-child �⋆ i is created [step 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Further, the number of ED samples ni in the refinement-child �i is determined [step 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' On the other hand, if the identified sub-domain �i does not contain enough samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' ni < nsim), the inherited PCE from the previous iteration is not sufficiently local (it was trained over a domain that has since been divided) and it is necessary to add new samples to �i before constructing a new local PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The second task of the proposed algorithm is sequential sampling and adaptive PCE construction in sub- domain �i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Recall that this domain may be either 10 (i) a refinement-child that was just divided but does not contain a sufficient number of points (ni < nsim) or, (ii) an inheriting-child that now does not contain at least nsim ED samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Next, a set of local candidates is generated in region �i [step 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' To ensure sufficient assessment of the coverage of the domain, the number of local candidates is empirically recommended as nc,l ∈ 〈3P,5P〉 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' From these candidates, the standard Θc criterion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (12) is used to iteratively select the best candidates until there are nsim samples in �i [step 14-16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This sequential extension of the sample in �i is adaptive in the sense that the pairwise distances in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (12) between candidates and existing ED points are updated after the addition of each new point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' However, because ni < nsim the local variance densities are estimated from the previously existing PCE, which cannot be updated until a sufficient number of samples are available in �i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The last step of each iteration is to construct the local PCE using scaled Legendre polynomials as basis func- tions (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (18)) [step 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Any non-intrusive technique can be used to estimate the coefficients βββ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' we use LARS and OLS for an adaptive construction of the local PCEs in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' At the end of the iteration, all sub-domains are re-numbered and a list of sub-domains with corresponding PCEs can be exported or the next iteration can be started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Adaptivity in PCE Construction and Domain Decomposition Adaptivity is central to the proposed DAL-PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In the proposed algorithm, there are two types of adaptivity employed: (i) adaptivity in PCE construction (selection of the optimal set of basis functions), and (ii) adaptivity in domain decomposition Since the PCE can be constructed by any regression technique in each sub-domain, PCE adaptivity is incorporated by sparse solvers and best model selection algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Least Angle Regression [32], orthogonal matching pursuit [33] or Bayesian compressive sensing [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although sparse solvers are often used for PCE with high p, this adaptivity is also important for reducing the number of basis functions (and thus the minimum number of ED samples) for high-dimensional examples or, in our case, for very low-size ED in each �i approximated by low-p local PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The second type of adaptivity is the proposed adaptivity in the domain decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' At any point in the iterative process, the existing ED samples can be used to construct local PCEs or a single global PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The DAL- PCE is not guaranteed to provide a better approximation than the global PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This can be measured via Q2, specifically by computing Q2 local from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (23) and Q2 global from a single global PCE according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' If Q2 local > Q2 global at a given iteration, the domain decomposition is deemed to be poor and the whole decomposition process is re-started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' That is, the complete geometrical decomposition is forgotten and all existing ED points are taken as an initial ED for a brand new run of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 3 which shows the decomposition (top) and the associated error (bottom) right before the restart a) at Nsim = 181, b) the new decomposition and error right after the restart, and c) the final decomposition/error which shows significant improvement over the global PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' These histories show the standard R2 error defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' It is not necessary to check this criterion at every iteration, but it is suggested to check it periodically, every nr steps, to ensure adequate local refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Stopping Criteria The proposed DAL-PCE algorithm can be fully automated by adding an adequate stopping criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' A simple but practical stopping criterion is based on computational budget, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' once the total number of model evaluations Nsim or number of iterations niter have reached a critical level/budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' One may also use a stopping criterion based on decomposition pattern, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' the smallest or the largest volumes of any subdomain, to ensure a desired resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Valuable stopping criterion can be also obtained directly from Q2, corresponding to a target/threshold level of achieved accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Regardless of the selected stopping criteria, it can easily be applied before step 5 of the proposed algorithm (start of each iteration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 11 Figure 3: Illustration of domain decomposition restart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' a) decomposition and error evolution prior to restart, b) rebuilt decomposition and error drop right after the restart, c) final decomposition and error showing that the restart unlocks a dramatic decrease in approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Numerical Experiments The proposed DAL-PCE is presented on four numerical examples of increasing complexity and which illus- trated different aspects of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The obtained results are compared (a) to the standard global PCE approach with adaptive maximum order p ∈ [5,25] and (b) to SSE [2], as current state-of-the-art non-intrusive surrogate modeling technique based on the domain decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The PCE is constructed using the UQPy pack- age [36] and the original implementation of SSE is used from the UQLab package [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' To compare methods, the relative mean squared errors ε are calculated for all three approximations ˜f on a validation set containing a large pool of 106 integration points generated by crude Monte Carlo according to: ε(XXX) := � �� f (XXX) − ˜f (XXX) �2� � � f (XXX) � , (24) where �[] and �[] are the mean value and variance operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' To show representative results of the proposed DAL-PCE algorithm, the calculations were repeated 100 times, and the same settings of the algorithm for all examples were selected as follows: maximum local polynomial degree p = 2, number of global candidates nc,g = 1000 M, number of local candidates nc,l = 5P, minimum number of samples for local PCE construction nsim = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='5P, minimum number of iterations before checking for restart nr = 20, and βββ are obtained by LARS and OLS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Minimum number of samples in sub-domains required to justify an expansions for SSE was set identically to DAL-PCE and polynomial order is adaptively selected in the range p ∈ [2,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Since the SSE is not a sequential approach, the presented results were obtained for 10 discrete sample sets of increasing size to compare convergence of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Note that all samples and candidates are generated by LHS for all compared approaches, though it was shown [1] that for the variance- based sequential sampling, it is significantly better to use advanced techniques such as Coherence D-optimal sampling [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' One-dimensional Toy Example The first example involves a simple 1D function [2] that is extremely difficult to approximate with PCE due to the third, highly nonlinear “exp” term: f (X) = −X + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='1sin(30X) + exp(−(50(X − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='65))2), X ∼ U[0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (25) The poor performance of a single global PCE learned from 200 samples is depicted by the blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 4c where it is clear that a single global PCE is not able to accurately approximate the function even for a high number of samples and high maximum polynomial order p ∈ [5,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This function was originally developed to demonstrate the efficiency of SSE based on domain decomposition and thus it is a natural choice for comparison of the proposed DAL-PCE and SSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 4a-b show a typical realization of the DAL-PCE where the algorithm sequentially decomposes the domain and adds additional samples to the ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Specifically shown are the 4th and 11th iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The boundaries of sub-domains are represented by blue vertical lines and red dots show the positions of samples in the ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Once the algorithm discovers the highly nonlinear region (the steep peak caused by exp), it progressively refines this region and adds more samples there as a result of the high variance density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Of course, these figures show only one realization of the algorithm and the decomposition is dependent on the initial ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Therefore, it is necessary to repeat the algorithm many times with random initial ED to assess convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 4d shows convergence Figure 4: (a), (b) The adapted domain and ED before (iteration 4) and after (iteration 11) exploration and discovery of the exponential part of the mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (c) Final surrogate models from global PCE and DAL-PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (d) Convergence plot comparing the mean square error for global PCE SSE, and DAL-PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The convergence plots for Global PCE and DAL-PCE show continuous mean value ±σ intervals from 100 repeated trials, while those for SSE are plotted for several discrete ED sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' of the error ε from 100 repeated trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The single global PCE is unable to accurately approximate the original function even when using high p and thus the ε does not converge, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Both methods based on domain decomposition (DAL-PCE and SSE) achieve great accuracy already for 200 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' However, the DAL-PCE consistently has 1–2 orders of magnitude higher accuracy than SSE for the given number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Moreover, increase in variance of ε is, in general, slower in DAL-PCE than in SSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Fast increment in variance of SSE can be seen also in the original paper [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Finally, we again observe that convergence is continuous with DAL-PCE, where convergence can only be assessed at discrete sample sizes with SSE through a new analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' All of these 13 Figure 5: Results for the 2-dimensional Singularity function: a) original mathematical model, b) approximation via DAL-PCE (background color), current domain division and the corresponding ED, c) local LOO-CV Q2 �i and Θi value for each sub-domain, d) convergence plots for DAL-PCE, Global PCE, and SSE showing the mean value and ±σ interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Convergence plots for SSE show the mean ±σ at discrete sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' advantages of the DAL-PCE can be attributed to the active learning, which both explores the space and exploits the behavior of the function to decompose the domain and add samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although active learning might lead to lower accuracy (higher ε) initially (for small nsim = 10–20) as it is dominated by exploration, it rapidly improves once it identifies important features and begins to favor exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Two-dimensional Singularity The second example involves a 2D function with mirrored quarter-circle arc line singularities [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The form of the function is give by: f (XXX) = 1 |0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='3 − X 2 1 − X 2 2| + δ − 1 |0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='3 − (1 − X1)2 − (1 − X2)2| + δ, XXX ∼ U[0,1]2, (26) where the strength of the singularities is controlled by the parameter δ, which we set as δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The singularities in this example represent a challenging task for a global PCE even with high order, due to the well-known Gibbs phenomenon [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' It is thus beneficial to identify the location of the singularity, locally decompose the domain, and construct low-order local PCEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 5 illustrates the decomposition and DAL-PCE approximation at a given stage of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Panel a) visualizes the true values of the function via a background color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The same coloring scheme is used in panel b) for the pointwise information available in the current ED (small circles) and for the function approximation via DAL-PCE by the background color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Panels b) and c) show also the final domain decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The symmetry 14 Figure 6: Results for the 2-dimensional discontinuiy function: a) original mathematical model, b) approximation via DAL-PCE and ED, c) local LOO-CV Q2 �i and Θi value for each sub-domain, d) convergence plots for DAL-PCE, Global PCE, and SEE showing the mean value and ±σ interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Convergence plots for SSE show the mean ±σ at discrete sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' in the decomposition documents the great convergence of the DAL-PCE thanks to an adaptive decomposition described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Plot c) shows the local Q2 �i error in each individual sub-domain (darker color corresponds to higher local error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' These local errors clearly show localization of the prediction error to very small areas near singularities, which are continually being refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The color of the small solid squares in the center of each sub-domains shows the Θi value for that sub-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Finally, the convergence plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 5d) shows that both DAL-PCE and SSE outperform the global PCE, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The SSE performs comparable to or slightly better than DAL-PCE for small Nsim, but the DAL-PCE begins to outperform SSE as Nsim grows thanks to the active learning approach that targets samples in the vicinity of the singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Note that the error converges for both SSE and DAL-PCE as we approach 1000 samples and does not seem to substantially reduce after this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This is due to the fundamental limitation of trying to approximate this singularity, even locally, with low-order polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' M-dimensional Discontinuity The third example investigates the role of dimensionality on the performance of the proposed DAL-PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The following discontinuous function is defined for an arbitrary number of input random variables M [26]: f (XXX) = � sin(X1π)sin(X2π) if x1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='5 and x2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='5 �M i=3 Xi otherwise , XXX ∼ U[0,1]M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (27) This function has a discontinuity in the first two input random variables, which can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' A single global PCE cannot accurately approximate the function because of the discontinuity, although the function f (XXX) 15 Figure 7: Convergence plots for the M-dimensional function: a) 3-dimensional version, b) 5-dimensional version, c) 6-dimensional version, and d) 8-dimensional version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Convergence plots for the DAL-PCE and global PCE show the mean value ±σ interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Convergence plots for SSE also show the mean ±σ, but at discrete sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' can be easily approximated by two separate PCEs in the two regions for which the definitions differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' But, this requires a priori knowledge of the discontinuity location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Since the location of the discontinuity is assumed to be unknown, this function is a good example for domain adaptation using DAL-PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The detailed results for a 2D version of this problem are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 6 in identical form as in the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Note that the local Q2 i errors Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 6c show perfect accuracy in the part of the input random space where f (XXX) = 0 and thus the associated sub-domains are not preferred for further decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The convergence plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 6d confirms that a single global PCE is not able to create an accurate approximation and adding more points to ED does not lead to significant improvements in the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The mean values of errors ε associated to the proposed DAL-PCE approach are significantly lower in comparison to SSE (1–2 orders of magnitude) similarly as in the first example, though the convergence trend is similar for both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' SSE, however, uses a random splitting routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This can lead to very high variance of results, since the accuracy is highly dependent on the pattern of the decomposed input random space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This clearly shows the advantage of an active learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The influence of dimensionality M on convergence of the DAL-PCE, SSE, and global PCE is studied in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 7 for a) 3, b) 5, c) 6, and d) 8 input random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' As the domain dimension increases, the linear part of the function f (XXX) occupies an increasing proportion of the domain while the discontinuity remain low-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The proposed DAL-PCE greatly improves the convergence because it is able to identify an ideal decomposition and local samples to resolve the discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' For low-dimensions (M = 2,3), SSE error ε shows a decreasing trend that is better than global PCE but has an extremely high variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This is caused by a lack of control in sample placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The domain decomposition in SSE is a product of sample location and without active learning to guide sample placement, SSE will sometimes produce a very good decomposition and sometimes a very poor decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Meanwhile, the proposed DAL-PCE errors have comparably low variance for low-dimensions and consistently have accuracy comparable to, or better than, the best SSE realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' As the dimension, M, increases the DAL-PCE is able to maintain a very high level of accuracy, while the accuracy degrades completely for the SSE such that it is comparable to the global PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The DAL-PCE is able to maintain its low error because the discontinuity remains low-dimensional and the active learning process is able to target this region for domain refinement and sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This means that the DAL-PCE remains largely independent of the problem dimension, and instead depends predominantly on the intrinsic dimension of the 16 Figure 8: Convergence plots for the modified M-dimensional function: a) 3-dimensional version, b) 5-dimensional version, c) 6-dimensional version, and d) 8-dimensional version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Convergence plots for the DAL-PCE and global PCE show the mean value ±σ interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Convergence plots for SSE also show the mean ±σ, but at discrete sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' discontinuous/nonlinear features of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The performance of SSE, on the other hand, degrades with dimension because its domain decomposition depends only on a set of a priori specified points that are not selected in a way that is aware of the important features of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Consequently, as the dimension increases the algorithm becomes less likely to refine the domain appropriately around an embedded low-dimensional feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' We remark that this desirable scalable convergence trend of the DAL-PCE is not likely a universal property, as the trend may break down in problems where the intrinsic dimension of the discontinuity/nonlinearity is high or where the discontinuity occupies a very small proportion of the domain – in which case exploration of the space to find the important feature may take a very large number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In the present example, the discontinuity in the function given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (27) lies at x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='5 and x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='5, which corresponds to the exact location where the domain will be split for both SSE and during the early iterations of the DAL-PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' One might argue that this presents an unreasonable advantage for the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' We therefore modified the function such that the discontinuity lies at x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='61 and x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 8 shows the convergence for the DAL-PCE and SSE for this modified function with varying dimension, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The absolute errors ε exhibit slower decrease, especially for dimensions M = 3 and M = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' However, the proposed active learning still leads to superior results (especially for higher dimensions as in the previous case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Note that there are visible spikes in the DAL-PCE convergence graph for the 3-dimensional example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although the results were statistically processed, these spikes are caused by the restart adaptivity occurring at the same Nsim in each replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In this case, the optimal decomposition pattern is very complicated and therefore the algorithm activates the restart adaptivity frequently (after multiples of nr steps), until it finds a suitable pattern to continue convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' SSE in the 3- and 5-dimensional cases has higher mean error and significantly lower variance in comparison to the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This is caused by the fact that the modified discontinuity location no longer lies along the boundary of the domain decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In the previous example, some SSE realizations achieved near-perfect accuracy because the domain was coincidentally divided along the discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This phenomenon is investigated more closely in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 9, which compares number of outliers in both versions of 3D examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In addition to the mean ±σ seen previously, the figure also shows standard boxplots for SSE (median along with lower and upper quartiles) and the corresponding number of “extreme” realizations produc- ing very high accuracy (top axis) for a) the original position of discontinuity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' and b) discontinuity at x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='61 17 Figure 9: Convergence plots for DAL-PCE and SSE with additional boxplots for SSE showing the median, lower and upper quartiles and outliers for: a) the 3D example with discontinuity at x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='5 and x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='5, b) the 3D example with discontinuity at x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='61 and x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' and x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' As can be seen, in panel a) there are many outliers producing ε < −7, which effectively decreases µ relative to the median while also significantly increasing the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In contrast DAL-PCE has no outliers and it leads to very consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In panel b), there are no outliers for either SSE or DAL-PCE and the results are thus consistent with low variance for both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Asymmetric shallow von Mises truss In this section, we demonstrate the relevance of the proposed method for a representative engineering exam- ple exhibiting discontinuous response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Consider the shallow two-bar planar truss subjected to a vertical load at its top joint, as presented in [50] and illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The truss is formed by two prismatic bars made of a hard wood (density 800 kg/m3, modulus of elasticity E = 12 GPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' There are two variables in the studied von Mises truss: (i) the loading vertical force F, and (ii) a half sine-wave imperfection of the left bar having magnitude δ, see the sketch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The load is applied dynamically as a step function at time zero for an unlimited duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The structure is modeled, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 10b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In particular, the mass of the bar is concentrated in 21 mass points, including the supports and the loading point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' These mass points are connected via 10 + 10 translational springs representing the normal stiffness of the true bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The pairs of the axial members are connected via rotational spring having zero moment for a zero angle between adjacent bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The only exceptions are the loading ans support points where there are no rotational springs attached (hinges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The damping is associated with the mass points via linear viscous damping coefficient set to 11 N · s/(kg · m) approximating the relative damping of about 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Explicit dynamics solver FyDiK [51, 52] was used to solve the equations of equilibrium at the mass points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The numerical solution lasts to up to two seconds, which is the time needed for almost complete stabilization of the solution (kinetic energy drops below a negligible threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Since the structure is very shallow, sudden application of the vertical force can cause snap-through buckling, wherein the loading point drops down between the supports and the members switch from a state of compression to tensile stresses in the final stable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' We specifically study the horizontal coordinate yF of the loading point after the dynamic response stabilizes to the final deformed shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The force F ∈ (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='6,772.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='6) kN and initial imperfection δ ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='4) m are treated as uniform random variables mapped to the unit square such that the model input X ∼ U[0,1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Because of the potential snap-through buckling, the solution is discontinuous as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 10c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' On each side of the discontinuity, the solution yF is smooth and slowly-varying having values near +1 m and -1 m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Note that the output is not symmetric with respect to δ = 0 because the dynamical response evolves differently for concave and convex initial displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The sharp boundary between the buckled and unbuckled regions, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 11a cause global PCE to produce poor approximations that are vulnerable to the Gibbs phenomenon, similar to the example in subsection 18 b) a) c) Figure 10: Asymmetric shallow von Mises truss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' a) Initial geometry with two random variables F and δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' b) illustrative sketch of the discrete dynamical model and the meaning of output variable yF, c) illustration of the discontinuous response function of the two input variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Figure 11: Results for the von Misses truss example: a) original mathematical model (numerical solution), b) approximation via DAL-PCE and ED, c) local LOO-CV Q2 �i and Θi value for each sub-domain, d) convergence plots for DAL-PCE, Global PCE, and SSE showing the mean value and ±σ interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' convergence plots for SSE show the mean ±σ at discrete sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This is shown by the convergence plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 11d comparing global PCE, DAL-PCE, and SSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Clearly, the complexity of this example and the complicated shape of the discontinuity limits the accuracy of all the surrogate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The proposed DAL-PCE achieves low accuracy for small sample sizes because the corresponding small number of sub-domains and low-order PCEs are unable to sufficiently approximate the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Therefore, the global PCE and SSE (with a low number of embedding levels) are initially better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' With increasing number of samples, the proposed DAL-PCE approach leads to superior results because the active learning is able to resolve the discontinuity as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 11b, which shows the domain decomposition and approximation after 2000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 11c shows the corresponding LOO-CV errors for each subdomain, demonstrating the errors are confined to small, localized regions near the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Discussion & Future Work The proposed DAL-PCE approach is a general methodology for the decomposition of the input random space and construction of localized PCEs using active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The proposed active learning is based on a novel Θ criterion that optimally balances global exploration with local exploitation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although this paper presents one specific learning algorithm, the methodology is general and amenable to modifications to reflect the specific user’s needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The whole process can be divided into two tasks: A) decomposition of the input random space and B) construction of localized PCEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' and both can be easily modified as discussed further: A) The most important sub-domain �i is identified by extended Θ according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (17) evaluated for a large number of global candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In this paper, we use standard LHS for candidate generation, but it may be beneficial to use different sampling methods that produce more uniform coverage of the whole input random space (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' [53, 54, 45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although it is generally possible to generate a large number of candidates, it might be challenging to uniformly cover the entire input random space, especially in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Thus, one can use any sampling technique suitable for a specific example, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Once the �i is identified via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' (17), it is either divided (providing it contains enough ED points) or the sample is extended inside it, to achieve a better PCE approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The simplest division occurs by splitting the volume into two parts of identical hypervolume in the direction of the highest first-order Sobol’ index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' However, the algorithm can accommodate various different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' For example, it is possible to divide the �i into a higher number of sub-domains, not just two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Moreover, instead of splitting the domain into parts of equal hypervolume, other criteria can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' For example, the cutting plane can be positioned so to split the domain variance into equal parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' B) The user can choose to employ any existing method to construct the non-intrusive PCEs, including various sparse solvers or adaptive algorithms, which may be preferable for certain applications [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' For example, we use LARS with OLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' However, it is generally more efficient to use active learning based on the Θ criterion for PCE as shown in [1], which employs variance-based sequential sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' This improvement can be integrated within the DAL-PCE to make local PCE more efficient in each subdomain, and thereby improving the overall convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The can be compounded by the use of advanced sampling techniques within the subdomains such as Coherence D-optimal sampling [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' As seen from the previous paragraphs, the whole algorithm can be adapted for specific needs reflecting the characteristics of a given mathematical model, such as dimensionality, sparsity, non-linearity etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=', by simply ex- changing components of the proposed algorithm for suitable existing (or new) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Note that even after the modification, the whole methodology based on Θ criterion is still valid and can be used for uncertainty quantification and surrogate modelling as described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Moreover, in comparison to SSE, the DAL- PCE sequentially adds points and divides the sub-domains one-by-one based on information obtained from the previous iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Another significant advantage of the DAL-PCE is that it provides estimates of the local errors, Q�i, associ- ated with each sub-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Since localized PCEs are constructed independently, local errors estimate the local accuracy of the surrogate model directly, and can be assembled to provide global error measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Naturally, local accuracy is very important information that can be used for further probabilistic analysis and active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although this paper does not propose any specific approach for further processing of this information, it could serve as a main ingredient for various active learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' For example, it could be directly used to predict uncertainty in industrial applications and possibly extend the ED in a sub-domain of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Finally, an important topic of further research is to study the behavior of the proposed criterion in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' In particular, the geometrical terms l M c,s and �i likely cause poor convergence in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Although some preliminary results focused on investigating of l M c,s in high dimensions was previously performed in the paper [1] proposing the original Θ criterion, it is still necessary to perform an extensive study of its behavior as well as investigating the influence of �i, which may need to be reformulated for high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Conclusion The paper presented a novel approach, domain adaptively localzed PCE, for the adaptive sequential con- struction of localized PCEs based on active learning and decomposition of the input random space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' It combines adaptive sequential sampling based on the recently proposed Θ criterion to maintain the balance between ex- ploration of the input random space and exploitation of the current characteristics of the PCE together with the adaptive sequential decomposition of the input random space creating sub-domains approximated by local sur- rogate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The methodology offers a general technique that can be easily adapted or modified for specific functions extending its applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' The performance of the proposed methodology was validated on several nu- merical examples of increasing complexity investigating different aspects of the algorithm and leading to superior results in comparison to a single global PCE and the recently proposed SSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Acknowledgments The first author acknowledge financial support provided by the Czech Science Foundation under project num- ber 22-00774S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Additionally, the major part of this research was conducted during the research stay of the first author at Johns Hopkins University supported by the project International Mobility of Researchers of Brno Uni- versity of Technology, Czechia under project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' EF18_053/0016962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' References [1] L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='advengsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='102709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' [55] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Voˇrechovský, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' Mašek, Distance-based optimal sampling in a hypercube: Energy potentials for high-dimensional and low-saturation designs, Advances in Engineering Software 149 (2020) 102880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='advengsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content='102880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} +page_content=' 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FRT4oBgHgl3EQfvjiF/content/2301.13635v1.pdf'} diff --git a/C9FKT4oBgHgl3EQfYi5Z/content/tmp_files/2301.11799v1.pdf.txt b/C9FKT4oBgHgl3EQfYi5Z/content/tmp_files/2301.11799v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..45ee72d7fda20fb1386ab52e95eea118af44920b --- /dev/null +++ b/C9FKT4oBgHgl3EQfYi5Z/content/tmp_files/2301.11799v1.pdf.txt @@ -0,0 +1,1326 @@ + + + + + Số chuyên san (11/2022): 1 – 11 + 1 + + +MỐI TƯƠNG QUAN CỦA CÁC NHÂN TỐ ẢNH HƯỞNG +TỚI VIỆC SỬ DỤNG ỨNG DỤNG BLUEZONE +Nguyễn Thế Vịnh1*, Nguyễn Tuấn Anh1, Nguyễn Hồng Tân1, Lương Khắc Định2 +1Khoa Công nghệ thông tin, Trường ĐH Công nghệ thông tin và Truyền thông, ĐH Thái Nguyên +2Khoa Công nghệ thông tin, Trường ĐH Hạ Long +* Email: vinhnt@ictu.edu.vn +Ngày nhận bài: 11/6/2022 +Ngày nhận bài sửa sau phản biện: 09/11/2022 +Ngày chấp nhận đăng: DD/MM/YYYY +TÓM TẮT +Sự xuất hiện của đại dịch Covid-19 đã gây ra nhiều tác động tiêu cực đến mọi mặt của đời +sống. Chính phủ đã áp dụng nhiều biện pháp để giảm thiểu sự ảnh hưởng và lây truyền của dịch +bệnh. Trong số đó có việc áp dụng chuyển đổi số đối với việc quản lý và truy vết người bị nhiễm +Covid thông qua phần mềm Bluezone (nay là PC-Covid). Tuy nhiên, việc cài đặt và sử dụng +Bluezone lại không được như kỳ vọng. Vì vậy, nghiên cứu này tìm hiểu những nhân tố chính và +sự ảnh hưởng của chúng tới ý định hành vi của người dùng về việc sử dụng phần mềm truy vết +Bluezone. Phiếu khảo sát được gửi tới người dùng thông qua công cụ Google Form. Kết quả +phân tích các nhân tố khám phá trên 224 đối tượng khảo sát cho thấy, có bốn nhân tố chính ảnh +hưởng tới hành vi của người dùng, trong đó: sự tin tưởng và kỳ vọng hiệu quả, kỳ vọng nỗ lực, +ảnh hưởng xã hội có tác động tích cực đến ý định hành vi của việc sử dụng phần mềm truy vết +Bluezone; trong khi rủi ro về quyền riêng tư có ảnh hưởng tiêu cực đến hành vi này. +Từ khóa: EFA, SEM, UTAUT, tin tưởng, quyền riêng tư, Covid-19. +FACTORS INFLUENCING TO USE OF BLUEZONE +ABSTRACT +The emergence of the Covid-19 pandemic has been causing many negative impacts on all +aspects of life. The government has taken many measures to minimize the impact and +transmission of the disease. Among them is the application of digital transformation to the +management and tracing of people infected with Covid through the Bluezone app (now PC- +Covid). However, using and installing Bluezone is not as expected. Therefore, this study aims +to understand the main factors and their influence on the behavioral intention of users about +using Bluezone. Surveys are sent to users through the Google Form tool. Experimental results +through analysis of exploratory factors on 224 survey subjects show that there are 4 main factors +affecting user behavior. Structural equation modeling indicates that trust, performance +expectations, effort expectations, and social influence have a positive impact on behavioral +intention of using Bluezone. Meanwhile, privacy risks have a negative effect on this behavior. +Keywords: EFA, SEM, UTAUT, trust, privacy, Covid-19. + +ap chi khoa hoc +DAI HOC HA LONGTAP CHI KHOAHOC DAI HOCHALONG +Scientific JournalofHa Long Vniversity +KHOAHOC +DAIHOCHALONG +http://uhl.edu.vnl +Hac de thanh cong + + + +2 Số 01(2021): 1 – 11 + + + +KHOA HỌC TỰ NHIÊN +1. ĐẶT VẤN ĐỀ +Đại dịch Covid-19 xuất hiện vào cuối năm +2019 và bùng phát mạnh mẽ trong thời gian +qua đã có những ảnh hưởng tiêu cực tới tất cả +các quốc gia trên toàn thế giới (Whitelaw và +c.s., 2020). Đứng trước vấn đề đó, chính phủ +các quốc gia trên thế giới đã tiến hành nhiều +biện pháp cấp bách nhằm hạn chế tầm ảnh +hưởng, lây lan của dịch bệnh (Nguyen và c.s., +2021). Song song với các biện pháp tuyên +truyền đến người dân về ý thức phòng chống +dịch thông qua các phương tiện truyền thông, +chính phủ Việt Nam cũng tiến hành nhiều +biện pháp hỗ trợ nhằm truy vết tiếp xúc và +cảnh báo người nhiễm Covid-19 (Le và c.s., +2021). Cụ thể, Bộ Y tế và Bộ Thông tin và +Truyền thông đã phối hợp tạo ra ứng dụng +Bluezone. Bluezone được coi là “cần thiết +trong quá trình sinh hoạt hàng ngày, khi mọi +người có tiếp xúc, ứng dụng trên điện thoại +của họ sẽ tự “nói chuyện” với nhau” +(baochinhphu.vn, 2020). Ứng dụng Bluezone +được kỳ vọng là sẽ giúp ích cho các cơ quan +nhà nước có thể nhanh chóng truy vết và quản +lý được các ca nhiễm trong cộng đồng, người +dân có thể nắm bắt được thông tin kịp thời để +phòng dịch (Nguyen và c.s., 2021). +Mặc dù Bluezone được kỳ vọng sẽ mang +lại hiệu quả tích cực cao và nhiều người sẽ sử +dụng, nhưng số liệu thống kê thực tế lại +không được như mong muốn (Nguyen và c.s., +2021). Tính đến 27 tháng 5 năm 2021, cả +nước chỉ ghi nhận 33,48 triệu lượt tải (khoảng +34,7% so với tổng dân số), trong đó tập trung +chủ yếu ở hai địa phương lớn là Hà Nội (3,1 +triệu lượt cài đặt) và Thành phố Hồ Chí Minh +(2,83 triệu lượt cài đặt). Ở chiều ngược lại, +các tỉnh khác như Điện Biên, Kon Tum, Lai +Châu, Bắc Kạn lại ghi nhận số lượng người +tải ứng dụng Bluezone thấp nhất. Vì vậy, câu +hỏi đặt ra là: Những yếu tố nào ảnh hưởng tới +việc sử dụng phần mềm Bluezone? +Trả lời được câu hỏi nghiên cứu trên đóng +vai trò quan trọng trong việc khuyến khích +người dân tham gia, hỗ trợ phòng chống dịch +trên môi trường số (Nguyen & Nguyen, 2022; +Whitelaw và c.s., 2020). Có nhiều nghiên cứu +trên thế giới tìm hiểu các yếu tố ảnh hưởng +tới việc sử dụng phần mềm truy vết nói chung +(Mbunge, 2020; Whitelaw và c.s., 2020), +nhưng chưa có nghiên cứu nào được thực +hiện ở Việt Nam trả lời cho câu hỏi trên một +cách đầy đủ. Vì vậy nghiên cứu này có vị trí +riêng biệt và cần thiết trong bối cảnh hiện +nay, đặc biệt khi đại dịch Covid-19 vẫn chưa +có dấu hiệu kết thúc do sự xuất hiện của các +biến chủng mới. Nghiên cứu của Nguyen và +c.s. (2021) mới chỉ dừng lại ở việc trích xuất +được các nhân tố mà chưa xem xét đến mối +tương quan giữa các nhân tố đó tới ý định sử +dụng phần mềm Bluezone như thế nào. Chính +vì vậy, nghiên cứu này được mở rộng bằng +cách áp dụng mô hình phương trình cấu trúc +nhằm đánh giá mối quan hệ giữa các yếu tố +tới ý định sử dụng phần mềm Bluezone. Kết +quả của bài báo được kỳ vọng sẽ có những +đóng góp tích cực trong lĩnh vực nghiên cứu +bao gồm: 1) việc khám phá ra các nhân tố +chính ảnh hưởng tới ý định sử dụng phần +mềm Bluezone, 2) đánh giá mối quan hệ giữa +các yếu tố tới ý định sử dụng phần mềm +Bluezone. Kết quả nghiên cứu sẽ là tài liệu +tham khảo cho các nghiên cứu tương tự và là +một trong các chỉ báo giúp các nhà quản lý +điều chỉnh chính sách phù hợp nhằm nâng cao +hiệu quả của ứng dụng truy vết. +2. MÔ HÌNH NGHIÊN CỨU VÀ CƠ SỞ +LÝ THUYẾT +2.1. Tổng quan về mô hình nghiên cứu +Sự phát triển không ngừng của các thiết bị +mới và phần mềm mới đã giúp cho người +dùng trải nghiệm và giải quyết các vấn đề +trong cuộc sống dễ dàng hơn. Tuy nhiên, +không phải mọi công nghệ mới đều được +người dùng chấp nhận và sử dụng. Để giảm +thiểu các rủi ro trên, nhiều mô hình chấp nhận +công nghệ được phát triển và áp dụng rộng rãi +như: mô hình SOR – stimulus (kích thích), +organism (chủ thể), response (phản hồi) – mô +tả cách mà sinh vật, con người phản ứng, đáp +lại với kích thích từ môi trường (Mehrabian +& Russell, 1974), mô hình chấp nhận công +nghệ – Technology Acceptance Model +(TAM) (Davis, 1985), mô hình lý thuyết chấp +nhận công nghệ hợp nhất (UTAUT). UTAUT +được phát triển bằng việc kết hợp và tinh +chỉnh tám mô hình trước đây thành một mô +hình duy nhất để mô tả hành vi của người + + + + + + Số 02 (2022): 1 – 11 + 3 + +KHOA HỌC TỰ NHIÊN +dùng với một hệ thống công nghệ thông tin +(Venkatesh và c.s., 2003). Mô hình UTAUT +chỉ ra có 4 yếu tố chính ảnh hưởng đến hành vi +của người dùng bao gồm: kỳ vọng hiệu quả +(performance expectancy), kì vọng nỗ lực +(effort expectancy), ảnh hưởng xã hội (social +influence), và các điều kiện thuận lợi +(facilitating conditions). Ngoài ra còn có các +yếu tố khác điều chỉnh đến ý định sử dụng như +giới tính, độ tuổi, sự tự nguyện và kinh nghiệm. +UTAUT được áp dụng rộng rãi trong nhiều lĩnh +vực khác nhau (Jung và c.s., 2020, 2021; +Nguyen, 2022). Trong nghiên cứu này, chúng +tôi mở rộng mô hình UTAUT với hai nhân tố +mới là sự riêng tư (privacy) và độ tin cậy (trust) +được tham khảo từ những nghiên cứu tương tự +(Arfi và c.s., 2021; Chopdar, 2022). +2.2. Cơ sở lý thuyết +Kỳ +vọng +hiệu +quả +(Performance +Expectancy) được định nghĩa là mức độ mà +một cá nhân tin rằng việc sử dụng hệ thống sẽ +giúp họ đạt được hiệu quả trong công việc +(Venkatesh và c.s., 2003). Năm yếu tố từ các +mô hình khác nhau liên quan đến kỳ vọng +hiệu quả là nhận thức phần mềm hữu ích, +động lực bên ngoài, sự phù hợp với công việc, +lợi thế tương đối và kỳ vọng kết quả. +Kỳ vọng nỗ lực (Effort Expectancy) được +định nghĩa là mức độ dễ dàng liên quan đến +việc sử dụng hệ thống (Venkatesh và c.s., +2003). Ba yếu tố từ các mô hình khác nhau +liên quan đến kỳ vọng nỗ lực là nhận thức dễ +sử dụng, độ phức tạp (mô hình sử dụng máy +tính) và tính dễ dùng (mô hình khuếch tán +đổi mới). +Ảnh hưởng xã hội (Social Influence) được +định nghĩa là mức độ mà một cá nhân nhận +thấy rằng những người khác quan trọng tin +rằng họ nên sử dụng hệ thống mới (Venkatesh +và c.s., 2003). Ba yếu tố từ các mô hình khác +nhau liên quan đến ảnh hưởng xã hội là chuẩn +chủ quan, yếu tố xã hội và hình ảnh. +Các điều kiện thuận lợi (Facilitating +Conditions) được định nghĩa là “Mức độ mà +một cá nhân tin rằng có sẵn cơ sở hạ tầng kỹ +thuật và tổ chức để hỗ trợ việc sử dụng hệ +thống” (Venkatesh và c.s., 2003). Venkatesh +cho rằng các điều kiện thuận lợi không ảnh +hưởng đến ý định hành vi, nhưng ảnh hưởng +đến hành vi sử dụng. Các điều kiện thuận lợi +liên quan đến sự sẵn có của nguồn lực và hỗ +trợ cho các cá nhân sử dụng công nghệ. +Rủi ro về quyền riêng tư (Privacy Risk) +được hiểu là mối quan ngại của người dùng +về việc tiết lộ thông tin cá nhân (Arfi và c.s., +2021; Chopdar, 2022; Li, 2011). Nhiều +nghiên cứu đã chỉ ra rằng rủi ro về quyền +riêng tư có ảnh hưởng tới độ tin cậy của người +dùng và gián tiếp ảnh hưởng đến ý định sử +dụng hệ thống (Arfi và c.s., 2021; Bansal và +c.s., 2010; Chopdar, 2022). +Sự tin tưởng (Trust) phản ánh sự sẵn sàng +ở trong tình trạng dễ bị tổn thương dựa trên +kỳ vọng tích cực đối với hành vi trong tương +lai của yếu tố ngoại vi (Arfi và c.s., 2021; +Chopdar, 2022). Nhiều nghiên cứu đã chỉ ra +rằng sự tin tưởng có ảnh hưởng tới ý định +hành vi và nhận thức rủi ro (Arfi và c.s., 2021; +Chopdar, 2022). +3. PHƯƠNG PHÁP NGHIÊN CỨU +3.1. Đối tượng nghiên cứu +Phiếu khảo sát được tạo ra và gửi đến người +dùng thông qua ứng dụng Zalo và mạng xã hội +Facebook trong khoảng thời gian từ ngày +18/6/2021 đến ngày 21/6/2021. Số lượng ước +lượng người dùng tham gia khảo sát là 400 +người, tỷ lệ phản hồi là 73,75% (295 phản +hồi), nhóm nghiên cứu loại bỏ 25 phản hồi do +người dùng không cài đặt ứng dụng Bluezone, +41 câu trả lời không hợp lệ do chỉ chọn một +lựa chọn duy nhất, 5 phản hồi không hoàn +thành khảo sát. Tổng số dữ liệu cuối cùng để +đưa vào phân tích là 224 (75,93%). Bảng 1 +tổng hợp dữ liệu từ phiếu khảo sát, tỷ lệ nam +chiếm 16,07%, trong khi đó tỷ lệ nữ chiếm +83,48%. Hơn một nửa đối tượng tham gia điều +tra là sinh viên, học sinh trong độ tuổi từ 10 – +20 (52,68%), 27,23% nằm trong độ tuổi từ 21 +– 30, 11,16% nằm trong độ tuổi 31 – 40%, số +còn lại trên 41 tuổi chiếm 8,93%. Khu vực sinh +sống của người dùng ứng dụng Bluezone chủ +yếu tập trung ở khu vực thị xã, nông thôn và +miền núi (52,23%), còn lại là ở các khu vực +thành phố (28,57%) và quận /huyện (19,20%). +Kết quả của phiếu khảo sát này cũng phù hợp +với đặc tính vùng miền của tỉnh Thái Nguyên +– là tỉnh miền núi. + +ap chi khoa hoc +DAI HOC HA LONG + + + +4 Số 01(2021): 1 – 11 + + + +KHOA HỌC TỰ NHIÊN +3.2. Công cụ khảo sát +Sau khi nghiên cứu các câu hỏi dùng cho +việc khảo sát dựa trên mô hình nghiên cứu +(Arfi và c.s., 2021; Chopdar, 2022), 18 câu +hỏi được nhóm tác giả lựa chọn và đưa vào +nghiên cứu (xem +Bảng 2). Thang điểm Likert năm điểm (1 += Hoàn toàn không đồng ý, 2 = Không đồng +ý, 3 = Trung lập, 4 = Đồng ý, 5 = Hoàn toàn +đồng ý) được sử dụng cho mỗi câu hỏi. +Bảng 1. Thông tin chung về đối tượng khảo sát +Thông tin chung +Số lượng +% + +Giới tính + + + +Nam +36 +16,07 +Nữ +187 +83,48 +Không xác định +1 +0,45 +Độ tuổi + + +10 – 20 +118 +52,68 +21 – 30 +61 +27,23 +31 – 40 +25 +11,16 +Trên 40 tuổi +20 +8,93 +Khu vực sinh sống + + +Thành phố +64 +28,57 +Quận/huyện +43 +19,20 +Thị xã, nông thôn +117 +52,23 +Tổng +224 +100 +3.3. Phân tích các nhân tố khám phá +Phân tích nhân tố khám phá (Explatory +Factor Analysis - EFA) là một phương pháp +thống kê dùng để rút gọn nhiều biến đo lường +phụ thuộc lẫn nhau (đo được) thành một tập +biến ít hơn (gọi là các nhân tố – không đo +được trực tiếp) mà vẫn chứa đựng hầu hết nội +dung thông tin của tập biến ban đầu (Hair Jr +và c.s., 2009). EFA giả định rằng mỗi chỉ số +trong một tập hợp các chỉ số là một hàm tuyến +tính của một hoặc nhiều nhân tố chung và một +nhân tố duy nhất. Các nhân tố chung là các +yếu tố tiềm ẩn không thể quan sát được có ảnh +hưởng đến nhiều hơn một chỉ số trong một +tập hợp các chỉ số (Fabrigar & Wegener, +2012). Các nhân tố duy nhất là các biến tiềm +ẩn được giả định chỉ ảnh hưởng đến một chỉ +số từ một tập hợp các chỉ số và không tính +đến mối tương quan giữa các chỉ số. Mục tiêu +của mô hình nhân tố chung là tìm hiểu cấu +trúc mối tương quan giữa các chỉ số bằng +cách ước tính các mô hình mối quan hệ giữa +các chỉ số và các nhân tố tiềm ẩn được lập chỉ +mục gọi là tải nhân tố. +Bảng 2. Bảng câu hỏi sử dụng khảo sát +Mã Câu hỏi + +Kỳ vọng hiệu quả (Venkatesh và c.s., 2003) + +PE1 Sử dụng phần mềm Bluezone giúp tôi nắm +bắt thông tin về Covid nhanh hơn. + +PE2 Sử dụng phần mềm Bluezone giúp tôi +nâng cao hiệu quả về phòng tránh Covid. + +PE3 Sử dụng phần mềm Bluezone giúp tôi nắm +bắt kịp thời các thông tin cần thiết nơi tôi +sinh sống. + +Kỳ vọng nỗ lực (Venkatesh và c.s., 2003) + +EE1 Học cách sử dụng phần mềm Bluezone là +tương đối dễ với tôi. + +EE2 Các chức năng và thao tác của Bluezone là +rõ ràng và dễ hiểu. + +EE3 Phần mềm Bluezone là dễ sử dụng. + +EE4 Tôi dễ dàng sử dụng thành thạo phần mềm +Bluezone. + +Ảnh hưởng xã hội (Venkatesh và c.s., 2003) + +SI1 Người thân trong gia đình tôi cho rằng tôi +nên sử dụng phần mềm Bluezone. + +SI2 Bạn bè và đồng nghiệp tôi cho rằng tôi +nên sử dụng phần mềm Bluezone. + +SI3 Tôi sử dụng phần mềm Bluezone là do +được tuyên truyền từ các phương tiện +truyền thông. + +Các điều kiện thuận lợi (Venkatesh và c.s., 2003) + +FC1 Tôi có thiết bị để cài đặt phần mềm +Bluezone (ví dụ: điện thoại, máy tính +bảng). + +FC2 Phần mềm Bluezone tương thích với các +thiết bị của tôi. + +FC3 Tôi có sự hỗ trợ khi gặp trục trặc với phần +mềm Bluezone. + +Rủi ro về quyền riêng tư (Arfi và c.s., 2021; +Chopdar, 2022) + +PR1 Tôi nghĩ rằng việc sử dụng Bluezone sẽ +khiến quyền riêng tư của tôi gặp rủi ro. + +PR2 Dữ liệu cá nhân của tôi có thể bị rò rỉ khi +sử dụng phần mềm Bluezone. + +Sự tin tưởng (Trust) (Arfi và c.s., 2021; +Chopdar, 2022) + +T1 +Tôi tin rằng thông tin mà Bluezone cung +cấp là đáng tin cậy. + +T2 +Tôi tin tưởng việc sử dụng phần mềm +Bluezone. + +T3 +Bluezone cung cấp các chức năng mà +người dùng cần. + +Nếu giá trị trung bình của một câu được tìm +thấy là gần với 1 hoặc 5 thì nhóm nghiên cứu + + + + + + Số 02 (2022): 1 – 11 + 5 + +KHOA HỌC TỰ NHIÊN +loại bỏ câu trả lời đó ra khỏi bảng số liệu vì nó +có thể làm giảm tiêu chuẩn tương quan giữa các +mục còn lại (J. J. Kim, 2011). Sau bước này, +tính chuẩn mực trong phân phối đã được kiểm +tra bằng cách kiểm tra độ lệch (skewness) và độ +nhọn (kurtosis) trước khi tiến hành phân tích +nhân tố khám phá. Vì tính chuẩn mực của phân +phối đã được xác nhận, nên việc phân tích nhân +tố khám phá được tiến hành thông qua việc sử +dụng phần mềm SPSS 26 (Statistical Package +for the Social Sciences). +Tiến trình phân tích nhân tố khám phá được +bắt đầu bằng việc thu thập các giá trị riêng +(eigenvalues) cho mỗi nhân tố. Tiếp theo, thang +đo Kaiser-Meyer-Olkin (KMO) được sử dụng +để đo về mức độ phù hợp của dữ liệu cho việc +phân tích nhân tố (Goretzko và c.s., 2021). Giá +trị của KMO thay đổi giữa 0 và 1 và các giá trị +trên 0,5 thường được coi là đủ cho EFA +(Goretzko và c.s., 2021; Schneeweiss & +Mathes, 1995). Mức độ tương quan giữa các +câu hỏi có đủ lớn để phân tích nhân tố có ý +nghĩa thống kê hay không được kiểm tra thông +qua phương pháp Bartlett. Chỉ khi kiểm định +Bartlett có ý nghĩa thống kê (sig. < 0,05) thì các +phân tích tiếp theo mới được tiến hành. +3.4. Mô hình phương trình cấu trúc +Sau khi có kết quả từ phân tích nhân tố +khám phá, các nhân tố tìm được sẽ được sử +dụng để tìm hiểu sự tác động của chúng đối +với ý định hành vi của việc sử dụng phần +mềm Bluezone. Mô hình phương trình cấu +trúc (Structural Equation Modeling – SEM) +được sử dụng để tìm hiểu sự tác động của các +biến độc lập (nhân tố) đối với biến phụ thuộc +(ý định hành vi) (Kline, 2015). SEM là một +mô hình cấu trúc tuyến tính bao gồm các mô +hình thống kê nhằm tìm lời giải thích mối +quan hệ giữa các biến số (Kline, 2015). SEM +được ứng dụng rộng rãi trong nhiều lĩnh vực +với các tên gọi khác nhau như phân tích cấu +trúc hiệp phương sai, phân tích biến ẩn, hoặc +mô hình nhân quả. Mục đích của SEM là +kiểm tra lý thuyết bằng cách chỉ định một mô +hình đại diện cho các dự đoán của lý thuyết +đó trong số các cấu trúc hợp lý được đo bằng +các biến quan sát thích hợp. +4. KẾT QUẢ NGHIÊN CỨU +4.1. Phân tích nhân tố khám phá +EFA được thực hiện trên 18 câu hỏi với +vòng quay Varimax. Kết quả phân tích từ +phần mềm SPSS cho phép nhóm nghiên cứu +trích xuất được giá trị đặc trưng cho từng +nhân tố. Phép đo KMO đã xác minh tính thích +hợp của việc lấy mẫu cho phép phân tích với +giá trị là 0,889 (xem Bảng 3), cao hơn đề xuất +của J. O. Kim & Mueller (1978) là 0,6. +Bảng 3. Kiểm định KMO và Barlett +Kaiser-Meyer-Olkin +0,889 +Kiểm định +Bartlett +Chi-Square +2825,528 +df +153 +Sig. +0,000 +Kiểm định Bartlett (Bartlett's test of +sphericity) cho kết quả χ2 (153) = 2825,528, +ρ < 0,000, chỉ ra rằng mối tương quan giữa +các hạng mục câu hỏi là đủ lớn để tiến hành +phân tích nhân tố khám phá. +Số liệu từ +Bảng 4 cho thấy có bốn nhân tố chính +được hình thành từ tập 18 câu hỏi với giá trị +đặc trưng lớn hơn 1. Nói cách khác, 18 câu +hỏi này đóng góp 70,269% tầm quan trọng +của các yếu tố tác động đến việc sử dụng ứng +dụng Bluezone, 29,731% còn lại là do các +yếu tố khác. Tỷ lệ phần trăm được giải thích +theo từng nhân tố là: nhân tố 1 (46,749%), +nhân tố 2 (10,563%), nhân tố 3 (6,587%) và +nhân tố 4 (3,369%). +Dữ liệu trong Bảng 5 cho thấy có sự dịch +chuyển về hạng mục câu hỏi giữa các nhân tố +chính. Trong mô hình ban đầu, chúng tôi giả +định rằng có sáu nhân tố chính ảnh hưởng tới +việc sử dụng phần mềm Bluezone, tuy nhiên +kết quả phân tích chỉ ra bốn nhân tố cơ bản +phản ánh mối tương quan giữa các câu hỏi. Có +một điểm đáng chú ý trong kết quả phân tích +đó là nhóm nhân tố chính thứ hai và thứ tư vẫn +giữ nguyên theo giả định ban đầu của nhóm +tác giả, trong khi nhóm nhân tố chính thứ nhất +được hình thành bằng việc kết hợp giữa hai +yếu tố sự tin cậy (trust) và kỳ vọng hiệu quả +(Performance Expectancy) – đặt lại tên là Hiệu + +ap chi khoa hoc +DAI HOC HA LONG + + + +6 Số 01(2021): 1 – 11 + + + +KHOA HỌC TỰ NHIÊN +quả tin cậy; Nhóm nhân tố thứ 3 được hình +thành bằng việc kết hợp giữa ảnh hưởng xã hội +và các điều kiện thuận lợi – đặt lại tên là Xã +hội và Kỳ vọng hiệu quả. Hạng mục FC3 (Tôi +có sự hỗ trợ khi gặp trục trặc với phần mềm +Bluezone) bị loại bỏ sau quá trình phân tích. +Bảng 4. Các nhân tố chính +Bảng 5. Ma trận nhân tố xoay + +1 +2 +3 +4 +T3 +0,721 + + + +PE2 +0,712 + + + +PE3 +0,690 + + + +T2 +0,649 + + + +PE1 +0,575 + + + +T1 +0,481 + + + +FC3 + + + + +EE1 + +0,769 + + +EE2 + +0,739 + + +EE3 + +0,688 + + +EE4 + +0,664 + + +SI2 + + +0,796 + +SI1 + + +0,671 + +FC1 + + +0,614 + +FC2 + + +0,566 + +SI3 + + +0,375 + +PR1 + + + +0,905 +PR2 + + + +0,872 +4.2. Mô hình phương trình cấu trúc +Dựa vào kết quả của phân tích nhân tố +khám phá, nhóm nghiên cứu đưa ra các giả +thiết sau: +H1. Sự tin tưởng và kỳ vọng hiệu quả có +ảnh hưởng tích cực tới ý định hành vi của việc +sử dụng phần mềm Bluezone. +H2. Kỳ vọng nỗ lực có ảnh hưởng tích cực tới ý +định hành vi của việc sử dụng phần mềm Bluezone. +H3. Ảnh hưởng xã hội có tác động tích cực tới ý +định hành vi của việc sử dụng phần mềm Bluezone. +H4. Rủi ro về quyền riêng tư có ảnh hưởng +tiêu cực tới ý định hành vi của việc sử dụng +phần mềm Bluezone. +Chỉ có các hạng mục có hệ số trong Bảng +5 lớn hơn 0,6 được giữ lại trong phân tích. Số +mẫu tối thiểu cần thiết để phân tích có ý nghĩa +thống kê theo công cụ tính toán của (Soper, +2022) là 166 (với 4 biến tiềm ẩn và 13 biến +quan sát được). Số mẫu trong nghiên cứu là +224 lớn hơn so với số mẫu tối thiểu. Kỹ thuật +phân tích thành phần có cấu trúc tổng quát +(GSCA) được sử dụng để phân tích mô hình +nghiên cứu được đề xuất do khả năng xử lý +với kích thước mẫu nhỏ trong khi cần phân +phối chuẩn nghiêm ngặt (Hwang & Takane, +2014). GSCA là một thành phần dựa trên mô +hình phương trình cấu trúc có thể được sử +dụng để mô phỏng các đường dẫn Bình +phương tối thiểu một phần (PLS). Nghiên cứu +này sử dụng phần mềm GSCA Pro trong việc +ước lượng các tham số (Hwang và c.s., 2021). + +Tính nhất quán của dữ liệu và các phép +đo giá trị hội tụ cho mỗi nhân tố được thể hiện +trong Bảng 6. Dillon-Goldstein’s rho được sử +dụng để đánh giá cho các yêu cầu về tính nhất +quán và độ tin cậy bên trong của mỗi nhân tố +(Hwang & Takane, 2014). +Nhân tố +Giá trị đặc trưng khởi tạo +Tổng bình phương của +hệ số tải nhân tố +Tổng bình phương +của hệ số tải nhân +tố xoay +Tổng +% Phương +sai +% Tích lũy +Tổng +% Phương +sai +% Tích lũy +Tổng +1 +8,415 +46,749 +46,749 +8,068 +44,823 +44,823 +3,765 +2 +1,901 +10,563 +57,312 +1,682 +9,343 +54,165 +3,326 +3 +1,186 +6,587 +63,900 +0,911 +5,062 +59,228 +2,652 +4 +1,146 +3,369 +70,269 +0,760 +4,220 +63,447 +1,677 +5 +0,973 +5,405 +75,674 + + + + +6 +0,728 +4,043 +79,717 + + + + + + + + + + Số 02 (2022): 1 – 11 + 7 + +KHOA HỌC TỰ NHIÊN +Bảng 6. Độ tin cậy của thang đo +Nhân tố +Rho +AVE +Sự tin tưởng và kỳ +vọng hiệu quả +0,912 +0,722 +Kỳ vọng nỗ lực +0,940 +0,796 +Ảnh hưởng xã hội +0,897 +0,746 +Rủi ro về quyền +riêng tư +0,947 +0,899 +Ý định hành vi +0,090 +0,750 +Hầu hết tất cả các giá trị, nằm trong +khoảng từ 0,897 đến 0,947, đều lớn hơn 0,7, +trên mức ước tính độ tin cậy có thể chấp nhận +được (Hwang & Takane, 2014). Chúng tôi +cũng đã xem xét giá trị phương sai trung bình +được trích xuất (Average Variance Extracted +– AVE) của mỗi biến tiềm ẩn để xác định xem +biến có hội tụ hay không. Tất cả các giá trị +AVE đều lớn hơn 0,5 (Hwang & Takane, +2014), nằm trong khoảng từ 0,722 đến 0,899, +cho thấy độ tin cậy hội tụ. +Bảng 7. Ước lượng hệ số tải (loadings) + +Ước lượng +SE +95%CI_LB +95%CI_UB +PE2 +0,876 +0,022 +0,826 +0,911 +PE3 +0,850 +0,031 +0,786 +0,904 +T2 +0,833 +0,031 +0,782 +0,893 +T3 +0,839 +0,027 +0,763 +0,887 +EE1 +0,849 +0,032 +0,789 +0,907 +EE2 +0,912 +0,017 +0,873 +0,939 +EE3 +0,915 +0,020 +0,867 +0,947 +EE4 +0,890 +0,019 +0,851 +0,932 +SI1 +0,896 +0,014 +0,869 +0,921 +SI2 +0,943 +0,008 +0,924 +0,955 +FC1 +0,739 +0,050 +0,621 +0,822 +PR1 +0,949 +0,008 +0,934 +0,968 +PR2 +0,948 +0,009 +0,931 +0,967 +BI1 +0,893 +0,029 +0,836 +0,939 +BI2 +0,869 +0,029 +0,808 +0,917 +BI3 +0,835 +0,043 +0,716 +0,889 +Bảng 7 cho thấy hệ số tải của các hạng +mục cùng với các tham số khác như sai số +chuẩn (SE), khoảng tin cậy dưới (CI_LB) và +khoảng tin cậy trên (CI_UB). Phương pháp +Boostrap thực hiện với số mẫu lặp lại là 100 +lần, giá trị trung bình của 100 lần lặp này +được dùng để ước lượng giá trị gần đúng của +tổng thể. Ở mức 0,05 alpha, ước tính tham số +được coi là có ý nghĩa thống kê nếu 95% +khoảng tin cậy không bao gồm giá trị 0. Kết +quả Bảng 7 cho thấy tất cả các hạng mục đều +đáng tin cậy và các ước lượng tải đều có ý +nghĩa thống kê. + +Kết quả phân tích từ phần mềm GSCA +Pro cho các kết quả như: độ phù hợp của mô +hình (Model FIT) là 0,59; độ phù hợp điều +chỉnh của mô hình (Adjusted FIT - AFIT) là +0,586. Cả FIT và FIT điều chỉnh (AFIT) đều +được sử dụng để điều tra sự khác biệt trong +dữ liệu được giải thích bởi một cấu hình mô +hình nhất định. Các giá trị FIT nằm trong +khoảng từ 0 đến 1. Các đặc điểm và ý nghĩa +của FIT và AFIT tương đương với R2 và R2 +điều chỉnh trong hồi quy tuyến tính. Kết quả +thực nghiệm của FIT và AFIT cho thấy mô +hình lần lượt chiếm khoảng 59% và 58,6% +tổng phương sai của tất cả các biến. + +ap chi khoa hoc +DAI HOC HA LONG + + + +8 Số 01(2021): 1 – 11 + + + +KHOA HỌC TỰ NHIÊN +Bảng 8. Ước tính hệ số đường dẫn + +Ước lượng +SE +95%CI_LB 95%CI_UB +Sự tin tưởng và kỳ vọng hiệu quả +→ Ý định hành vi sử dụng Bluezone (H1) +0,218* +0,105 +0,029 +0,049 +Kỳ vọng nỗ lực +→ Ý định hành vi sử dụng Bluezone (H2) +0,116* +0,084 +0,05 +0,290 +Ảnh hưởng xã hội +→ Ý định hành vi sử dụng Bluezone (H3) +0,137* +0,097 +0,043 +0,320 +Rủi ro về quyền riêng tư +→ Ý định hành vi sử dụng Bluezone (H4) +-0,06* +0.63 +0.06 +0.185 + * có ý nghĩa thống kê ở mức 0,05 +Bảng 8 trình bày các ước tính của hệ số +đường dẫn của mô hình phương trình cấu +trúc, cùng với sai số chuẩn và khoảng tin cậy +95% cận dưới và cận trên. Kết quả thực +nghiệm cho thấy sự tin tưởng và kỳ vọng hiệu +quả có ảnh hưởng tích cực tới ý định hành vi +sử dụng phần mềm Bluezone (H1 = 0,218; SE += 0,105; CIs = 0,029 – 0,049). Tương tự, kỳ +vọng nỗ lực có ảnh hưởng tích cực tới ý định +hành vi (H2 = 0,016; SE = 0,084; CIs = 0,05 +– 0,29). Ảnh hưởng xã hội cũng đóng góp vào +ý định hành vi một cách tích cực (H3 = 0,137; +SE = 0,097; CIs = 0,043 – 0,32) và cuối cùng +rủi ro về quyền riêng tư có ảnh hưởng tiêu cực +tới ý định hành vi sử dụng Bluezone của +người dùng (H4 = -0,06; SE = 0,63; CIs = +0,06 – 0,185). +5. THẢO LUẬN +Đã hơn hai năm kể từ khi đại dịch Covid- +19 xuất hiện, mặc dù số lượng ca nhiễm và tử +vong đã giảm đáng kể so với giai đoạn đầu +nhưng vẫn chưa có dấu hiệu nào cho thấy sự +kết thúc của đại dịch này. Cùng với các biện +pháp cách ly xã hội, tiêm vắc xin, khai báo +trực tiếp bằng văn bản, việc ứng dụng công +nghệ thông tin trong hỗ trợ đại dịch cũng đã +và đang đem lại nhiều lợi ích nhất định. Hầu +như các hoạt động xã hội đều đã được số hóa +như họp trực tuyến, đặt hàng trực tuyến, +thanh toán trực tuyến, giảng dạy trực tuyến... +đến truy vết bằng công nghệ số. Các hoạt +động này, cho dù không có đại dịch xảy ra, +cũng là xu hướng tất yếu trong chuyển đổi số, +nhưng sự xuất hiện của đại dịch khiến cho +quá trình này được chuyển đổi nhanh hơn. +Phần mềm Bluezone là sản phẩm kịp thời để +ứng phó nhanh với đại dịch. Tuy nhiên, số +lượng người dùng sử dụng liên tục lại không +được như kỳ vọng. Điều đó dẫn đến ứng dụng +công nghệ thông tin này chưa phát huy được +hết sức mạnh. Do đó, ý thức về việc tích cực +tham gia vào việc sử dụng ứng dụng +Bluezone (hay PC-Covid) vẫn cần phải được +nâng cao để giúp các nhà chức trách nhanh +chóng tìm ra các giải pháp kịp thời. +Kết quả thực nghiệm từ mô hình phương +trình cấu trúc cho thấy, cả bốn nhân tố thu +được từ phân tích các nhân tố khám phá đều +có ảnh hưởng tích cực hoặc tiêu cực tới ý định +hành vi của người dùng đối với phần mềm +Bluezone. Cụ thể, sự tin tưởng và kỳ vọng +hiệu quả, kỳ vọng nỗ lực, ảnh hưởng xã hội +có tác động tích cực đến ý định hành vi của +việc sử dụng phần mềm truy vết Bluezone. +Trong khi đó, rủi ro về quyền riêng tư có ảnh +hưởng tiêu cực đến hành vi này. +Về mặt lý thuyết, kết quả của nghiên cứu +này một lần nữa xác thực các mối quan hệ +nguyên nhân – hậu quả đã được nghiên cứu +và xác định ở trong mô hình phương trình cấu +trúc, qua đó tạo thêm nhiều minh chứng cho +sự tồn tại và ảnh hưởng của các nhân tố này. +Những độc giả quan tâm hoặc các nhà nghiên +cứu khác có thể tham khảo kết quả trên cho +các nghiên cứu tương tự. +Về mặt thực tiễn, kết quả nghiên cứu là cơ +sở để các nhà phát triển phần mềm, người +quản lý đưa ra các chiến lược và giải pháp phù +hợp để tăng cường ý định hành vi sử dụng + + + + + + Số 02 (2022): 1 – 11 + 9 + +KHOA HỌC TỰ NHIÊN +phần mềm truy vết Bluezone. Cụ thể, đối với +các nhân tố có ảnh hưởng tích cực, cần phải +liên tục và cập nhật phần mềm sao cho nó +thực sự mang lại hiệu quả hay nói cách khác, +dữ liệu có được sử dụng tối ưu cho các nhà +quản lý hay không. Hơn nữa, phần mềm phải +nên thiết kế dễ sử dụng để bất kỳ ai cũng có +thể tự thao tác. Ảnh hưởng xã hội cho thấy +phương tiện truyền thông, gia đình, bạn bè và +đồng nghiệp đóng vai trò quan trọng tới ý +định hành vi, do đó việc tuyên truyền cũng +nên tiếp tục được duy trì thông qua các +phương tiện truyền thông khác nhau. Vì rủi ro +về quyền riêng tư cũng đóng vai trò quyết +định tới ý định, hành vi của người sử dụng, +do đó các nhà quản lý, các nhà phát triển phần +mềm, an ninh mạng cũng phải có các kỹ +thuật, cơ chế, chính sách sử dụng và bảo vệ +một cách phù hợp để giúp người dùng yên +tâm hơn về dữ liệu cá nhân của mình. +Ngoài các yếu tố tích cực, nghiên cứu này +cũng tồn tại một số giới hạn. Thứ nhất, việc +lấy mẫu là không hoàn toàn ngẫu nhiên vì đối +tượng tham gia nghiên cứu nằm trong mạng +lưới của tác giả. Do đó việc khái quát hóa đến +một số lượng người dùng lớn hơn cần phải +được xem xét một cách kỹ lưỡng. Thứ hai, +việc khảo sát chỉ được thực hiện trong một +khoảng thời gian nhất định nên hành vi của +đối tượng tham gia nghiên cứu có thể không +nhất quán trong tương lai. Thứ ba, chỉ có một +số các nhân tố được đưa vào phân tích trong +mô hình phương trình cấu trúc, có thể tồn tại +nhiều nhân tố khác cũng có tầm ảnh hưởng +tới việc sử dụng Bluezone, do đó chúng tôi +khuyến nghị các nhà nghiên cứu quan tâm tìm +hiểu thêm các nhân tố mới này. +6. KẾT LUẬN +Nghiên cứu này khám phá các nhân tố +và đánh giá sự ảnh hưởng của các nhân tố +đó tới ý định hành vi của người dùng trong +việc sử dụng phần mềm truy vết Bluezone. +Mô hình lý thuyết thống nhất về chấp nhận +và sử dụng công nghệ được mở rộng thêm +hai nhân tố mới bao gồm sự tin tưởng và rủi +ro về quyền riêng tư. Kết quả khảo sát từ +224 người dùng cho thấy có bốn nhân tố +chính ảnh hưởng tới việc sử dụng phần +mềm truy vết, trong đó có 3 nhân tố ảnh +hưởng tích cực tới ý định hành vi, trong khi +đó nhân tố rủi ro về quyền riêng tư có ảnh +hưởng theo chiều ngược lại. Kết quả nghiên +cứu đóng góp về mặt lý thuyết bằng cách +giải thích sự ảnh hưởng của các nhân tố đối +với ý định hành vi một cách tinh gọn hơn +(giảm chiều từ 6 nhân tố xuống còn 4 nhân +tố trong EFA) và xác thực các mối quan hệ +nguyên nhân – hậu quả thông qua mô hình +phương trình cấu trúc. Đồng thời, kết quả +nghiên cứu cũng có thể được sử dụng trong +thực tiễn giúp các nhà quản lý, nhà phát +triển phần mềm, an ninh môi trường mạng +có thêm cơ sở để tiếp tục hoàn thiện phần +mềm truy vết Covid-19. +LỜI CẢM ƠN + +Nhóm tác giả trân trọng cảm ơn các +bạn bè, đồng nghiệp trong việc tham gia khảo +sát. Cảm ơn TS. Nguyễn Hải Minh (ICTU) vì +đã phổ biến phiếu khảo sát đến các sinh viên +trong trường. +TÀI LIỆU THAM KHẢO +Arfi, W. 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(2021). +GSCA Pro 1.0 User’s Manual. +Hwang, +H., +& +Takane, +Y. +(2014). +Generalized +Structured +Component +Analysis: A Component-Based Approach +to +Structural +Equation +Modeling. +Chapman +and +Hall/CRC. +https://doi.org/10.1201/b17872 +Jung, K., Nguyen, T. V., Piscarac, D., & +Yoo, S.-C. (2020). Meet the Virtual Jeju +Dol Harubang—The Mixed VR/AR +Application for Cultural Immersion in +Korea’s +Main +Heritage. +ISPRS +International +Journal +of +Geo- +Information, +9(6), +Art. +6. +https://doi.org/10.3390/ijgi9060367 +Jung, K., Nguyen, V. T., & Lee, J. (2021). +BlocklyXR: An Interactive Extended +Reality Toolkit for Digital Storytelling. +Applied +Sciences, +11(3), +Art. +3. +https://doi.org/10.3390/app11031073 +Kim, J. J. (2011). Developing an instrument +to measure social presence in distance +higher education. British Journal of +Educational Technology, 42(5), 763–777. +https://doi.org/10.1111/j.1467- +8535.2010.01107.x +Kim, J. O., & Mueller, C. W. (1978). Factor +Analysis: +Statistical +Methods +and +Practical Issues. SAGE Publications, Inc. +Kline, R. B. (2015). Principles and Practice +of Structural Equation Modeling (Fourth +edition). The Guilford Press. +Le, T.-A. T., Vodden, K., Wu, J., & Atiwesh, +G. (2021). Policy Responses to the +COVID-19 +Pandemic +in +Vietnam. +International Journal of Environmental +Research and Public Health, 18(2), Art. 2. +https://doi.org/10.3390/ijerph18020559 +Li, Y. (2011). Empirical Studies on Online +Information Privacy Concerns: Literature +Review and an Integrative Framework. +Communications of the Association for +Information +Systems, +28(1). +https://doi.org/10.17705/1CAIS.02828 +Mbunge, E. (2020). Integrating emerging +technologies into COVID-19 contact +tracing: Opportunities, challenges and +pitfalls. Diabetes & Metabolic Syndrome: +Clinical Research & Reviews, 14(6), +1631–1636. +https://doi.org/10.1016/j.dsx.2020.08.029 +Mehrabian, A., & Russell, J. A. (James A. +(1974). An approach to environmental +psychology. Cambridge, M.I.T. Press. +http://archive.org/details/approachtoenvir +o00albe +Nguyen, T. V. (2022). The perceptions of +social media users of digital detox apps +considering personality traits. Education +and Information Technologies, 27(7), +9293–9316. +https://doi.org/10.1007/s10639-022- +11022-7 +Nguyen, T. V., Anh, N., Tan, N., & Dinh, L. +(2021). Tìm hiểu các yếu tố ảnh hưởng tới + +ap chi khoa hoc +DAI HOC HA LONG + + + + Số 02 (2022): 1 – 11 + 11 + +KHOA HỌC TỰ NHIÊN +việc sử dụng ứng dụng Bluezone tại Việt +Nam. Hội thảo quốc gia lần thứ XXIV: +Một số vấn đề chọn lọc của Công nghệ +thông tin và truyền thông, Thái Nguyên. +Nguyen, T. V., & Nguyen, T. H. C. (2022). +Factors Influencing Intention to use the +COVID-19 Contact Tracing Application. +Journal of Computer Science, 18(6), 453– +462. +https://doi.org/10.3844/jcssp.2022.453.462 +Schneeweiss, H., & Mathes, H. (1995). +Factor +Analysis +and +Principal +Components. Journal of Multivariate +Analysis, +55(1), +105–124. +https://doi.org/10.1006/jmva.1995.1069 +Soper, D. S. (2022). A-priori Sample Size +Calculator for Structural Equation Models. +https://www.danielsoper.com/statcalc/cal +culator.aspx?id=89 +Venkatesh, V., Morris, M. G., Davis, G. B., & +Davis, F. D. (2003). User Acceptance of +Information Technology: Toward a Unified +View. MIS Quarterly, 27(3), 425–478. +https://doi.org/10.2307/30036540 +Whitelaw, S., Mamas, M. A., Topol, E., & +Spall, H. G. C. V. (2020). Applications of +digital +technology +in +COVID-19 +pandemic planning and response. The +Lancet Digital Health, 2(8), e435–e440. +https://doi.org/10.1016/S2589- +7500(20)30142-4 + +ap chi khoa hoc +DAI HOC HA LONG \ No newline at end of file diff --git a/C9FKT4oBgHgl3EQfYi5Z/content/tmp_files/load_file.txt b/C9FKT4oBgHgl3EQfYi5Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..43cbed6ce054431949a91a3dd66175cc08e142c1 --- /dev/null +++ b/C9FKT4oBgHgl3EQfYi5Z/content/tmp_files/load_file.txt @@ -0,0 +1,622 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf,len=621 +page_content='Số chuyên san (11/2022): 1 – 11 1 MỐI TƯƠNG QUAN CỦA CÁC NHÂN TỐ ẢNH HƯỞNG TỚI VIỆC SỬ DỤNG ỨNG DỤNG BLUEZONE Nguyễn Thế Vịnh1*, Nguyễn Tuấn Anh1, Nguyễn Hồng Tân1, Lương Khắc Định2 1Khoa Công nghệ thông tin, Trường ĐH Công nghệ thông tin và Truyền thông, ĐH Thái Nguyên 2Khoa Công nghệ thông tin, Trường ĐH Hạ Long * Email: vinhnt@ictu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='vn Ngày nhận bài: 11/6/2022 Ngày nhận bài sửa sau phản biện: 09/11/2022 Ngày chấp nhận đăng: DD/MM/YYYY TÓM TẮT Sự xuất hiện của đại dịch Covid-19 đã gây ra nhiều tác động tiêu cực đến mọi mặt của đời sống.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chính phủ đã áp dụng nhiều biện pháp để giảm thiểu sự ảnh hưởng và lây truyền của dịch bệnh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Trong số đó có việc áp dụng chuyển đổi số đối với việc quản lý và truy vết người bị nhiễm Covid thông qua phần mềm Bluezone (nay là PC-Covid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tuy nhiên, việc cài đặt và sử dụng Bluezone lại không được như kỳ vọng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Vì vậy, nghiên cứu này tìm hiểu những nhân tố chính và sự ảnh hưởng của chúng tới ý định hành vi của người dùng về việc sử dụng phần mềm truy vết Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Phiếu khảo sát được gửi tới người dùng thông qua công cụ Google Form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả phân tích các nhân tố khám phá trên 224 đối tượng khảo sát cho thấy, có bốn nhân tố chính ảnh hưởng tới hành vi của người dùng, trong đó: sự tin tưởng và kỳ vọng hiệu quả, kỳ vọng nỗ lực, ảnh hưởng xã hội có tác động tích cực đến ý định hành vi của việc sử dụng phần mềm truy vết Bluezone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' trong khi rủi ro về quyền riêng tư có ảnh hưởng tiêu cực đến hành vi này.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Từ khóa: EFA, SEM, UTAUT, tin tưởng, quyền riêng tư, Covid-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' FACTORS INFLUENCING TO USE OF BLUEZONE ABSTRACT The emergence of the Covid-19 pandemic has been causing many negative impacts on all aspects of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' The government has taken many measures to minimize the impact and transmission of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Among them is the application of digital transformation to the management and tracing of people infected with Covid through the Bluezone app (now PC- Covid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' However, using and installing Bluezone is not as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Therefore, this study aims to understand the main factors and their influence on the behavioral intention of users about using Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Surveys are sent to users through the Google Form tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Experimental results through analysis of exploratory factors on 224 survey subjects show that there are 4 main factors affecting user behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Structural equation modeling indicates that trust, performance expectations, effort expectations, and social influence have a positive impact on behavioral intention of using Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Meanwhile, privacy risks have a negative effect on this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Keywords: EFA, SEM, UTAUT, trust, privacy, Covid-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' ap chi khoa hoc DAI HOC HA LONGTAP CHI KHOAHOC DAI HOCHALONG Scientific JournalofHa Long Vniversity KHOAHOC DAIHOCHALONG http://uhl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='vnl Hac de thanh cong 2 Số 01(2021): 1 – 11 KHOA HỌC TỰ NHIÊN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' ĐẶT VẤN ĐỀ Đại dịch Covid-19 xuất hiện vào cuối năm 2019 và bùng phát mạnh mẽ trong thời gian qua đã có những ảnh hưởng tiêu cực tới tất cả các quốc gia trên toàn thế giới (Whitelaw và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Đứng trước vấn đề đó, chính phủ các quốc gia trên thế giới đã tiến hành nhiều biện pháp cấp bách nhằm hạn chế tầm ảnh hưởng, lây lan của dịch bệnh (Nguyen và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Song song với các biện pháp tuyên truyền đến người dân về ý thức phòng chống dịch thông qua các phương tiện truyền thông, chính phủ Việt Nam cũng tiến hành nhiều biện pháp hỗ trợ nhằm truy vết tiếp xúc và cảnh báo người nhiễm Covid-19 (Le và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Cụ thể, Bộ Y tế và Bộ Thông tin và Truyền thông đã phối hợp tạo ra ứng dụng Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Bluezone được coi là “cần thiết trong quá trình sinh hoạt hàng ngày, khi mọi người có tiếp xúc, ứng dụng trên điện thoại của họ sẽ tự “nói chuyện” với nhau” (baochinhphu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='vn, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ứng dụng Bluezone được kỳ vọng là sẽ giúp ích cho các cơ quan nhà nước có thể nhanh chóng truy vết và quản lý được các ca nhiễm trong cộng đồng, người dân có thể nắm bắt được thông tin kịp thời để phòng dịch (Nguyen và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Mặc dù Bluezone được kỳ vọng sẽ mang lại hiệu quả tích cực cao và nhiều người sẽ sử dụng, nhưng số liệu thống kê thực tế lại không được như mong muốn (Nguyen và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tính đến 27 tháng 5 năm 2021, cả nước chỉ ghi nhận 33,48 triệu lượt tải (khoảng 34,7% so với tổng dân số), trong đó tập trung chủ yếu ở hai địa phương lớn là Hà Nội (3,1 triệu lượt cài đặt) và Thành phố Hồ Chí Minh (2,83 triệu lượt cài đặt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ở chiều ngược lại, các tỉnh khác như Điện Biên, Kon Tum, Lai Châu, Bắc Kạn lại ghi nhận số lượng người tải ứng dụng Bluezone thấp nhất.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Vì vậy, câu hỏi đặt ra là: Những yếu tố nào ảnh hưởng tới việc sử dụng phần mềm Bluezone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Trả lời được câu hỏi nghiên cứu trên đóng vai trò quan trọng trong việc khuyến khích người dân tham gia, hỗ trợ phòng chống dịch trên môi trường số (Nguyen & Nguyen, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Whitelaw và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Có nhiều nghiên cứu trên thế giới tìm hiểu các yếu tố ảnh hưởng tới việc sử dụng phần mềm truy vết nói chung (Mbunge, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Whitelaw và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2020), nhưng chưa có nghiên cứu nào được thực hiện ở Việt Nam trả lời cho câu hỏi trên một cách đầy đủ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Vì vậy nghiên cứu này có vị trí riêng biệt và cần thiết trong bối cảnh hiện nay, đặc biệt khi đại dịch Covid-19 vẫn chưa có dấu hiệu kết thúc do sự xuất hiện của các biến chủng mới.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Nghiên cứu của Nguyen và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' (2021) mới chỉ dừng lại ở việc trích xuất được các nhân tố mà chưa xem xét đến mối tương quan giữa các nhân tố đó tới ý định sử dụng phần mềm Bluezone như thế nào.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chính vì vậy, nghiên cứu này được mở rộng bằng cách áp dụng mô hình phương trình cấu trúc nhằm đánh giá mối quan hệ giữa các yếu tố tới ý định sử dụng phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả của bài báo được kỳ vọng sẽ có những đóng góp tích cực trong lĩnh vực nghiên cứu bao gồm: 1) việc khám phá ra các nhân tố chính ảnh hưởng tới ý định sử dụng phần mềm Bluezone, 2) đánh giá mối quan hệ giữa các yếu tố tới ý định sử dụng phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả nghiên cứu sẽ là tài liệu tham khảo cho các nghiên cứu tương tự và là một trong các chỉ báo giúp các nhà quản lý điều chỉnh chính sách phù hợp nhằm nâng cao hiệu quả của ứng dụng truy vết.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' MÔ HÌNH NGHIÊN CỨU VÀ CƠ SỞ LÝ THUYẾT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tổng quan về mô hình nghiên cứu Sự phát triển không ngừng của các thiết bị mới và phần mềm mới đã giúp cho người dùng trải nghiệm và giải quyết các vấn đề trong cuộc sống dễ dàng hơn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tuy nhiên, không phải mọi công nghệ mới đều được người dùng chấp nhận và sử dụng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Để giảm thiểu các rủi ro trên, nhiều mô hình chấp nhận công nghệ được phát triển và áp dụng rộng rãi như: mô hình SOR – stimulus (kích thích), organism (chủ thể), response (phản hồi) – mô tả cách mà sinh vật, con người phản ứng, đáp lại với kích thích từ môi trường (Mehrabian & Russell, 1974), mô hình chấp nhận công nghệ – Technology Acceptance Model (TAM) (Davis, 1985), mô hình lý thuyết chấp nhận công nghệ hợp nhất (UTAUT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' UTAUT được phát triển bằng việc kết hợp và tinh chỉnh tám mô hình trước đây thành một mô hình duy nhất để mô tả hành vi của người Số 02 (2022): 1 – 11 3 KHOA HỌC TỰ NHIÊN dùng với một hệ thống công nghệ thông tin (Venkatesh và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Mô hình UTAUT chỉ ra có 4 yếu tố chính ảnh hưởng đến hành vi của người dùng bao gồm: kỳ vọng hiệu quả (performance expectancy), kì vọng nỗ lực (effort expectancy), ảnh hưởng xã hội (social influence), và các điều kiện thuận lợi (facilitating conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ngoài ra còn có các yếu tố khác điều chỉnh đến ý định sử dụng như giới tính, độ tuổi, sự tự nguyện và kinh nghiệm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' UTAUT được áp dụng rộng rãi trong nhiều lĩnh vực khác nhau (Jung và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Nguyen, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Trong nghiên cứu này, chúng tôi mở rộng mô hình UTAUT với hai nhân tố mới là sự riêng tư (privacy) và độ tin cậy (trust) được tham khảo từ những nghiên cứu tương tự (Arfi và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chopdar, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Cơ sở lý thuyết Kỳ vọng hiệu quả (Performance Expectancy) được định nghĩa là mức độ mà một cá nhân tin rằng việc sử dụng hệ thống sẽ giúp họ đạt được hiệu quả trong công việc (Venkatesh và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Năm yếu tố từ các mô hình khác nhau liên quan đến kỳ vọng hiệu quả là nhận thức phần mềm hữu ích, động lực bên ngoài, sự phù hợp với công việc, lợi thế tương đối và kỳ vọng kết quả.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kỳ vọng nỗ lực (Effort Expectancy) được định nghĩa là mức độ dễ dàng liên quan đến việc sử dụng hệ thống (Venkatesh và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ba yếu tố từ các mô hình khác nhau liên quan đến kỳ vọng nỗ lực là nhận thức dễ sử dụng, độ phức tạp (mô hình sử dụng máy tính) và tính dễ dùng (mô hình khuếch tán đổi mới).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ảnh hưởng xã hội (Social Influence) được định nghĩa là mức độ mà một cá nhân nhận thấy rằng những người khác quan trọng tin rằng họ nên sử dụng hệ thống mới (Venkatesh và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ba yếu tố từ các mô hình khác nhau liên quan đến ảnh hưởng xã hội là chuẩn chủ quan, yếu tố xã hội và hình ảnh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Các điều kiện thuận lợi (Facilitating Conditions) được định nghĩa là “Mức độ mà một cá nhân tin rằng có sẵn cơ sở hạ tầng kỹ thuật và tổ chức để hỗ trợ việc sử dụng hệ thống” (Venkatesh và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Venkatesh cho rằng các điều kiện thuận lợi không ảnh hưởng đến ý định hành vi, nhưng ảnh hưởng đến hành vi sử dụng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Các điều kiện thuận lợi liên quan đến sự sẵn có của nguồn lực và hỗ trợ cho các cá nhân sử dụng công nghệ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Rủi ro về quyền riêng tư (Privacy Risk) được hiểu là mối quan ngại của người dùng về việc tiết lộ thông tin cá nhân (Arfi và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chopdar, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Li, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Nhiều nghiên cứu đã chỉ ra rằng rủi ro về quyền riêng tư có ảnh hưởng tới độ tin cậy của người dùng và gián tiếp ảnh hưởng đến ý định sử dụng hệ thống (Arfi và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Bansal và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chopdar, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Sự tin tưởng (Trust) phản ánh sự sẵn sàng ở trong tình trạng dễ bị tổn thương dựa trên kỳ vọng tích cực đối với hành vi trong tương lai của yếu tố ngoại vi (Arfi và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chopdar, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Nhiều nghiên cứu đã chỉ ra rằng sự tin tưởng có ảnh hưởng tới ý định hành vi và nhận thức rủi ro (Arfi và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chopdar, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' PHƯƠNG PHÁP NGHIÊN CỨU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Đối tượng nghiên cứu Phiếu khảo sát được tạo ra và gửi đến người dùng thông qua ứng dụng Zalo và mạng xã hội Facebook trong khoảng thời gian từ ngày 18/6/2021 đến ngày 21/6/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Số lượng ước lượng người dùng tham gia khảo sát là 400 người, tỷ lệ phản hồi là 73,75% (295 phản hồi), nhóm nghiên cứu loại bỏ 25 phản hồi do người dùng không cài đặt ứng dụng Bluezone, 41 câu trả lời không hợp lệ do chỉ chọn một lựa chọn duy nhất, 5 phản hồi không hoàn thành khảo sát.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tổng số dữ liệu cuối cùng để đưa vào phân tích là 224 (75,93%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Bảng 1 tổng hợp dữ liệu từ phiếu khảo sát, tỷ lệ nam chiếm 16,07%, trong khi đó tỷ lệ nữ chiếm 83,48%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Hơn một nửa đối tượng tham gia điều tra là sinh viên, học sinh trong độ tuổi từ 10 – 20 (52,68%), 27,23% nằm trong độ tuổi từ 21 – 30, 11,16% nằm trong độ tuổi 31 – 40%, số còn lại trên 41 tuổi chiếm 8,93%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Khu vực sinh sống của người dùng ứng dụng Bluezone chủ yếu tập trung ở khu vực thị xã, nông thôn và miền núi (52,23%), còn lại là ở các khu vực thành phố (28,57%) và quận /huyện (19,20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả của phiếu khảo sát này cũng phù hợp với đặc tính vùng miền của tỉnh Thái Nguyên – là tỉnh miền núi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' ap chi khoa hoc DAI HOC HA LONG 4 Số 01(2021): 1 – 11 KHOA HỌC TỰ NHIÊN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Công cụ khảo sát Sau khi nghiên cứu các câu hỏi dùng cho việc khảo sát dựa trên mô hình nghiên cứu (Arfi và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chopdar, 2022), 18 câu hỏi được nhóm tác giả lựa chọn và đưa vào nghiên cứu (xem Bảng 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Thang điểm Likert năm điểm (1 = Hoàn toàn không đồng ý, 2 = Không đồng ý, 3 = Trung lập, 4 = Đồng ý, 5 = Hoàn toàn đồng ý) được sử dụng cho mỗi câu hỏi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Bảng 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Thông tin chung về đối tượng khảo sát Thông tin chung Số lượng % Giới tính Nam 36 16,07 Nữ 187 83,48 Không xác định 1 0,45 Độ tuổi 10 – 20 118 52,68 21 – 30 61 27,23 31 – 40 25 11,16 Trên 40 tuổi 20 8,93 Khu vực sinh sống Thành phố 64 28,57 Quận/huyện 43 19,20 Thị xã, nông thôn 117 52,23 Tổng 224 100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Phân tích các nhân tố khám phá Phân tích nhân tố khám phá (Explatory Factor Analysis - EFA) là một phương pháp thống kê dùng để rút gọn nhiều biến đo lường phụ thuộc lẫn nhau (đo được) thành một tập biến ít hơn (gọi là các nhân tố – không đo được trực tiếp) mà vẫn chứa đựng hầu hết nội dung thông tin của tập biến ban đầu (Hair Jr và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' EFA giả định rằng mỗi chỉ số trong một tập hợp các chỉ số là một hàm tuyến tính của một hoặc nhiều nhân tố chung và một nhân tố duy nhất.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Các nhân tố chung là các yếu tố tiềm ẩn không thể quan sát được có ảnh hưởng đến nhiều hơn một chỉ số trong một tập hợp các chỉ số (Fabrigar & Wegener, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Các nhân tố duy nhất là các biến tiềm ẩn được giả định chỉ ảnh hưởng đến một chỉ số từ một tập hợp các chỉ số và không tính đến mối tương quan giữa các chỉ số.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Mục tiêu của mô hình nhân tố chung là tìm hiểu cấu trúc mối tương quan giữa các chỉ số bằng cách ước tính các mô hình mối quan hệ giữa các chỉ số và các nhân tố tiềm ẩn được lập chỉ mục gọi là tải nhân tố.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Bảng 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Bảng câu hỏi sử dụng khảo sát Mã Câu hỏi Kỳ vọng hiệu quả (Venkatesh và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2003) PE1 Sử dụng phần mềm Bluezone giúp tôi nắm bắt thông tin về Covid nhanh hơn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' PE2 Sử dụng phần mềm Bluezone giúp tôi nâng cao hiệu quả về phòng tránh Covid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' PE3 Sử dụng phần mềm Bluezone giúp tôi nắm bắt kịp thời các thông tin cần thiết nơi tôi sinh sống.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kỳ vọng nỗ lực (Venkatesh và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2003) EE1 Học cách sử dụng phần mềm Bluezone là tương đối dễ với tôi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' EE2 Các chức năng và thao tác của Bluezone là rõ ràng và dễ hiểu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' EE3 Phần mềm Bluezone là dễ sử dụng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' EE4 Tôi dễ dàng sử dụng thành thạo phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ảnh hưởng xã hội (Venkatesh và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2003) SI1 Người thân trong gia đình tôi cho rằng tôi nên sử dụng phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' SI2 Bạn bè và đồng nghiệp tôi cho rằng tôi nên sử dụng phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' SI3 Tôi sử dụng phần mềm Bluezone là do được tuyên truyền từ các phương tiện truyền thông.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Các điều kiện thuận lợi (Venkatesh và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2003) FC1 Tôi có thiết bị để cài đặt phần mềm Bluezone (ví dụ: điện thoại, máy tính bảng).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' FC2 Phần mềm Bluezone tương thích với các thiết bị của tôi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' FC3 Tôi có sự hỗ trợ khi gặp trục trặc với phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Rủi ro về quyền riêng tư (Arfi và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chopdar, 2022) PR1 Tôi nghĩ rằng việc sử dụng Bluezone sẽ khiến quyền riêng tư của tôi gặp rủi ro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' PR2 Dữ liệu cá nhân của tôi có thể bị rò rỉ khi sử dụng phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Sự tin tưởng (Trust) (Arfi và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chopdar, 2022) T1 Tôi tin rằng thông tin mà Bluezone cung cấp là đáng tin cậy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' T2 Tôi tin tưởng việc sử dụng phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' T3 Bluezone cung cấp các chức năng mà người dùng cần.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Nếu giá trị trung bình của một câu được tìm thấy là gần với 1 hoặc 5 thì nhóm nghiên cứu Số 02 (2022): 1 – 11 5 KHOA HỌC TỰ NHIÊN loại bỏ câu trả lời đó ra khỏi bảng số liệu vì nó có thể làm giảm tiêu chuẩn tương quan giữa các mục còn lại (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kim, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Sau bước này, tính chuẩn mực trong phân phối đã được kiểm tra bằng cách kiểm tra độ lệch (skewness) và độ nhọn (kurtosis) trước khi tiến hành phân tích nhân tố khám phá.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Vì tính chuẩn mực của phân phối đã được xác nhận, nên việc phân tích nhân tố khám phá được tiến hành thông qua việc sử dụng phần mềm SPSS 26 (Statistical Package for the Social Sciences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tiến trình phân tích nhân tố khám phá được bắt đầu bằng việc thu thập các giá trị riêng (eigenvalues) cho mỗi nhân tố.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tiếp theo, thang đo Kaiser-Meyer-Olkin (KMO) được sử dụng để đo về mức độ phù hợp của dữ liệu cho việc phân tích nhân tố (Goretzko và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Giá trị của KMO thay đổi giữa 0 và 1 và các giá trị trên 0,5 thường được coi là đủ cho EFA (Goretzko và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Schneeweiss & Mathes, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Mức độ tương quan giữa các câu hỏi có đủ lớn để phân tích nhân tố có ý nghĩa thống kê hay không được kiểm tra thông qua phương pháp Bartlett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chỉ khi kiểm định Bartlett có ý nghĩa thống kê (sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' < 0,05) thì các phân tích tiếp theo mới được tiến hành.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Mô hình phương trình cấu trúc Sau khi có kết quả từ phân tích nhân tố khám phá, các nhân tố tìm được sẽ được sử dụng để tìm hiểu sự tác động của chúng đối với ý định hành vi của việc sử dụng phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Mô hình phương trình cấu trúc (Structural Equation Modeling – SEM) được sử dụng để tìm hiểu sự tác động của các biến độc lập (nhân tố) đối với biến phụ thuộc (ý định hành vi) (Kline, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' SEM là một mô hình cấu trúc tuyến tính bao gồm các mô hình thống kê nhằm tìm lời giải thích mối quan hệ giữa các biến số (Kline, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' SEM được ứng dụng rộng rãi trong nhiều lĩnh vực với các tên gọi khác nhau như phân tích cấu trúc hiệp phương sai, phân tích biến ẩn, hoặc mô hình nhân quả.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Mục đích của SEM là kiểm tra lý thuyết bằng cách chỉ định một mô hình đại diện cho các dự đoán của lý thuyết đó trong số các cấu trúc hợp lý được đo bằng các biến quan sát thích hợp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' KẾT QUẢ NGHIÊN CỨU 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Phân tích nhân tố khám phá EFA được thực hiện trên 18 câu hỏi với vòng quay Varimax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả phân tích từ phần mềm SPSS cho phép nhóm nghiên cứu trích xuất được giá trị đặc trưng cho từng nhân tố.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Phép đo KMO đã xác minh tính thích hợp của việc lấy mẫu cho phép phân tích với giá trị là 0,889 (xem Bảng 3), cao hơn đề xuất của J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kim & Mueller (1978) là 0,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Bảng 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kiểm định KMO và Barlett Kaiser-Meyer-Olkin 0,889 Kiểm định Bartlett Chi-Square 2825,528 df 153 Sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=" 0,000 Kiểm định Bartlett (Bartlett's test of sphericity) cho kết quả χ2 (153) = 2825,528, ρ < 0,000, chỉ ra rằng mối tương quan giữa các hạng mục câu hỏi là đủ lớn để tiến hành phân tích nhân tố khám phá." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Số liệu từ Bảng 4 cho thấy có bốn nhân tố chính được hình thành từ tập 18 câu hỏi với giá trị đặc trưng lớn hơn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Nói cách khác, 18 câu hỏi này đóng góp 70,269% tầm quan trọng của các yếu tố tác động đến việc sử dụng ứng dụng Bluezone, 29,731% còn lại là do các yếu tố khác.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tỷ lệ phần trăm được giải thích theo từng nhân tố là: nhân tố 1 (46,749%), nhân tố 2 (10,563%), nhân tố 3 (6,587%) và nhân tố 4 (3,369%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Dữ liệu trong Bảng 5 cho thấy có sự dịch chuyển về hạng mục câu hỏi giữa các nhân tố chính.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Trong mô hình ban đầu, chúng tôi giả định rằng có sáu nhân tố chính ảnh hưởng tới việc sử dụng phần mềm Bluezone, tuy nhiên kết quả phân tích chỉ ra bốn nhân tố cơ bản phản ánh mối tương quan giữa các câu hỏi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Có một điểm đáng chú ý trong kết quả phân tích đó là nhóm nhân tố chính thứ hai và thứ tư vẫn giữ nguyên theo giả định ban đầu của nhóm tác giả, trong khi nhóm nhân tố chính thứ nhất được hình thành bằng việc kết hợp giữa hai yếu tố sự tin cậy (trust) và kỳ vọng hiệu quả (Performance Expectancy) – đặt lại tên là Hiệu ap chi khoa hoc DAI HOC HA LONG 6 Số 01(2021): 1 – 11 KHOA HỌC TỰ NHIÊN quả tin cậy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Nhóm nhân tố thứ 3 được hình thành bằng việc kết hợp giữa ảnh hưởng xã hội và các điều kiện thuận lợi – đặt lại tên là Xã hội và Kỳ vọng hiệu quả.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Hạng mục FC3 (Tôi có sự hỗ trợ khi gặp trục trặc với phần mềm Bluezone) bị loại bỏ sau quá trình phân tích.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Bảng 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Các nhân tố chính Bảng 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ma trận nhân tố xoay 1 2 3 4 T3 0,721 PE2 0,712 PE3 0,690 T2 0,649 PE1 0,575 T1 0,481 FC3 EE1 0,769 EE2 0,739 EE3 0,688 EE4 0,664 SI2 0,796 SI1 0,671 FC1 0,614 FC2 0,566 SI3 0,375 PR1 0,905 PR2 0,872 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Mô hình phương trình cấu trúc Dựa vào kết quả của phân tích nhân tố khám phá, nhóm nghiên cứu đưa ra các giả thiết sau: H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Sự tin tưởng và kỳ vọng hiệu quả có ảnh hưởng tích cực tới ý định hành vi của việc sử dụng phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kỳ vọng nỗ lực có ảnh hưởng tích cực tới ý định hành vi của việc sử dụng phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ảnh hưởng xã hội có tác động tích cực tới ý định hành vi của việc sử dụng phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Rủi ro về quyền riêng tư có ảnh hưởng tiêu cực tới ý định hành vi của việc sử dụng phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chỉ có các hạng mục có hệ số trong Bảng 5 lớn hơn 0,6 được giữ lại trong phân tích.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Số mẫu tối thiểu cần thiết để phân tích có ý nghĩa thống kê theo công cụ tính toán của (Soper, 2022) là 166 (với 4 biến tiềm ẩn và 13 biến quan sát được).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Số mẫu trong nghiên cứu là 224 lớn hơn so với số mẫu tối thiểu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kỹ thuật phân tích thành phần có cấu trúc tổng quát (GSCA) được sử dụng để phân tích mô hình nghiên cứu được đề xuất do khả năng xử lý với kích thước mẫu nhỏ trong khi cần phân phối chuẩn nghiêm ngặt (Hwang & Takane, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' GSCA là một thành phần dựa trên mô hình phương trình cấu trúc có thể được sử dụng để mô phỏng các đường dẫn Bình phương tối thiểu một phần (PLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Nghiên cứu này sử dụng phần mềm GSCA Pro trong việc ước lượng các tham số (Hwang và c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tính nhất quán của dữ liệu và các phép đo giá trị hội tụ cho mỗi nhân tố được thể hiện trong Bảng 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Dillon-Goldstein’s rho được sử dụng để đánh giá cho các yêu cầu về tính nhất quán và độ tin cậy bên trong của mỗi nhân tố (Hwang & Takane, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Nhân tố Giá trị đặc trưng khởi tạo Tổng bình phương của hệ số tải nhân tố Tổng bình phương của hệ số tải nhân tố xoay Tổng % Phương sai % Tích lũy Tổng % Phương sai % Tích lũy Tổng 1 8,415 46,749 46,749 8,068 44,823 44,823 3,765 2 1,901 10,563 57,312 1,682 9,343 54,165 3,326 3 1,186 6,587 63,900 0,911 5,062 59,228 2,652 4 1,146 3,369 70,269 0,760 4,220 63,447 1,677 5 0,973 5,405 75,674 6 0,728 4,043 79,717 Số 02 (2022): 1 – 11 7 KHOA HỌC TỰ NHIÊN Bảng 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Độ tin cậy của thang đo Nhân tố Rho AVE Sự tin tưởng và kỳ vọng hiệu quả 0,912 0,722 Kỳ vọng nỗ lực 0,940 0,796 Ảnh hưởng xã hội 0,897 0,746 Rủi ro về quyền riêng tư 0,947 0,899 Ý định hành vi 0,090 0,750 Hầu hết tất cả các giá trị, nằm trong khoảng từ 0,897 đến 0,947, đều lớn hơn 0,7, trên mức ước tính độ tin cậy có thể chấp nhận được (Hwang & Takane, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Chúng tôi cũng đã xem xét giá trị phương sai trung bình được trích xuất (Average Variance Extracted – AVE) của mỗi biến tiềm ẩn để xác định xem biến có hội tụ hay không.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tất cả các giá trị AVE đều lớn hơn 0,5 (Hwang & Takane, 2014), nằm trong khoảng từ 0,722 đến 0,899, cho thấy độ tin cậy hội tụ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Bảng 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ước lượng hệ số tải (loadings) Ước lượng SE 95%CI_LB 95%CI_UB PE2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='876 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='022 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='826 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='911 PE3 0,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='029 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='836 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='939 BI2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='869 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='029 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='808 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='917 BI3 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='835 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='043 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='716 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='889 Bảng 7 cho thấy hệ số tải của các hạng mục cùng với các tham số khác như sai số chuẩn (SE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' khoảng tin cậy dưới (CI_LB) và khoảng tin cậy trên (CI_UB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Phương pháp Boostrap thực hiện với số mẫu lặp lại là 100 lần, giá trị trung bình của 100 lần lặp này được dùng để ước lượng giá trị gần đúng của tổng thể.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ở mức 0,05 alpha, ước tính tham số được coi là có ý nghĩa thống kê nếu 95% khoảng tin cậy không bao gồm giá trị 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả Bảng 7 cho thấy tất cả các hạng mục đều đáng tin cậy và các ước lượng tải đều có ý nghĩa thống kê.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả phân tích từ phần mềm GSCA Pro cho các kết quả như: độ phù hợp của mô hình (Model FIT) là 0,59;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' độ phù hợp điều chỉnh của mô hình (Adjusted FIT - AFIT) là 0,586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Cả FIT và FIT điều chỉnh (AFIT) đều được sử dụng để điều tra sự khác biệt trong dữ liệu được giải thích bởi một cấu hình mô hình nhất định.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Các giá trị FIT nằm trong khoảng từ 0 đến 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Các đặc điểm và ý nghĩa của FIT và AFIT tương đương với R2 và R2 điều chỉnh trong hồi quy tuyến tính.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả thực nghiệm của FIT và AFIT cho thấy mô hình lần lượt chiếm khoảng 59% và 58,6% tổng phương sai của tất cả các biến.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' ap chi khoa hoc DAI HOC HA LONG 8 Số 01(2021): 1 – 11 KHOA HỌC TỰ NHIÊN Bảng 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ước tính hệ số đường dẫn Ước lượng SE 95%CI_LB 95%CI_UB Sự tin tưởng và kỳ vọng hiệu quả → Ý định hành vi sử dụng Bluezone (H1) 0,218* 0,105 0,029 0,049 Kỳ vọng nỗ lực → Ý định hành vi sử dụng Bluezone (H2) 0,116* 0,084 0,05 0,290 Ảnh hưởng xã hội → Ý định hành vi sử dụng Bluezone (H3) 0,137* 0,097 0,043 0,320 Rủi ro về quyền riêng tư → Ý định hành vi sử dụng Bluezone (H4) -0,06* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='185 * có ý nghĩa thống kê ở mức 0,05 Bảng 8 trình bày các ước tính của hệ số đường dẫn của mô hình phương trình cấu trúc, cùng với sai số chuẩn và khoảng tin cậy 95% cận dưới và cận trên.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả thực nghiệm cho thấy sự tin tưởng và kỳ vọng hiệu quả có ảnh hưởng tích cực tới ý định hành vi sử dụng phần mềm Bluezone (H1 = 0,218;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' SE = 0,105;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' CIs = 0,029 – 0,049).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tương tự, kỳ vọng nỗ lực có ảnh hưởng tích cực tới ý định hành vi (H2 = 0,016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' SE = 0,084;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' CIs = 0,05 – 0,29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ảnh hưởng xã hội cũng đóng góp vào ý định hành vi một cách tích cực (H3 = 0,137;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' SE = 0,097;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' CIs = 0,043 – 0,32) và cuối cùng rủi ro về quyền riêng tư có ảnh hưởng tiêu cực tới ý định hành vi sử dụng Bluezone của người dùng (H4 = -0,06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' SE = 0,63;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' CIs = 0,06 – 0,185).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' THẢO LUẬN Đã hơn hai năm kể từ khi đại dịch Covid- 19 xuất hiện, mặc dù số lượng ca nhiễm và tử vong đã giảm đáng kể so với giai đoạn đầu nhưng vẫn chưa có dấu hiệu nào cho thấy sự kết thúc của đại dịch này.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Cùng với các biện pháp cách ly xã hội, tiêm vắc xin, khai báo trực tiếp bằng văn bản, việc ứng dụng công nghệ thông tin trong hỗ trợ đại dịch cũng đã và đang đem lại nhiều lợi ích nhất định.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Hầu như các hoạt động xã hội đều đã được số hóa như họp trực tuyến, đặt hàng trực tuyến, thanh toán trực tuyến, giảng dạy trực tuyến.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' đến truy vết bằng công nghệ số.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Các hoạt động này, cho dù không có đại dịch xảy ra, cũng là xu hướng tất yếu trong chuyển đổi số, nhưng sự xuất hiện của đại dịch khiến cho quá trình này được chuyển đổi nhanh hơn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Phần mềm Bluezone là sản phẩm kịp thời để ứng phó nhanh với đại dịch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Tuy nhiên, số lượng người dùng sử dụng liên tục lại không được như kỳ vọng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Điều đó dẫn đến ứng dụng công nghệ thông tin này chưa phát huy được hết sức mạnh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Do đó, ý thức về việc tích cực tham gia vào việc sử dụng ứng dụng Bluezone (hay PC-Covid) vẫn cần phải được nâng cao để giúp các nhà chức trách nhanh chóng tìm ra các giải pháp kịp thời.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả thực nghiệm từ mô hình phương trình cấu trúc cho thấy, cả bốn nhân tố thu được từ phân tích các nhân tố khám phá đều có ảnh hưởng tích cực hoặc tiêu cực tới ý định hành vi của người dùng đối với phần mềm Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Cụ thể, sự tin tưởng và kỳ vọng hiệu quả, kỳ vọng nỗ lực, ảnh hưởng xã hội có tác động tích cực đến ý định hành vi của việc sử dụng phần mềm truy vết Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Trong khi đó, rủi ro về quyền riêng tư có ảnh hưởng tiêu cực đến hành vi này.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Về mặt lý thuyết, kết quả của nghiên cứu này một lần nữa xác thực các mối quan hệ nguyên nhân – hậu quả đã được nghiên cứu và xác định ở trong mô hình phương trình cấu trúc, qua đó tạo thêm nhiều minh chứng cho sự tồn tại và ảnh hưởng của các nhân tố này.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Những độc giả quan tâm hoặc các nhà nghiên cứu khác có thể tham khảo kết quả trên cho các nghiên cứu tương tự.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Về mặt thực tiễn, kết quả nghiên cứu là cơ sở để các nhà phát triển phần mềm, người quản lý đưa ra các chiến lược và giải pháp phù hợp để tăng cường ý định hành vi sử dụng Số 02 (2022): 1 – 11 9 KHOA HỌC TỰ NHIÊN phần mềm truy vết Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Cụ thể, đối với các nhân tố có ảnh hưởng tích cực, cần phải liên tục và cập nhật phần mềm sao cho nó thực sự mang lại hiệu quả hay nói cách khác, dữ liệu có được sử dụng tối ưu cho các nhà quản lý hay không.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Hơn nữa, phần mềm phải nên thiết kế dễ sử dụng để bất kỳ ai cũng có thể tự thao tác.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ảnh hưởng xã hội cho thấy phương tiện truyền thông, gia đình, bạn bè và đồng nghiệp đóng vai trò quan trọng tới ý định hành vi, do đó việc tuyên truyền cũng nên tiếp tục được duy trì thông qua các phương tiện truyền thông khác nhau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Vì rủi ro về quyền riêng tư cũng đóng vai trò quyết định tới ý định, hành vi của người sử dụng, do đó các nhà quản lý, các nhà phát triển phần mềm, an ninh mạng cũng phải có các kỹ thuật, cơ chế, chính sách sử dụng và bảo vệ một cách phù hợp để giúp người dùng yên tâm hơn về dữ liệu cá nhân của mình.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Ngoài các yếu tố tích cực, nghiên cứu này cũng tồn tại một số giới hạn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Thứ nhất, việc lấy mẫu là không hoàn toàn ngẫu nhiên vì đối tượng tham gia nghiên cứu nằm trong mạng lưới của tác giả.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Do đó việc khái quát hóa đến một số lượng người dùng lớn hơn cần phải được xem xét một cách kỹ lưỡng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Thứ hai, việc khảo sát chỉ được thực hiện trong một khoảng thời gian nhất định nên hành vi của đối tượng tham gia nghiên cứu có thể không nhất quán trong tương lai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Thứ ba, chỉ có một số các nhân tố được đưa vào phân tích trong mô hình phương trình cấu trúc, có thể tồn tại nhiều nhân tố khác cũng có tầm ảnh hưởng tới việc sử dụng Bluezone, do đó chúng tôi khuyến nghị các nhà nghiên cứu quan tâm tìm hiểu thêm các nhân tố mới này.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' KẾT LUẬN Nghiên cứu này khám phá các nhân tố và đánh giá sự ảnh hưởng của các nhân tố đó tới ý định hành vi của người dùng trong việc sử dụng phần mềm truy vết Bluezone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Mô hình lý thuyết thống nhất về chấp nhận và sử dụng công nghệ được mở rộng thêm hai nhân tố mới bao gồm sự tin tưởng và rủi ro về quyền riêng tư.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả khảo sát từ 224 người dùng cho thấy có bốn nhân tố chính ảnh hưởng tới việc sử dụng phần mềm truy vết, trong đó có 3 nhân tố ảnh hưởng tích cực tới ý định hành vi, trong khi đó nhân tố rủi ro về quyền riêng tư có ảnh hưởng theo chiều ngược lại.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Kết quả nghiên cứu đóng góp về mặt lý thuyết bằng cách giải thích sự ảnh hưởng của các nhân tố đối với ý định hành vi một cách tinh gọn hơn (giảm chiều từ 6 nhân tố xuống còn 4 nhân tố trong EFA) và xác thực các mối quan hệ nguyên nhân – hậu quả thông qua mô hình phương trình cấu trúc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Đồng thời, kết quả nghiên cứu cũng có thể được sử dụng trong thực tiễn giúp các nhà quản lý, nhà phát triển phần mềm, an ninh môi trường mạng có thêm cơ sở để tiếp tục hoàn thiện phần mềm truy vết Covid-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' LỜI CẢM ƠN Nhóm tác giả trân trọng cảm ơn các bạn bè, đồng nghiệp trong việc tham gia khảo sát.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Cảm ơn TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Nguyễn Hải Minh (ICTU) vì đã phổ biến phiếu khảo sát đến các sinh viên trong trường.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' TÀI LIỆU THAM KHẢO Arfi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', Nasr, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', Kondrateva, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', & Hikkerova, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' The role of trust in intention to use the IoT in eHealth: Application of the modified UTAUT in a consumer context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' Technological Forecasting and Social Change, 167, 120688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='2307/30036540 Whitelaw, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', Mamas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', Topol, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=', & Spall, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content=' C.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} +page_content='1016/S2589- 7500(20)30142-4 ap chi khoa hoc DAI HOC HA LONG' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf'} diff --git a/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf b/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2f0ca574015c0123cbb441648895ddfa7c38aa52 --- /dev/null +++ b/CNE4T4oBgHgl3EQfFgxh/content/2301.04886v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:46f93baa2e035b3bc9e40ccfcd280f14b2e68750128f939c00f88b2755bbab6f +size 216387 diff --git a/CNE4T4oBgHgl3EQfFgxh/vector_store/index.faiss b/CNE4T4oBgHgl3EQfFgxh/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..bf7ae19b21c4c4776924c52e1fc77196ea709548 --- /dev/null +++ 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sha256:9338aaee8a3a5ada5447606dd01b2aa5899e258df8cd8e9596cf6c98ce9ef73a +size 11479871 diff --git a/D9E1T4oBgHgl3EQfqQU2/content/tmp_files/2301.03340v1.pdf.txt b/D9E1T4oBgHgl3EQfqQU2/content/tmp_files/2301.03340v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd64b8332ed8d7783957794cc229bc684fadaf74 --- /dev/null +++ b/D9E1T4oBgHgl3EQfqQU2/content/tmp_files/2301.03340v1.pdf.txt @@ -0,0 +1,879 @@ +arXiv:2301.03340v1 [physics.atom-ph] 9 Jan 2023 +Formation of strongly shifted EIT resonances using "forbidden" transitions of Cesium +Armen Sargsyan,1 Ara Tonoyan,1 Rodolphe Momier,1, 2, ∗ Claude Leroy,2 and David Sarkisyan1 +1Institute for Physical Research, NAS of Armenia, Ashtarak-2, 0203 Armenia +2Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS 6303, +Université Bourgogne Franche-Comté, 21000 Dijon, France +(Dated: January 10, 2023) +Atomic transitions satisfying Fe − Fg = ∆F = ±2 (where Fe stands for excited and Fg stands +for ground state) of alkali atoms have zero probability in zero magnetic field (they are so-called +"forbidden" transitions) but experience a large probabilty increase in an external magnetic field. +These transitions are called magnetically induced (MI) transitions. In this paper, we use for the first +time the σ+ (∆mF = ++ 1) MI transitions Fg = 3 → Fe = 5 of Cesium as probe radiation to form +EIT resonances in strong magnetic fields (1 - 3 kG) while the coupling radiation frequency is resonant +with Fg = 4 → Fe = 5 σ+ transitions. The experiment is performed using a nanometric-thin cell +filled with Cs vapor and a strong permanent magnet. The thickness of the vapor column is 852 nm, +corresponding to the Cs D2 line transition wavelength. Due to the large frequency shift slope of the +MI transitions (∼ 4 MHz/G), it is possible to form contrasted and strongly frequency-shifted EIT +resonances. Particularly, a strong 12 GHz frequency shift is observed when applying an external +magnetic field of ∼ 3 kG. Preliminary calculations performed considering Doppler-broadened three +level systems in a nanocell are in reasonable agreement with the experimental measurements. +I. +INTRODUCTION +Optical processes occurring in Rubidium, Cesium, +Potassium and Sodium vapors confined in optical cells +have important applications such as optical atomic +clocks, optical atomic magnetometers, atomic gyro- +scopes, markers of atomic transition frequencies, as de- +scribed for example in [1–6]. Therefore, the study of the +peculiarities of atomic transitions (in particular Zeeman +transitions in an external magnetic field) of alkali atoms +is of utmost importance. It is well known that the appli- +cation of a strong magnetic field can significantly change +the probabilities (intensities) of the Zeeman transitions, +as shown in [7–13]. High interest has recently been fo- +cused on atomic transitions between ground and excited +levels that satisfy the condition Fe − Fg = ∆F = ±2 +(these transitions are so-called forbidden by the selection +rules, thus their probability is zero when no external mag- +netic field is applied). However, the probabilities of these +transitions in a magnetic field increase significantly. For +this reason, we refer to these transitions as Magnetically +Induced (MI) transitions [8, 11, 12]. +This giant increase in the probabilities of the MI transi- +tions is due to the “mixing” of magnetic sublevels |F, mF ⟩ +of the ground (Fg) or excited (Fe) levels with sublevels +having the same magnetic quantum number mF . This +mixing is the strongest for D2 lines of alkali atoms, as +up to four states |Fe, 0⟩ can experience mixing, thus re- +sulting in a 4×4 block in the magnetic Hamiltonian, as +described in [7, 8, 11, 12]. +Magnetically-induced transitions are of great interest +because, over a wide range of magnetic field, their proba- +bilities can be much higher than the probabilities of usual +∗ rodolphe.momier@u-bourgogne.fr +("allowed", satisfying the selection rule on F) transitions. +It is important to note that the slope of the frequency +shifts (obtained by diagonalizing the magnetic Hamilto- +nian [7]) as a function of the magnetic field B in strong +magnetic fields can reach up to around 4 MHz/G, which +is 3 times larger than in the case of ordinary transitions. +Thus, the frequency shift of MI transitions in strong mag- +netic fields can reach several tens of GHz, which can be +useful for working in higher frequency ranges, for exam- +ple for the frequency stabilisation of lasers on strongly +shifted frequencies [14, 15]. +In [11, 12], we established the following rule for the +probabilities of MI transitions: +the probabilities and +number of MI transitions with ∆F = +2 are maximal +for σ+ radiation, whereas the probabilities and number +of MI transitions with ∆F = −2 are maximal for σ− +radiation. The difference between the intensities of MI +transitions for the σ+ and σ−-polarized radiation beams +can reach several orders of magnitude. +It +has +been +recently +demonstrated +that +electromagnetically-induced +transparency +(EIT) +resonances can be formed using Λ-system made of +∆F += ++2 MI transitions only if both probe and +coupling beam are σ+-polarized. +This statement was +experimentally and theoretically verified for 87Rb (MI +transitions Fg = 1 → Fe = 3) and 85Rb (MI transitions +Fg = 2 → Fe = 4) [16, 17]. However, if the Λ-system +is formed by MI transitions satisfying ∆F = −2, then +both probe and coupling radiation must be σ−-polarized +in order to form EIT resonances. +This statement +was experimentally and theoretically verified for Cs +(MI transitions Fg = 4 → Fe = 2). +This is a direct +consequence of magnetically-induced circular dichroism +[18]. +In this work, we consider seven σ+ MI transitions of +Cs (Fg = 3 → Fe = 5, see Fig. 1). The probabilities of +these transitions increase highly in the range 0.3 - 3 kG + +2 +0 ++1 ++2 ++3 +-1 +-2 +-3 +0 ++1 ++2 ++3 +-1 +-2 +1 +2 +3 +4 +5 +7 +6 ++4 +0 ++1 ++2 ++3 +-1 +-2 +-3 +FIG. 1. Scheme of Cs D2 line σ+ transitions between Fg = 3, 4 +and Fe = 5. The probe frequency νp is scanned across the +MI transitions labelled 1-7 (Fg = 3 → Fe = 5). The coupling +frequencies νcn are resonant with Fg = 4 → Fe = 5 transi- +tions, forming seven Λ-systems. Only the states involved in +the process under consideration are shown. Note that |F, mF ⟩ +is just a notation for visualization, as the atomic states are +better described in the uncoupled basis |J, mJ, I, mI⟩ in high +magnetic fields. +and we used these transitions to form EIT resonances +in strong B-fields. +A nanometric-thin cell (NC) filled +with Cs vapor (thickness L ≈ 850 nm, approximately +the resonant wavelength of Cs D2 line [19]) has been +used. The advantages of using thin cells, including strong +reduction of Doppler broadening, are noted in [12, 17, 20]. +A. +Probabilities and frequency shifts of the MI +transitions of Cs D2 line +The curves in Fig. 2 were calculated using a known the- +oretical model depicting the changes of transition proba- +bilities as a function of the external magnetic field. The +block-diagonal (each block corresponding to a given value +of the magnetic quantum number) magnetic Hamiltonian +is built for each value of the magnetic field and then diag- +onalized in order to calculate the probability coefficients. +This model was presented in a number of papers, e.g. +[7, 11, 13]. +The evolution of the probabilities of MI transitions (la- +belled 1 to 7, see Fig. 1) with respect to the magnetic field +B is shown in Fig. 2a). Note that in the range 0.3 - 2 +kG the probabilities of the MI transitions labeled 5, 6 +and 7 are the strongest among all transitions occurring +from Fg = 3 [8, 12]. The frequency shift slope of the +MI transitions, obtained through the eigenvalues of the +Hamiltonian, is quite large (∼ 4 MHz/G) while for usual +transitions the slope is 3 times smaller. Despite the fact +that the probabilities of the MI transitions decrease as +B increases, they can still be recorded easily at 7 kG. +As noted below, this is due to the fact that these tran- +sitions are formed far on the high-frequency wing where +there are no intersections with other transitions (spec- +tra are presented for Na in [21], but Cs behaves almost +identically). +The evolution of the probabilities of the corresponding +seven coupling transitions Fg = 4 → Fg = 5 (Ac1 to Ac7) +that are used to form seven Λ-systems (see Fig. 1) with +respect to the magnetic field are shown in Fig. 2b). In the +case of σ− polarization, the probability of the strongest +Fg = 4 → Fe = 5 σ− transition already tends to zero for +B > 300 G, as shown in Fig. 2c). Thus, both the probe +and the coupling beams must be σ+-polarized in order +to form EIT resonances. +B. +Qualitative description of the EIT process +For a qualitative description of the EIT process, we +present a formula from [3, 22]. The ratio of absorption at +the probe radiation frequency νp at which EIT resonance +is observed (in the presence of νc radiation) to absorp- +tion (when there is no coupling radiation), assuming low +radiation intensity νp and zero frequency detuning of the +coupling radiation, is described by the expression: +α(Ωc) +α(0) = +K +1 + Ω2c/4Γ21γN +, +(1) +where K is a constant including the Doppler width, +γN is the natural width of the level (γN/2π ≃ 5.2 MHz +for the 62P3/2 level of the Cs atom), Ωc is the Rabi fre- +quency for the coupling radiation and Γ21 is the dephas- +ing rate of the coherence between the two ground states +of the Λ-system, which is caused in particular by colli- +sions of atoms with the windows of the nanocell. The +case α(Ωc) = 0 corresponds to complete transparency +(the contrast of the EIT resonance reaches 100%) and a +large amplitude of the EIT resonance, which decreases +with an increase in Γ21. The spectral width of the EIT +resonance can be described by the simple expression [3]: +γEIT ≃ 2Γ21 + Ω2 +c/γN . +(2) +It follows from formula (1) that in order to obtain small +value of α(Ωc) (which means high electromagnetically in- +duced transparency of the medium), it is necessary to in- +crease Ωc, however, an increase in Ωc leads to an increase +in the spectral width of the EIT resonance. Therefore, +it is necessary to find a compromise for Ωc. Estimates +can be obtained from Ωc/2π = aγN(I/8)1/2 where I is +the laser intensity in mW/cm2, γN ∼ 5 MHz, and a +is a fit parameter (for our case a is of ∼ 0.5) [23] and +Ωc ∼ 15 MHz. +II. +EXPERIMENT +A. +Experimental setup +The layout of the experimental setup is shown on +Fig. 3. Two extended cavity diode lasers are tuned in the +vicinity of the Cs D2 line, with a wavelength λ ≃ 852 nm. +The Λ-systems shown in Fig. 1 are formed by scanning +the frequency νp of a VitaWave laser (δνp ∼ 1 MHz) [24] + +3 +1 +2 +3 +4 +5 +6 +7 +a) +b) +c) +FIG. 2. Magnetic field dependence of the Zeeman transition intensities of the D2 line of Cs. a) Fg = 3 → Fe = 5 σ+ MI +transitions. b) Fg = 4 → Fe = 5 σ+ transitions. c) Transition |4, −1⟩ → |5, −2⟩ (σ−). This transition forms a Λ-system with +transition 7 as shown in panel a) and in the inset (see Fig. 1). Its probability tends to 0 as the magnetic field increases, thus +forming EIT resonances at high magnetic fields requires both probe and coupling beams to be σ+-polarized. +FI +FI +SO +BS +PD +ECDL 1 +ECDL 2 +probe +coupling +C +Ref. channel +Meas. channel +PBS2 +PBS1 +PBS3 +PBS4 +M +IF PD +NC +PM +FIG. 3. +Scheme of the experimental setup. +ECDL: CW +narrow-band external-cavity diode lasers with λ = 852 nm +(resonant with Cs D2 line). FI: Faraday insulators. PBSi: +polarizing beam splitters. BS: beam splitter. IF: interference +filter. C: saturated absorption spectroscopy unit for frequency +reference. NC: nanocell placed in oven. PM: permanent mag- +net. PD: photodiodes. SO: 4-channel digital oscilloscope. +in the vicinity of the MI transitions Fg = 3 → Fe = 5, +while keeping the frequency νc from a MOGLabs “cat- +eye” laser (δνp ≃ 0.1 MHz) on resonance with one of the +4 → 5 transitions. A fraction about 10% of the coupling +radiation power was sent to a frequency stabilization unit +based on the DAVLL method [25]. Probe radiation has +vertical polarization, while the coupling radiation has +horizontal polarization. In the case of a longitudinal B- +field, linearly polarized laser radiation can be considered +as consisting of σ+ and σ− radiations. The use of mutu- +ally perpendicular polarizations allows by using PBS4 to +direct only probe radiation to the photo-receiver, while +cutting off the coupling radiation. As noted above, in +the case of MI transitions with ∆F = +2 for the for- +mation of the EIT resonance, both probe and coupling +radiations must have σ+ polarization. A photograph of +the Cs nanocell is shown in Fig. 3. Interference fringes +are formed by the reflection of light on the inner surfaces +of windows (made of sapphire). The region correspond- +Coupling off +7 +6 +5 +4 +3 +EIT 7 +EIT 6 +EIT 5 +EIT 4 +EIT 3 +(1) +(2) +(3) +(4) +(5) +(6) +FIG. 4. Probe transmission spectra of the Cs nanocell (L = +λ = 852 nm). Spectra labelled 1 to 5 show five EIT reso- +nances, labelled EIT 3 to EIT 7, while the probe frequency is +scanned across transitions 3 to 7 (see Fig. 1). The coupling +and probe powers are respectively 10 and 0.05 mW and the ex- +ternal longitudinal magnetic field is B = 1400 G. Spectrum n° +6 corresponds to the case where coupling is off. Small VSOP +peaks are visible on each atomic resonance. Zero frequency +corresponds to the transition frequency of Cs D2 line. +ing to a thickness L ≈ λ ∼ 850 nm is outlined by an +oval. The design of the Cs-filled NC used in our experi- +ments is similar to that of extremely thin cell described +in [26]. Earlier it was demonstrated in [16, 17, 27] that +the use of a nanocell (NC) with thickness L = λ makes it +easy to record contrasted EIT resonances, which is due +to the low absorption of the NC, while the disadvan- +tage is broadening of the EIT resonance caused by fre- +quent inelastic collisions of atoms with the windows of +the NC. Studies of the EIT resonances were done using +a strong neodymium–iron–boron alloy ring-shaped per- +manent magnet (PM). Due to the small thickness of the +vapor column, the high-gradient field produced by magne +can be considered uniform across the interaction region. +The PM was placed after the rear window of the NC, +with the axis aligned along the probe beam propagation + +4 +direction. The magnetic field in the NC was simply var- +ied by longitudinal displacement of the PM, calibrated +using a Teslameter HT201 magnetometer. +B. +Experimental results: using MI transitions to +form EIT resonances +Curves 1 to 5 in Fig. 4 show the experimental trans- +mission spectra of the probe radiation which contain the +resonances EIT 3 to EIT 7 (numbers 3-7 means that MI +transitions with numbers 3-7 are involved, respectively) +in a longitudinal magnetic field B = 1400 G. The NC +thickness is L = λ = 852 nm and the temperature of +the reservoir is 100 ◦C (to prevent Cs vapor condensa- +tion on the windows, the temperature of the windows is +slightly higher). The coupling and the probe powers are +20 mW and 0.1 mW, respectively. Note that since only +σ+ radiations participate to the formation of the EIT +resonances (see Fig. 1), only half of the power of these +radiations must be considered, meaning 10 mW and 50 +µW, respectively. Curve n° 6 is a probe spectrum when +the coupling is blocked. Since the cell thickness is L = λ, +small peaks formed by velocity selective optical pump- +ing (VSOP) resonances are located exactly at the atomic +transitions frequencies, as described in [9]. +The amplitude of the EIT resonance is a factor ∼10 +larger than the amplitude of the VSOP resonance, +whereas the spectral width of the EIT resonance is a +factor of 1.5 smaller, which is characteristic of the coher- +ent EIT process [17]. Note that the contrast of the EIT +resonance defined as the ratio of the EIT resonance am- +plitude divided by the peak absorption of the Cs vapor +when the coupling is blocked reached 40-50 % which is +typical when a nanocell is used [27]. +In Fig. 6, curves 1 to 4 are probe transmission spectra +which contain EIT 6, EIT 5, EIT 4 and EIT 3 resonances +for B = 1770 G. Curve n° 5 shows only the probe spec- +trum when the coupling is blocked. In Fig. 7, lines 1 to 3 +show the probe transmission spectra which contain EIT +6, EIT 4 and EIT 3 resonances for B = 2880 G. Line +n° 4 shows only the probe spectrum when the coupling +is blocked. The inset shows the profile of EIT 6 reso- +nance fitted with a Gaussian profile with a FWHM of ∼ +35 MHz. There is also a small VSOP resonance which is +formed when the coupling is blocked. The typical FWHM +of VSOP resonances is 40-50 MHz. +Preliminary theoretical calculations (shown in the +right part of the inset of Fig. 7 were obtained by solv- +ing the Liouville equations of motion for an ensemble of +three-level Λ-systems (as presented in Fig. 5), taking into +account the geometry of the nanocell (coherence dephas- +ing rate determined by the time of flight of the atoms), its +Fabry-Perot nature (reflections of the fields on the inner +surfaces of the cell) and Doppler broadening, following +the procedure described in [28]. The Rabi frequencies of +the probe and coupling lasers are respectively Ωc = 1.5γN +and Ωp = 0.06γN. Reasonable agreement between theory +and experiment regarding the width and depth of the EIT +resonance is obtained and the VSOP resonance is seen. +Small discrepancies (assymetry of the profile and ampli- +tude of the VSOP resonance) can arise notably from the +need of considering neighboring Zeeman sublevels (not +shown in Fig. 1, and therefore more than three levels, to +obtain more accurate results. +FIG. 5. Scheme of the three-level Λ-system used in the calcu- +lations. The total decay rate Γ33 of state |3⟩ is 1/2(γ31 + γ32) +[29]. +The dephasing rate of coherence between the ground +states is Γ21 = (2πt)−1 where t is the time of flight of +the atoms through the cell (at the most probable velocity +u = +� +2kBT/M where T is the vapor temperature and M the +atomic mass). +The amplitude of resonance n° 6 is ∼ 50 times greater +than that of the VSOP resonance and is spectrally nar- +rower than the latter (this is a manifestation of the co- +herent EIT process [2, 17]). In Fig. 8 the solid lines in- +dicate the calculated dependences of the frequency shifts +for transitions 1–7 (Fig. 1) and Fg = 3 → Fe = 4 (marked +with dotted oval) to the magnetic field B. +The black +squares represent the experimental results. As mentioned +earlier, due to the high value of the frequency shift slope +for B > 3 kG, the group of MI transitions 1–7 is com- +pletely separated in frequency from Fg = 3 → Fe = 4 +transitions. +The curves in the inset of Fig. 8 show experimental and +theoretical spectra (calculated by combining the models +presented in [7] and [30]) of the seven MI transitions ab- +sorption for B = 6 kG when frequency shift reaches ∼ 30 +GHz. Note that the amplitude of transition 6 is slightly +bigger than that of transition 7 (while for B < 5 kG the +amplitude of transition 7 is bigger, see Fig. 2a), because +of the “mixing” effect. Note that one of the remarkable +features of the σ+ MI transitions 3 → 5′ is that they +are still well recorded for a magnetic field B ≈ 8 kG. +They are located in the high frequency wing of the spec- +trum presented in Fig. 18 of paper [31] and for this case +the frequency shift reaches 34 GHz. Using our theoret- +ically calculated curves for MI transitions 3 → 5′ we +checked the frequency position of these MI transitions +and found good agreement with the experimental curves +presented in Fig. 18. In paper [31] the 3 → 5′transitions +are not identified. Therefore, it is important to inform + +5 +Coupling off +EIT 6 +EIT 5 +EIT 4 +EIT 3 +(1) +(2) +(3) +(4) +(5) +6 +5 +4 +3 +FIG. 6. +Probe transmission spectra of the Cs nanocell +(L = λ ≈ 850 nm). Spectra 1 to 4 exhibit four EIT reso- +nances, labelled EIT 3 to EIT 6, while the probe frequency +is scanned across transitions 3 to 6. The external longitudi- +nal magnetic field is B = 1770 G. Spectrum n° 5 is a probe +transmission spectrum when the coupling is off. Small VSOP +peaks are visible on each atomic transition. Zero frequency +corresponds to the transition frequency of Cs D2 line. +Coupling off +Coupling off +(1) +(2) +(3) +(4) +Experiment +Coupling off +EIT 6 +EIT 4 +EIT 3 +6 +5 +4 +3 +EIT 6 +Theory +FIG. 7. Probe transmission spectra of the Cs nanocell (L = +λ = 852 nm). +Lines 1 to 3 show four EIT resonances, la- +belled EIT 4, EIT 5 and EIT 6. The external longitudinal +magnetic field is B = 2880 G. Line 4 is a probe transmission +spectrum when the coupling is off. The left part of the inset +is a zoom on EIT 6, fitted with a Gaussian profile (FWHM +35 MHz). The right curves are calculated. Red: coupling on, +black: coupling off. Small VSOP peaks are visible on each +atomic transitions formed by the probe radiation. Their typ- +ical linewidth is 40-50 MHz. Zero frequency corresponds to +the transition frequency of Cs D2 line. +scientists working in the field of laser spectroscopy of al- +kali metal atoms about the MI atomic transitions. The +above-mentioned MI transitions can be exploited in such +high B-fields as new frequency markers, for using new fre- +quency ranges, as well as for the frequency stabilization +of lasers at strongly shifted frequencies from the initial +transition in unperturbed atoms [13, 14]. +Exp. +Theory +6 +5 +4 +3 2 1 +7 +6 +5 +4 +3 +2 +1 +7 +FIG. 8. Red solid lines: frequency shift of transitions 1 to 7 +(see figure 1) as a function of the magnetic field. The black +squares with error bars represent experimental measurements, +the inaccuracy is around 1 %. Black dashed lines: frequency +shift of Fg = 3 → Fe = 4 transitions. For B > 3 kG, both +groups are well separated in frequency. Inset: theoretical and +experimental absorption spectra for B = 6 kG, the frequency +shift reaches 30 GHz from the Cs D2 line transition frequency. +III. +CONCLUSION +In this paper, we used for the first time forbidden +transitions of Cs (Fg = 3 → Fe = 5, more precisely +σ+(∆mF = +1) transitions) to create Λ-system allowing +the formation of EIT resonances. This was done in an ex- +ternal magnetic field, as such transitions have zero proba- +bility in the absence of magnetic field. A nanometric-thin +cell filled with Cs vapor was used, with a thickness corre- +sponding to the resonant wavelength of Cs D2 line (≈ 850 +nm), and the magnetic field was varied by longitudinal +displacement of the permanent magnet along the prop- +agation direction (Fig. 3). As expected, when the cou- +pling is blocked, small VSOP resonances are formed right +at the different transitions’ frequencies, while coupling +radiation allows for the formation of EIT resonances, +spectrally narrower and with a bigger amplitude. +We +formed EIT resonances with 6 out the 7 transitions de- +picted in Fig. 1. This was possible up to 3 kG thanks +to the big value of the frequency shift, reaching up to 4 +MHz/G, therefore leading to EIT resonances shifted 12 +GHz apart from the Cs D2 line transition frequency [31]. +This result is of great interest, as the highly-shifted spec- +tra can serve as frequency references [14, 15], especially +taking into account that these transitions are still easily +recorded up to 8 kG when the frequency shift reaches 35 +GHz. As for the theoretical description, further investi- +gation is necessary, mainly in order to take into account +the effect of neighbouring states, and thus including more +levels in the model. The complexity of the manifold and +the number of coupled equations make it a challenging + +6 +and computationally-intensive task. However, reasonable +agreement was already achieved by simply considering an +ensemble of three-level systems. To the best of our knowl- +edge, there are no reports on obtaining EIT resonances +in Λ-systems in such strong fields using usual transitions +of alkali atoms. We note that much narrower EIT reso- +nances can be attained by using cm-long cells (to lower +the effect of inelastic collisions of atoms with the win- +dows), and by using coherently coupled probe and cou- +pling radiations derived from a single narrow-band laser +beam [3]. +ACKNOWLEDGMENTS +This work was supported by the Science Committee of +the Republic of Armenia, in the frame of research project +n° 21T-1C005, and by the NATO Science for Peace and +Security Project under grant G5794. +DATA AVAILABILITY STATEMENT +Data underlying the results presented in this paper are +not publicly available at this time but may be obtained +from the authors upon reasonable request. +[1] J. +Kitching, +Chip-scale +atomic +devices, +Applied Physics Reviews 5, 031302 (2018). +[2] J. Vanier, Atomic clocks based on coherent population +trapping: a review, Applied Physics B 81, 421 (2005). +[3] M. Fleischhauer, A. Imamoglu, and J. P. 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B 20, 793 (2003). +[31] H. +Stærkind, +K. +Jensen, +J. +H. +Müller, +V. +O. +Boer, +E. +T. +Petersen, +and +E. +S. +Polzik, +Precision Measurement of the Excited State Landé g-factor and Diamagnetic Shift of the Cesium D2 Line +(2022), arXiv:2208.00077. + diff --git a/D9E1T4oBgHgl3EQfqQU2/content/tmp_files/load_file.txt b/D9E1T4oBgHgl3EQfqQU2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..385be63487eb237f41e2652acf32e7135ad742b2 --- /dev/null +++ b/D9E1T4oBgHgl3EQfqQU2/content/tmp_files/load_file.txt @@ -0,0 +1,457 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf,len=456 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='03340v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='atom-ph] 9 Jan 2023 Formation of strongly shifted EIT resonances using "forbidden" transitions of Cesium Armen Sargsyan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='1 Ara Tonoyan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='1 Rodolphe Momier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' ∗ Claude Leroy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='2 and David Sarkisyan1 1Institute for Physical Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' NAS of Armenia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Ashtarak-2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 0203 Armenia 2Laboratoire Interdisciplinaire Carnot de Bourgogne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' UMR CNRS 6303,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Université Bourgogne Franche-Comté,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 21000 Dijon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' France (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 2023) Atomic transitions satisfying Fe − Fg = ∆F = ±2 (where Fe stands for excited and Fg stands for ground state) of alkali atoms have zero probability in zero magnetic field (they are so-called "forbidden" transitions) but experience a large probabilty increase in an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' These transitions are called magnetically induced (MI) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' In this paper, we use for the first time the σ+ (∆mF = + 1) MI transitions Fg = 3 → Fe = 5 of Cesium as probe radiation to form EIT resonances in strong magnetic fields (1 - 3 kG) while the coupling radiation frequency is resonant with Fg = 4 → Fe = 5 σ+ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The experiment is performed using a nanometric-thin cell filled with Cs vapor and a strong permanent magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The thickness of the vapor column is 852 nm, corresponding to the Cs D2 line transition wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Due to the large frequency shift slope of the MI transitions (∼ 4 MHz/G), it is possible to form contrasted and strongly frequency-shifted EIT resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Particularly, a strong 12 GHz frequency shift is observed when applying an external magnetic field of ∼ 3 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Preliminary calculations performed considering Doppler-broadened three level systems in a nanocell are in reasonable agreement with the experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' INTRODUCTION Optical processes occurring in Rubidium, Cesium, Potassium and Sodium vapors confined in optical cells have important applications such as optical atomic clocks, optical atomic magnetometers, atomic gyro- scopes, markers of atomic transition frequencies, as de- scribed for example in [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Therefore, the study of the peculiarities of atomic transitions (in particular Zeeman transitions in an external magnetic field) of alkali atoms is of utmost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' It is well known that the appli- cation of a strong magnetic field can significantly change the probabilities (intensities) of the Zeeman transitions, as shown in [7–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' High interest has recently been fo- cused on atomic transitions between ground and excited levels that satisfy the condition Fe − Fg = ∆F = ±2 (these transitions are so-called forbidden by the selection rules, thus their probability is zero when no external mag- netic field is applied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' However, the probabilities of these transitions in a magnetic field increase significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' For this reason, we refer to these transitions as Magnetically Induced (MI) transitions [8, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' This giant increase in the probabilities of the MI transi- tions is due to the “mixing” of magnetic sublevels |F, mF ⟩ of the ground (Fg) or excited (Fe) levels with sublevels having the same magnetic quantum number mF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' This mixing is the strongest for D2 lines of alkali atoms, as up to four states |Fe, 0⟩ can experience mixing, thus re- sulting in a 4×4 block in the magnetic Hamiltonian, as described in [7, 8, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Magnetically-induced transitions are of great interest because, over a wide range of magnetic field, their proba- bilities can be much higher than the probabilities of usual ∗ rodolphe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='momier@u-bourgogne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='fr ("allowed", satisfying the selection rule on F) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' It is important to note that the slope of the frequency shifts (obtained by diagonalizing the magnetic Hamilto- nian [7]) as a function of the magnetic field B in strong magnetic fields can reach up to around 4 MHz/G, which is 3 times larger than in the case of ordinary transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Thus, the frequency shift of MI transitions in strong mag- netic fields can reach several tens of GHz, which can be useful for working in higher frequency ranges, for exam- ple for the frequency stabilisation of lasers on strongly shifted frequencies [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' In [11, 12], we established the following rule for the probabilities of MI transitions: the probabilities and number of MI transitions with ∆F = +2 are maximal for σ+ radiation, whereas the probabilities and number of MI transitions with ∆F = −2 are maximal for σ− radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The difference between the intensities of MI transitions for the σ+ and σ−-polarized radiation beams can reach several orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' It has been recently demonstrated that electromagnetically-induced transparency (EIT) resonances can be formed using Λ-system made of ∆F = +2 MI transitions only if both probe and coupling beam are σ+-polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' This statement was experimentally and theoretically verified for 87Rb (MI transitions Fg = 1 → Fe = 3) and 85Rb (MI transitions Fg = 2 → Fe = 4) [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' However, if the Λ-system is formed by MI transitions satisfying ∆F = −2, then both probe and coupling radiation must be σ−-polarized in order to form EIT resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' This statement was experimentally and theoretically verified for Cs (MI transitions Fg = 4 → Fe = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' This is a direct consequence of magnetically-induced circular dichroism [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' In this work, we consider seven σ+ MI transitions of Cs (Fg = 3 → Fe = 5, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The probabilities of these transitions increase highly in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='3 - 3 kG 2 0 +1 +2 +3 1 2 3 0 +1 +2 +3 1 2 1 2 3 4 5 7 6 +4 0 +1 +2 +3 1 2 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Scheme of Cs D2 line σ+ transitions between Fg = 3, 4 and Fe = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The probe frequency νp is scanned across the MI transitions labelled 1-7 (Fg = 3 → Fe = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The coupling frequencies νcn are resonant with Fg = 4 → Fe = 5 transi- tions, forming seven Λ-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Only the states involved in the process under consideration are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Note that |F, mF ⟩ is just a notation for visualization, as the atomic states are better described in the uncoupled basis |J, mJ, I, mI⟩ in high magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' and we used these transitions to form EIT resonances in strong B-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' A nanometric-thin cell (NC) filled with Cs vapor (thickness L ≈ 850 nm, approximately the resonant wavelength of Cs D2 line [19]) has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The advantages of using thin cells, including strong reduction of Doppler broadening, are noted in [12, 17, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Probabilities and frequency shifts of the MI transitions of Cs D2 line The curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 2 were calculated using a known the- oretical model depicting the changes of transition proba- bilities as a function of the external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The block-diagonal (each block corresponding to a given value of the magnetic quantum number) magnetic Hamiltonian is built for each value of the magnetic field and then diag- onalized in order to calculate the probability coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' This model was presented in a number of papers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' [7, 11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The evolution of the probabilities of MI transitions (la- belled 1 to 7, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 1) with respect to the magnetic field B is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Note that in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='3 - 2 kG the probabilities of the MI transitions labeled 5, 6 and 7 are the strongest among all transitions occurring from Fg = 3 [8, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The frequency shift slope of the MI transitions, obtained through the eigenvalues of the Hamiltonian, is quite large (∼ 4 MHz/G) while for usual transitions the slope is 3 times smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Despite the fact that the probabilities of the MI transitions decrease as B increases, they can still be recorded easily at 7 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' As noted below, this is due to the fact that these tran- sitions are formed far on the high-frequency wing where there are no intersections with other transitions (spec- tra are presented for Na in [21], but Cs behaves almost identically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The evolution of the probabilities of the corresponding seven coupling transitions Fg = 4 → Fg = 5 (Ac1 to Ac7) that are used to form seven Λ-systems (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 1) with respect to the magnetic field are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' In the case of σ− polarization, the probability of the strongest Fg = 4 → Fe = 5 σ− transition already tends to zero for B > 300 G, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Thus, both the probe and the coupling beams must be σ+-polarized in order to form EIT resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Qualitative description of the EIT process For a qualitative description of the EIT process, we present a formula from [3, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The ratio of absorption at the probe radiation frequency νp at which EIT resonance is observed (in the presence of νc radiation) to absorp- tion (when there is no coupling radiation), assuming low radiation intensity νp and zero frequency detuning of the coupling radiation, is described by the expression: α(Ωc) α(0) = K 1 + Ω2c/4Γ21γN , (1) where K is a constant including the Doppler width, γN is the natural width of the level (γN/2π ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='2 MHz for the 62P3/2 level of the Cs atom), Ωc is the Rabi fre- quency for the coupling radiation and Γ21 is the dephas- ing rate of the coherence between the two ground states of the Λ-system, which is caused in particular by colli- sions of atoms with the windows of the nanocell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The case α(Ωc) = 0 corresponds to complete transparency (the contrast of the EIT resonance reaches 100%) and a large amplitude of the EIT resonance, which decreases with an increase in Γ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The spectral width of the EIT resonance can be described by the simple expression [3]: γEIT ≃ 2Γ21 + Ω2 c/γN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' (2) It follows from formula (1) that in order to obtain small value of α(Ωc) (which means high electromagnetically in- duced transparency of the medium), it is necessary to in- crease Ωc, however, an increase in Ωc leads to an increase in the spectral width of the EIT resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Therefore, it is necessary to find a compromise for Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Estimates can be obtained from Ωc/2π = aγN(I/8)1/2 where I is the laser intensity in mW/cm2, γN ∼ 5 MHz, and a is a fit parameter (for our case a is of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='5) [23] and Ωc ∼ 15 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' EXPERIMENT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Experimental setup The layout of the experimental setup is shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Two extended cavity diode lasers are tuned in the vicinity of the Cs D2 line, with a wavelength λ ≃ 852 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The Λ-systems shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 1 are formed by scanning the frequency νp of a VitaWave laser (δνp ∼ 1 MHz) [24] 3 1 2 3 4 5 6 7 a) b) c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Magnetic field dependence of the Zeeman transition intensities of the D2 line of Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' a) Fg = 3 → Fe = 5 σ+ MI transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' b) Fg = 4 → Fe = 5 σ+ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' c) Transition |4, −1⟩ → |5, −2⟩ (σ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' This transition forms a Λ-system with transition 7 as shown in panel a) and in the inset (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Its probability tends to 0 as the magnetic field increases, thus forming EIT resonances at high magnetic fields requires both probe and coupling beams to be σ+-polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' FI FI SO BS PD ECDL 1 ECDL 2 probe coupling C Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' channel Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' channel PBS2 PBS1 PBS3 PBS4 M IF PD NC PM FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Scheme of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' ECDL: CW narrow-band external-cavity diode lasers with λ = 852 nm (resonant with Cs D2 line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' FI: Faraday insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' PBSi: polarizing beam splitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' BS: beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' IF: interference filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' C: saturated absorption spectroscopy unit for frequency reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' NC: nanocell placed in oven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' PM: permanent mag- net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' PD: photodiodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' SO: 4-channel digital oscilloscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' in the vicinity of the MI transitions Fg = 3 → Fe = 5, while keeping the frequency νc from a MOGLabs “cat- eye” laser (δνp ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='1 MHz) on resonance with one of the 4 → 5 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' A fraction about 10% of the coupling radiation power was sent to a frequency stabilization unit based on the DAVLL method [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Probe radiation has vertical polarization, while the coupling radiation has horizontal polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' In the case of a longitudinal B- field, linearly polarized laser radiation can be considered as consisting of σ+ and σ− radiations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The use of mutu- ally perpendicular polarizations allows by using PBS4 to direct only probe radiation to the photo-receiver, while cutting off the coupling radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' As noted above, in the case of MI transitions with ∆F = +2 for the for- mation of the EIT resonance, both probe and coupling radiations must have σ+ polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' A photograph of the Cs nanocell is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Interference fringes are formed by the reflection of light on the inner surfaces of windows (made of sapphire).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The region correspond- Coupling off 7 6 5 4 3 EIT 7 EIT 6 EIT 5 EIT 4 EIT 3 (1) (2) (3) (4) (5) (6) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Probe transmission spectra of the Cs nanocell (L = λ = 852 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Spectra labelled 1 to 5 show five EIT reso- nances, labelled EIT 3 to EIT 7, while the probe frequency is scanned across transitions 3 to 7 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The coupling and probe powers are respectively 10 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='05 mW and the ex- ternal longitudinal magnetic field is B = 1400 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Spectrum n° 6 corresponds to the case where coupling is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Small VSOP peaks are visible on each atomic resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Zero frequency corresponds to the transition frequency of Cs D2 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' ing to a thickness L ≈ λ ∼ 850 nm is outlined by an oval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The design of the Cs-filled NC used in our experi- ments is similar to that of extremely thin cell described in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Earlier it was demonstrated in [16, 17, 27] that the use of a nanocell (NC) with thickness L = λ makes it easy to record contrasted EIT resonances, which is due to the low absorption of the NC, while the disadvan- tage is broadening of the EIT resonance caused by fre- quent inelastic collisions of atoms with the windows of the NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Studies of the EIT resonances were done using a strong neodymium–iron–boron alloy ring-shaped per- manent magnet (PM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Due to the small thickness of the vapor column, the high-gradient field produced by magne can be considered uniform across the interaction region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The PM was placed after the rear window of the NC, with the axis aligned along the probe beam propagation 4 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The magnetic field in the NC was simply var- ied by longitudinal displacement of the PM, calibrated using a Teslameter HT201 magnetometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Experimental results: using MI transitions to form EIT resonances Curves 1 to 5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 4 show the experimental trans- mission spectra of the probe radiation which contain the resonances EIT 3 to EIT 7 (numbers 3-7 means that MI transitions with numbers 3-7 are involved, respectively) in a longitudinal magnetic field B = 1400 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The NC thickness is L = λ = 852 nm and the temperature of the reservoir is 100 ◦C (to prevent Cs vapor condensa- tion on the windows, the temperature of the windows is slightly higher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The coupling and the probe powers are 20 mW and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='1 mW, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Note that since only σ+ radiations participate to the formation of the EIT resonances (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 1), only half of the power of these radiations must be considered, meaning 10 mW and 50 µW, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Curve n° 6 is a probe spectrum when the coupling is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Since the cell thickness is L = λ, small peaks formed by velocity selective optical pump- ing (VSOP) resonances are located exactly at the atomic transitions frequencies, as described in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The amplitude of the EIT resonance is a factor ∼10 larger than the amplitude of the VSOP resonance, whereas the spectral width of the EIT resonance is a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='5 smaller, which is characteristic of the coher- ent EIT process [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Note that the contrast of the EIT resonance defined as the ratio of the EIT resonance am- plitude divided by the peak absorption of the Cs vapor when the coupling is blocked reached 40-50 % which is typical when a nanocell is used [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 6, curves 1 to 4 are probe transmission spectra which contain EIT 6, EIT 5, EIT 4 and EIT 3 resonances for B = 1770 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Curve n° 5 shows only the probe spec- trum when the coupling is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 7, lines 1 to 3 show the probe transmission spectra which contain EIT 6, EIT 4 and EIT 3 resonances for B = 2880 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Line n° 4 shows only the probe spectrum when the coupling is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The inset shows the profile of EIT 6 reso- nance fitted with a Gaussian profile with a FWHM of ∼ 35 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' There is also a small VSOP resonance which is formed when the coupling is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The typical FWHM of VSOP resonances is 40-50 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Preliminary theoretical calculations (shown in the right part of the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 7 were obtained by solv- ing the Liouville equations of motion for an ensemble of three-level Λ-systems (as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 5), taking into account the geometry of the nanocell (coherence dephas- ing rate determined by the time of flight of the atoms), its Fabry-Perot nature (reflections of the fields on the inner surfaces of the cell) and Doppler broadening, following the procedure described in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The Rabi frequencies of the probe and coupling lasers are respectively Ωc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='5γN and Ωp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content='06γN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Reasonable agreement between theory and experiment regarding the width and depth of the EIT resonance is obtained and the VSOP resonance is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Small discrepancies (assymetry of the profile and ampli- tude of the VSOP resonance) can arise notably from the need of considering neighboring Zeeman sublevels (not shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 1, and therefore more than three levels, to obtain more accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Scheme of the three-level Λ-system used in the calcu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The total decay rate Γ33 of state |3⟩ is 1/2(γ31 + γ32) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The dephasing rate of coherence between the ground states is Γ21 = (2πt)−1 where t is the time of flight of the atoms through the cell (at the most probable velocity u = � 2kBT/M where T is the vapor temperature and M the atomic mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The amplitude of resonance n° 6 is ∼ 50 times greater than that of the VSOP resonance and is spectrally nar- rower than the latter (this is a manifestation of the co- herent EIT process [2, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 8 the solid lines in- dicate the calculated dependences of the frequency shifts for transitions 1–7 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 1) and Fg = 3 → Fe = 4 (marked with dotted oval) to the magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The black squares represent the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' As mentioned earlier, due to the high value of the frequency shift slope for B > 3 kG, the group of MI transitions 1–7 is com- pletely separated in frequency from Fg = 3 → Fe = 4 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The curves in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 8 show experimental and theoretical spectra (calculated by combining the models presented in [7] and [30]) of the seven MI transitions ab- sorption for B = 6 kG when frequency shift reaches ∼ 30 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Note that the amplitude of transition 6 is slightly bigger than that of transition 7 (while for B < 5 kG the amplitude of transition 7 is bigger, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 2a), because of the “mixing” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Note that one of the remarkable features of the σ+ MI transitions 3 → 5′ is that they are still well recorded for a magnetic field B ≈ 8 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' They are located in the high frequency wing of the spec- trum presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 18 of paper [31] and for this case the frequency shift reaches 34 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Using our theoret- ically calculated curves for MI transitions 3 → 5′ we checked the frequency position of these MI transitions and found good agreement with the experimental curves presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' In paper [31] the 3 → 5′transitions are not identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Therefore, it is important to inform 5 Coupling off EIT 6 EIT 5 EIT 4 EIT 3 (1) (2) (3) (4) (5) 6 5 4 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Probe transmission spectra of the Cs nanocell (L = λ ≈ 850 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Spectra 1 to 4 exhibit four EIT reso- nances, labelled EIT 3 to EIT 6, while the probe frequency is scanned across transitions 3 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The external longitudi- nal magnetic field is B = 1770 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Spectrum n° 5 is a probe transmission spectrum when the coupling is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Small VSOP peaks are visible on each atomic transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Zero frequency corresponds to the transition frequency of Cs D2 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Coupling off Coupling off (1) (2) (3) (4) Experiment Coupling off EIT 6 EIT 4 EIT 3 6 5 4 3 EIT 6 Theory FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Probe transmission spectra of the Cs nanocell (L = λ = 852 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Lines 1 to 3 show four EIT resonances, la- belled EIT 4, EIT 5 and EIT 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The external longitudinal magnetic field is B = 2880 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Line 4 is a probe transmission spectrum when the coupling is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The left part of the inset is a zoom on EIT 6, fitted with a Gaussian profile (FWHM 35 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The right curves are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Red: coupling on, black: coupling off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Small VSOP peaks are visible on each atomic transitions formed by the probe radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Their typ- ical linewidth is 40-50 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Zero frequency corresponds to the transition frequency of Cs D2 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' scientists working in the field of laser spectroscopy of al- kali metal atoms about the MI atomic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The above-mentioned MI transitions can be exploited in such high B-fields as new frequency markers, for using new fre- quency ranges, as well as for the frequency stabilization of lasers at strongly shifted frequencies from the initial transition in unperturbed atoms [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Theory 6 5 4 3 2 1 7 6 5 4 3 2 1 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Red solid lines: frequency shift of transitions 1 to 7 (see figure 1) as a function of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The black squares with error bars represent experimental measurements, the inaccuracy is around 1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Black dashed lines: frequency shift of Fg = 3 → Fe = 4 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' For B > 3 kG, both groups are well separated in frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' Inset: theoretical and experimental absorption spectra for B = 6 kG, the frequency shift reaches 30 GHz from the Cs D2 line transition frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' CONCLUSION In this paper, we used for the first time forbidden transitions of Cs (Fg = 3 → Fe = 5, more precisely σ+(∆mF = +1) transitions) to create Λ-system allowing the formation of EIT resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' This was done in an ex- ternal magnetic field, as such transitions have zero proba- bility in the absence of magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' A nanometric-thin cell filled with Cs vapor was used, with a thickness corre- sponding to the resonant wavelength of Cs D2 line (≈ 850 nm), and the magnetic field was varied by longitudinal displacement of the permanent magnet along the prop- agation direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' As expected, when the cou- pling is blocked, small VSOP resonances are formed right at the different transitions’ frequencies, while coupling radiation allows for the formation of EIT resonances, spectrally narrower and with a bigger amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' We formed EIT resonances with 6 out the 7 transitions de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' This was possible up to 3 kG thanks to the big value of the frequency shift, reaching up to 4 MHz/G, therefore leading to EIT resonances shifted 12 GHz apart from the Cs D2 line transition frequency [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' This result is of great interest, as the highly-shifted spec- tra can serve as frequency references [14, 15], especially taking into account that these transitions are still easily recorded up to 8 kG when the frequency shift reaches 35 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' As for the theoretical description, further investi- gation is necessary, mainly in order to take into account the effect of neighbouring states, and thus including more levels in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' The complexity of the manifold and the number of coupled equations make it a challenging 6 and computationally-intensive task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' However, reasonable agreement was already achieved by simply considering an ensemble of three-level systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' To the best of our knowl- edge, there are no reports on obtaining EIT resonances in Λ-systems in such strong fields using usual transitions of alkali atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' We note that much narrower EIT reso- nances can be attained by using cm-long cells (to lower the effect of inelastic collisions of atoms with the win- dows), and by using coherently coupled probe and cou- pling radiations derived from a single narrow-band laser beam [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the Science Committee of the Republic of Armenia, in the frame of research project n° 21T-1C005, and by the NATO Science for Peace and Security Project under grant G5794.' metadata={'source': 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b/ENE1T4oBgHgl3EQfqQWR/content/tmp_files/2301.03341v1.pdf.txt @@ -0,0 +1,1100 @@ +Enantio-specific state transfer of chiral molecules through enantio-selective +shortcut-to-adiabaticity paths +Jian-Jian Cheng,1, 2 Chong Ye,3 and Yong Li1, 4, ∗ +1Center for Theoretical Physics and School of Science, Hainan University, Haikou 570228, China +2Beijing Computational Science Research Center, Beijing 100193, China +3Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, +School of Physics, Beijing Institute of Technology, 100081 Beijing, China +4Synergetic Innovation Center for Quantum Effects and Applications, +Hunan Normal University, Changsha 410081, China +(Dated: January 10, 2023) +An interesting method of fast enantio-specific state transfer is proposed for cyclic three-level +systems of chiral molecules. We show that the fast population transfer via shortcut to adiabaticity +can be accomplished for the cyclic three-level system of a general (chiral) molecule with invariant- +based inverse engineering of the coupling strengths. By choosing appropriate parameters, the two +enantiomers, which are initially prepared in their ground states in the three-level systems, will +evolve respectively along their enantio-selective shortcut-to-adiabaticity paths to different-energy +final states simultaneously, namely achieving the fast enantio-specific state transfer. +I. +INTRODUCTION +Since Pasteur first discovered chiral molecules in +1848, the theoretical and experimental studies of chiral +molecules have proliferated in chemistry [1], biotechnolo- +gies [2], and pharmaceutics [3]. Chiral molecules contain +two species, e.g. left- and right-handed ones [4], which are +often called enantiomers. The two enantiomers are mir- +ror images of each other but can be superposed on each +other via translations and rotations. The enantiodiscrim- +ination (as well as enantioseparation and enantioconver- +sion) [5–8] of chiral molecules remains an enormous chal- +lenge. The traditional method of enantiodiscrimination +is to break the mirror symmetry of the enantiomers by +using circularly polarized light [9]. Some commonly used +chiroptical methods of enantiodiscrimination are circular +dichroism [10], vibrating circular dichroism [11], optical +rotation [9], and Raman optical activity [12]. However, +these methods rely on the interference between electric- +dipole and weak magnetic-dipole (or electric-quadrupole) +transitions. +Alternatively, +enantiodiscrimination +methods +that +only use electric-dipole interactions [13, 14], have also +been proposed. +The left- and right-handed chiral +molecules can be modeled as cyclic three-level systems, +where three electromagnetic (optical or microwave) fields +couple respectively to three transitions via electric-dipole +interactions [15, 16]. Due to the intrinsic property of chi- +ral molecules, the product of the corresponding three cou- +pling strengths (Rabi frequencies) in the cyclic three-level +systems can differ in signs for the two enantiomers [15, +16]. +So the corresponding overall phases in the cyclic +three-level systems differ by π with the enantiomers. +Based on such cyclic three-level systems, one can use +different schemes, such as enantio-selective three-wave +∗ yongli@hainanu.edu.cn +mixing [17–21], enantio-selective absorption [22], enantio- +selective AC stark effect [23] and enantio-selective two- +dimensional spectra [24], to discriminate the left- and +right-handed molecules. Moreover, some more ingenious +sources of modern optics physics, such as frequency en- +tangled photons [25], quantized photons [26, 27], and cor- +related photons in cavities [28], have been introduced to +enhance the performance of enantiodiscrimination. +Beyond the enantiodiscrimination, the cyclic three- +level systems of chiral molecules have also been used in +some more ambitious issues, such as the enantio-specific +state transfer (ESST) [15, 29–38], enantioseparation [39– +42], and enantioconversion [16, 43–46]. The perfect ESST +of chiral molecules can be realized by transferring the left- +and right-handed chiral molecules from the same-energy +initial states to different-energy final states by choosing +suitable electromagnetic fields [15, 29–38]. Recently, the +feasibility of ESST based on the cyclic three-level sys- +tems has been demonstrated experimentally in gaseous +samples by using microwave fields [47–50]. +After the +achievement of the ESST, one can further realize the +enantiodiscrimination and spatial enantioseparation for +the chiral molecules [39, 40]. +In the original ESST method based on cyclic three- +level systems of chiral molecules [15], the ESST was re- +alized by using the adiabatic (and also diabatic) passage +technique, which makes the ESST process slow and com- +plicated. To overcome these defects, several theoretical +methods of fast ESST were proposed and developed [29– +38] based on cyclic three-level systems. Among them, an +ingenious method [31] was proposed to achieve the fast +ESST of chiral molecules by using the “shortcut to adi- +abaticity” (STA) concept via adding a counterdiabatic +field to accelerate the stimulated Raman adiabatic pas- +sage. +Motivated by Ref. [31], here we propose to achieve the +ESST by a different STA with invariant-based inverse +engineering [51–53], instead of the STA with adding the +counterdiabatic field [31]. +The invariant-based inverse +arXiv:2301.03341v1 [quant-ph] 9 Jan 2023 + +2 +engineering starts by introducing a Lewis-Riesenfeld in- +variant in a time-dependent system. The invariant can +be used to derive a law that governs the evolution state +for the designed Hamiltonian. By means of the invariant- +based inverse engineering of the time-dependent Hamil- +tonians with designing appropriate control parameters, +the left- and right-handed chiral molecules prepared ini- +tially in their corresponding ground states would evolve +(approximately) along their enantio-selective shortcut- +to-adiabaticity paths to different-energy final states. +II. +CYCLIC THREE-LEVEL SYSTEMS +A general chiral molecule can be modeled as the cyclic +three-level system by choosing appropriate three electro- +magnetic fields to couple with three electric-dipole tran- +sitions [15, 54]. Here, we only consider the case that all +the three electromagnetic fields couple resonantly with +the electric-dipole transitions respectively, as shown sim- +ilar to Fig. 1(a). In the basis of {|1⟩, |2⟩, |3⟩}, the Hamil- +tonian of the cyclic three-level system can be described +in the interaction picture as (ℏ = 1) [31] +ˆH(t) = +� +� +0 +Ωx(t) Ωz(t)e−iφ +Ωx(t) +0 +Ωy(t) +Ωz(t)eiφ Ωy(t) +0 +� +� +(1) +with |1⟩ = (1, 0, 0)T , |2⟩ = (0, 1, 0)T , |3⟩ = (0, 0, 1)T . +Here Ωj(t) (j = x, y, z) are the Rabi frequencies, which +can be controlled by varying the amplitudes of the ap- +plied electromagnetic fields. φ is the overall phase of the +three Rabi frequencies. Here we set φ = π/2. Without +loss of generality, we have assumed Ωj are real. Then the +Hamiltonian can be expressed as +ˆH(t) = Ωx(t) ˆKx + Ωy(t) ˆKy + Ωz(t) ˆKz. +(2) +Here, ˆKx, ˆKy, and ˆKz are the SU(2) angular-momentum +operators [55] +ˆKx = +� +� +0 1 0 +1 0 0 +0 0 0 +� +� , +ˆKy = +� +� +0 0 0 +0 0 1 +0 1 0 +� +� , +ˆKz = +� +� +0 0 −i +0 0 +0 +i 0 +0 +� +� . +(3) +They satisfy the commutation relations +[ ˆKx, ˆKy] = i ˆKz, [ ˆKy, ˆKz] = i ˆKx, [ ˆKz, ˆKx] = i ˆKy.(4) +The fact that Hamiltonian (2) is written as the sum of +three SU(2) operators, means it addresses the SU(2) al- +gebraic structure [53]. +For the two enantiomers of chiral molecules, the overall +phases in the cyclic three-level systems under consider- +ation differ by π [50]. For convenience, we specify that +the signs before Ωx and Ωz are equal for the two enan- +tiomers, while the sign before Ωy is opposite, as shown +in Fig. 1. +|3〉L +|3〉R +|2〉L +|2〉R +|1〉L +|1〉R +Ωzeiϕ +Ωy +Ωx +Ωzeiϕ +-Ωy +Ωx +(a) Left-handed +(b) Right-handed +FIG. 1. +(a) Left- and (b) right-handed chiral molecules of +cyclic three-level systems, where three electromagnetic fields +couple resonantly to the three electric-dipole transitions, re- +spectively, with Ωx, ±Ωy, and Ωzeiφ the corresponding Rabi +frequencies. +Therefore, the Hamiltonians of the cyclic three-level +systems for the two enantiomers in the basis {|m⟩L} and +{|m⟩R} (m = 1, 2, 3) can be described as +ˆHL,R(t) = Ωx(t) ˆKL,R +x +± Ωy(t) ˆKL,R +y ++ Ωz(t) ˆKL,R +z +. (5) +Here, the indices L and R [which correspond, respec- +tively, to the signs + and − in the right side of Eq. (5)], +denote the left- and right-handed chiral molecules, re- +spectively. +ˆKQ +j (j = x, y, z, Q = L, R) is just +ˆKj in +Eq. (3) for the two enantiomers. In this work, when refer- +ring to left- or right-handed chiral molecules, we will add +the index. When there is no index, we refer to general +molecules. +III. +INVARIANT DYNAMICS +Shortcut to adiabaticity (STA) is a fast route to ac- +celerate a slow adiabatic process by controlling the pa- +rameters of a system [56], while keeping the same initial +and final states as that in the adiabatic passage. A mo- +tivation to apply the STA technique is to manipulate the +quantum system on timescales shorter than decoherence +times. +There are two main STA techniques that have +been proposed theoretically and implemented experimen- +tally to inversely engineer the time-dependent Hamilto- +nian of a quantum system for accelerating slow adiabatic +process [52]. One is the counterdiabatic driving method +with adding an auxiliary field in a reference Hamilto- +nian to cancel the nonadiabatic coupling, where the dy- +namics follows exactly the adiabatic passage defined by +the reference Hamiltonian [52, 57, 58]. The other one is +the invariant-based inverse engineering method, which is +based on the Lewis-Riesenfeld invariant that carries the +eigenstates of a system from the initial state to the de- +sired final state [52], with keeping the same initial and +final states as those in the adiabatic passage, but without +following the adiabatic passage at the intermediate time +instants [51, 52]. In what follows, we focus on how to use + +3 +the latter STA technique to achieve the ESST of chiral +molecules. +Commonly a Lewis-Riesenfeld invariant for a Hamilto- +nian ˆH(t) is a Hermitian operator ˆI(t) that satisfies [59] +dˆI(t) +dt +≡ ∂ ˆI(t) +∂t +− i[ˆI(t), ˆH(t)] = 0, +(6) +so that its eigenvalues remain constant in time. Accord- +ing to the Lewis-Riesenfeld theory [51, 52, 59], if {|φn(t)⟩} +is a set of orthogonal eigenstates of the invariant ˆI(t), +the solution to the time-dependent Sch¨ordinger equation +can be constructed as |Ψ(t)⟩ = � +n cneiαn(t)|φn(t)⟩, with +cn being a time-independent coefficient. Here αn(t) = +� t +0⟨φn(t′)|[i∂t′ − ˆH(t′)]|φn(t′)⟩dt′ is the Lewis-Riesenfeld +phase [51, 52, 59]. +In general, ˆH(t) does not commute with the invari- +ant ˆI(t) at all time. We only require the invariant and +the Hamiltonian to commute at the initial and final time +instants, i.e., [ ˆH(0), ˆI(0)] = 0 and [ ˆH(τ), ˆI(τ)] = 0 [51– +53, 56]. The eigenstates of the Hamiltonian and the in- +variant coincide at the initial and final time instants but +may be different at the intermediate time. This leaves +large freedom to choose how the state evolves in the in- +termediate time. We can use Eq. (6) to find the Hamilto- +nian (2) that drives such a designed evolution of a given +state in the cyclic three-level system. Moreover, we con- +sider, respectively, the evolutions of the left- and right- +handed chiral molecules with cyclic three-level structures +by invariant-based inverse engineering of the Rabi fre- +quencies (equivalently the amplitude of the electromag- +netic fields). By choosing appropriate Rabi frequencies, +the fast ESST can be achieved by transferring the two +enantiomers from their ground states to different-energy +final states through their corresponding eigenstates of in- +variants, following their enantio-selective STA paths. +A. +Invariant dynamics for the left-handed chiral +molecules +We first consider the state transfer of the left-handed +chiral molecules with the cyclic three-level structures by +the invariant-based inverse engineering. Since ˆHL(t) in +Eq. (5) possesses the SU(2) algebraic structure, the cor- +responding invariant ˆIL(t) can be given as [53] +ˆIL =Ω0 +2 (cos γ sin β · ˆKL +x + cos γ cos β · ˆKL +y + sin γ · ˆKL +z ) +=Ω0 +2 +� +� +0 +cos γ sin β +−i sin γ +cos γ sin β +0 +cos γ cos β +i sin γ +cos γ cos β +0 +� +� +L +(7) +in the basis {|1⟩L, |2⟩L, |3⟩L}. Here, Ω0 is an arbitrary +constant with unit of frequency, and the time-dependent +auxiliary parameters γ and β satisfy the equations +˙γ = Ωx cos β − Ωy sin β, +˙β = (Ωx sin β + Ωy cos β) tan γ − Ωz. +(8) +The eigenstates of the invariant ˆIL(t), which satisfy +ˆIL(t)|φn(t)⟩L = λL +n|φn(t)⟩L (n = 0, ±), are +|φ0⟩L = +� +� +cos γ cos β +−i sin γ +− cos γ sin β +� +� +L +, +(9) +|φ±⟩L = +1 +√ +2 +� +� +sin γ cos β ± i sin β +i cos γ +− sin γ sin β ± i cos β +� +� +L +(10) +with the corresponding (time-independent) eigenval- +ues +λL +0 += +0 +and +λL +± += +±Ω0. +In +this +case, +the +Lewis-Riesenfeld +phases +are +αL +0 (t) += +0, +and +αL +±(t) = ∓ +� t +0[ ˙β(t′) sin β(t′) + Ωx(t′) sin β(t′) cos γ(t′) + +Ωy(t′) cos β(t′) cos γ(t′) + Ωz(t′) sin γ(t′)]dt′. +Here, we take Ωx(t) = Ωz(t) for simplicity. By using +Eq. (8), we have +Ωx = Ωz = +˙β sin β + ˙γ cos β tan γ +tan γ − sin β +, +Ωy = +˙β cos β + ˙γ(1 − tan γ sin β) +tan γ − sin β +. +(11) +Once the appropriate boundary conditions for γ and β +are fixed, one can insert a polynomial function to deter- +mine Ωx, Ωy, and Ωz. Our task is to design the Hamilto- +nian ˆHL(t) to drive the initial state |1⟩L to the final state +|3⟩L (up to a phase factor) along the invariant eigenstate +|φ0(t)⟩L in a given time τ. Therefore, based on the invari- +ant eigenstate |φ0(0)⟩L = (1, 0, 0)T +L = |1⟩L at the initial +instant time and |φ0(τ)⟩L = (0, 0, −1)T +L = −|3⟩L at the +final instant time τ, the boundary conditions for γ and +β can be given as +γ(0) = 0, β(0) = 0, +γ(τ) = 0, β(τ) = π +2 . +(12) +On one hand, one needs to impose the boundary con- +ditions to make ˆHL(t) and ˆIL(t) commute at t = 0 and +t = τ so that they have common eigenstates at these time +instants. On the other hand, one requires the Rabi fre- +quencies to vanish at the initial and final time instants to +make the electromagnetic fields turn on and off smoothly. +These requirements further imply the additional bound- +ary conditions +˙γ(0) = 0, ˙β(0) = 0, +˙γ(τ) = 0, ˙β(τ) = 0. +(13) +There are many interpolating functions consistent with +the boundary conditions at the initial and final time in- +stants. With these boundary conditions, we can simply +choose +γ(t) = 0, β(t) = 3π +2τ 2 t2 − π +τ 3 t3 + η. +(14) +Here the small value η is set to avoid the infinite values +of the Rabi frequencies at the initial time instant. Thus + +4 +the designed Rabi frequencies in Eq. (11) reduce to +Ωx = Ωz = 3πt +τ 2 +� t +τ − 1 +� +, +Ωy = 3πt +τ 2 +� t +τ − 1 +� +cot +� 3π +2τ 2 t2 − π +τ 3 t3 + η +� +. +(15) +(a) +Ωx (Ωz) +Ωy +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +t /τ +Rabi frequencies (2π /τ) +(b) +P1 +L +P3 +L +P2 +L +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +t /τ +population +FIG. 2. +(Color online) (a) The designed Rabi frequen- +cies for the left-handed chiral molecules with Ωx = Ωz (red +solid line) and Ωy (blue dashed line) given in Eq. (15). (b) +Time evolution of corresponding populations in |1⟩L (red solid +line), |2⟩L (black dotted line), and |3⟩L (blue dashed line) for +the left-handed chiral molecules with the initial state |1⟩L. +Here η = 0.02. +Fig. 2 shows the designed Rabi frequencies for the left- +handed chiral molecules and corresponding evolution of +the populations in the states |m⟩L (m = 1, 2, 3) for the +initial state |Ψ(0)⟩L = |1⟩L. In the ideal condition (i.e. +the case of η = 0), the left-handed chiral molecules will +evolve from the initial state |1⟩L (= |φ0(0)⟩L) to the +final target state −|3⟩L (up to a phase factor), along +the invariant eigenstate |φ0(t)⟩L. For the case of small +value η = 0.02 as shown in Fig. 2(b), the initial state +|1⟩L ≈ |φ0(0)⟩L, thus the populations in the initial state +|1⟩L with P L +1 (0) = 1 are finally transferred approxi- +mately to that in the target state |3⟩L with probabil- +ity P L +3 (τ) = 0.9991 for the left-handed chiral molecules. +Correspondingly, P L +2 (0) = 0 = P L +3 (0), P L +1 (τ) = 0.0005, +and P L +2 (τ) = 0.0004. +B. +Invariant dynamics for the right-handed chiral +molecules +Then we consider the state transfer of the right-handed +chiral molecules with the cyclic three-level structures +by the invariant-based inverse engineering. +Since the +Hamiltonian ˆHR(t) in Eq. (5) of the right-handed chi- +ral molecules has the same SU(2) algebraic structure as +ˆHL(t) of the left-handed ones, similarly the invariant +ˆIR(t) can be given in the basis {|1⟩R, |2⟩R, |3⟩R} as the +form +ˆIR= Ω0 +2 (cos ξ sin χ · ˆKR +x + cos ξ cos χ · ˆKR +y + sin ξ · ˆKR +z ) += Ω0 +2 +� +� +0 +cos ξ sin χ +−i sin ξ +cos ξ sin χ +0 +cos ξ cos χ +i sin ξ +cos ξ cos χ +0 +� +� +R +. +(16) +Here the time-dependent auxiliary parameters ξ(t) and +χ(t) satisfy the equations +˙ξ = Ωx cos χ + Ωy sin χ, +˙χ = (Ωx sin χ − Ωy cos χ) tan ξ − Ωz. +(17) +The eigenstates of the invariant ˆIR(t), which satisfy +ˆIR(t)|φn(t)⟩R = λR +n |φn(t)⟩R (n = 0, ±), are +|φ0⟩R = +� +� +cos ξ cos χ +−i sin ξ +− cos ξ sin χ +� +� +R +, +(18) +|φ±⟩R = +1 +√ +2 +� +� +sin ξ cos χ ± i sin χ +i cos ξ +− sin ξ sin χ ± i cos χ +� +� +R +(19) +with the corresponding eigenvalues λR +0 = 0 and λR +± = +±Ω0. Here the Lewis-Riesenfeld phase is αR +0 (t) = 0, and +αR +±(t) = ∓ +� t +0[ ˙χ(t′) sin χ(t′) + Ωx(t′) sin χ(t′) cos ξ(t′) − +Ωy(t′) cos χ(t′) cos ξ(t′) + Ωz(t′) sin ξ(t′)]dt′. +Here we still take Ωx = Ωz for simplicity. According +to Eq. (17), we have +Ωx = Ωz = ˙χ sin χ + ˙ξ cos χ tan ξ +tan ξ − sin χ +, +Ωy = ˙χ cos χ + ˙ξ(1 − tan ξ sin χ) +sin χ − tan ξ +. +(20) +Similar to the case of the left-handed chiral molecules +in the above subsection, once the functions χ and ξ are +fixed, we can construct Ωx, Ωy, and Ωz and thus the +Hamiltonian HR(t) can be determined. Here we aim to +design the Hamiltonian ˆHR(t) to make the system evolve +from the initial state |1⟩R to the finial state |2⟩R (up to +a phase factor) along the invariant eigenstate |φ0(t)⟩R +in a given time τ. +Therefore, based on the invariant +eigenstate |φ0(0)⟩R = (1, 0, 0)T +R = |1⟩R at the initial time +instant and |φ0(τ)⟩R = (0, −i, 0)T +R = −i|2⟩R at the final +time instant τ, the boundary conditions for ξ and χ can +be given as +ξ(0) = 0, χ(0) = 0, ξ(τ) = −π +2 . +(21) + +5 +(a) +Ωx (Ωz) +Ωy +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +t /τ +Rabi frequencies (2π /τ) +(b) +P1 +R +P2 +R +P3 +R +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +t /τ +population +FIG. 3. (Color online) (a) The designed Rabi frequencies for +the right-handed chiral molecules with Ωx = Ωz (red solid +line) and Ωy (blue dashed line) given in Eq. (24). (b) Time +evolution of corresponding populations in |1⟩R (red solid line), +|2⟩R (black dotted line), and |3⟩R (blue dashed line) for the +right-handed chiral molecules with the initial state |1⟩R. Here +η′ = −0.02. +Similarly, we set ˆHR(t) and ˆIR(t) commute at the ini- +tial and final time instants (so that they have the same +eigenstates at these time instants) and make the electro- +magnetic fields (equivalently the Rabi frequencies) turn +on and off smoothly for the right-handed chiral molecules. +Thus, the additional boundary conditions for ξ(t) and +χ(t) can be given as +˙ξ(0) = 0, ˙χ(0) = 0, +˙ξ(τ) = 0, ˙χ(τ) = 0. +(22) +Consistent with these boundary conditions, we can +choose +χ(t) = 0, ξ(t) = − 3π +2τ 2 t2 + π +τ 3 t3 + η′. +(23) +Here the small value η′ is set to avoid the infinite values +of the Rabi frequencies at the initial time instant. Thus +the designed Rabi frequencies in Eq. (20) reduce to +Ωx = Ωz = 3πt +τ 2 +� t +τ − 1 +� +, +Ωy = 3πt +τ 2 +� t +τ − 1 +� +cot +� 3π +2τ 2 t2 − π +τ 3 t3 − η′ +� +. +(24) +Fig. 3 shows the designed Rabi frequencies of the right- +handed chiral molecules and corresponding evolution of +the populations in the states |m⟩R (m = 1, 2, 3) for the +initial state |Ψ(0)⟩R = |1⟩R. +In the ideal condition (i.e. the case of η′ = 0), the +right-handed chiral molecules will evolve from the initial +state |1⟩R (= |φ0(0)⟩R) to the final target state −i|2⟩L +(up to a phase factor), along the invariant eigenstate +|φ0(t)⟩R. When we set the small value η′ = −0.02 as +shown in Fig. 3(b), the initial state |1⟩R ≈ |φ0(0)⟩R, thus +the populations in the initial state |1⟩R with P R +1 (0) = 1 +are finally transferred approximately to that in the tar- +get state |2⟩R with P R +2 (τ) = 0.9991 for the right-handed +chiral molecules. Correspondingly, P R +2 (0) = 0 = P R +3 (0), +P R +1 (τ) = 0.0005, and P R +3 (τ) = 0.0004. +C. +Achieving the fast enantio-specific state transfer +So far we have designed the desired evolution for the +left- and right-handed chiral molecules of the cyclic three- +level systems via the STA technique with invariant-based +inverse engineering in the above two subsections, respec- +tively. +By comparing Eq. (15) with Eq. (24), it can +be found that the two groups of designed Rabi frequen- +cies for the two enantiomers are exactly the same when +η = −η′. This means that the two enantiomers are driven +by the same three electromagnetic fields indeed. In this +case, the left-handed chiral molecule begins with |1⟩L and +terminates approximately at −|3⟩L, almost along the in- +variant eigenstate |φ0(t)⟩L, while the right-handed chi- +ral molecule begins with |1⟩R and terminates approxi- +mately at −i|2⟩R, almost along the invariant eigenstate +|φ0(t)⟩R simultaneously. +As also shown in Fig. 2 and +Fig. 3, the left- and right-handed chiral molecules pre- +pared in the same-energy initial states evolves (approx- +imately) to the different-energy final states via the dif- +ferent enantio-selective STA processes of invariant-based +inverse engineering, driven by the same electromagnetic +fields. Thus, the fast ESST via enantio-selective STA is +achieved (approximately). +In the above ESST method via the enantio-selective +STA with invariant-based inverse engineering, the enan- +tiomeric excess of the ESST can be defined as [23, 38] +ϵ ≡ +���P L +3 (τ) − P R +3 (τ) +P L +3 (τ) + P R +3 (τ) +���. +(25) +Although the small values η and η′ (e.g. +η = −η′ = +0.02) have been introduced to avoid the infinite Ωy at +the initial time instant, we still obtain a highly efficient +ESST with enantiomeric excess ϵ = 99.92% at the final +time instant (with most of left-chiral molecule staying +in |3⟩L and very few of the right-chiral molecule staying +in the same-energy state |3⟩R, as shown in Fig. 2 and +Fig. 3). +In general, the final populations are effected by the +small value η (or η′) and are independent of the param- +eter τ. As shown in Fig. 4(a), the population of the tar- + +6 +get state |3⟩L can be further decreased by increasing the +small value η, while the population of the other target +state |3⟩R would be commonly increased by increasing +the small value η. Therefore, it is possible to achieve a +better enantiomeric excess with relatively small value η. +According to Eq. (15) and Eq. (24), decreasing the small +amount η (or η′) implies the tradeoff of requiring larger +Rabi frequencies and laser intensities [53]. Here we define +Ωmax=Max{|Ωx(t)|, |Ωy(t)|, |Ωz(t)|} as the maximum ab- +solute value of the Rabi frequencies during the whole evo- +lution process. As shown in Fig. 4(b), the maximum ab- +solute value of the Rabi frequencies increase dramatically +when decreasing the small value η. +(a) +P3 +R +P3 +L +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.80 +0.85 +0.90 +0.95 +1.00 +0 +0.01 +0.02 +0.03 +0.04 +η +population +(b) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +5 +10 +50 +100 +500 +1000 +η +Ωmax (2π⨯MHz) +FIG. 4. (Color online) (a) The corresponding populations in +|3⟩L (red solid line) and |3⟩R (blue dashed line) at the final +time versus the small value η. The initial states are |1⟩L,R. +(b) The maximum absolute value of the Rabi frequencies Ωmax +versus the small value η with τ = 0.5 µs. +In experiments, the typical Rabi frequencies for the +transitions of chiral molecules are about 2π×10 MHz [18, +47, 48]. That means the evolution time can be shortened +to be 0.5 µs for the experimentally available Rabi fre- +quencies. Thus, the decoherence effects (typically being +about 5 ∼ 6 µs) [17, 47] will become negligable. +This +is the advantage of our ESST method since it allows to +manipulate the quantum system on the timescales much +shorter than the typical decoherence time. +Note that in the previous ESST method via STA [31], +an auxiliary counterdiabatic field has been applied. It +works as a shortcut to adiabaticity for canceling the +nonadiabatic coupling and induces perfect population +transfer between the states |1⟩L and |3⟩L for the left- +handed chiral molecules. +Simultaneously, it also acts +oppositely for strengthening the nonadiabatic coupling +for the right-handed chiral molecules and the population +transfer between the states |1⟩R and |3⟩R is canceled com- +pletely. Therefore, under such an ESST process, the left- +handed chiral molecule begins with |1⟩L and terminates +at −|3⟩L, following a STA path. But the right-handed +chiral molecule is subject to a free evolution, instead +of following the STA path. +By contrast, in our ESST +method via STA, the eigenstates of invariants for the two +enantiomers define their corresponding enantio-selective +STA paths. Thus, our ESST can be achieved with trans- +ferring the two enantiomers from their ground states to +different-energy final states along their enantio-selective +STA paths simultaneously, by choosing appropriate in- +tensities of the three electromagnetic fields (that is, the +Rabi frequencies). +IV. +CONCLUSION +In conclusion, we have proposed the fast ESST method +of chiral molecules via the STA technique with invariant- +based inverse engineering. 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Phys. 10, 1458 (1969). + diff --git a/ENE1T4oBgHgl3EQfqQWR/content/tmp_files/load_file.txt b/ENE1T4oBgHgl3EQfqQWR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3dced1395d54a078ea586383823264b45c89e3b --- /dev/null +++ b/ENE1T4oBgHgl3EQfqQWR/content/tmp_files/load_file.txt @@ -0,0 +1,784 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf,len=783 +page_content='Enantio-specific state transfer of chiral molecules through enantio-selective shortcut-to-adiabaticity paths Jian-Jian Cheng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2 Chong Ye,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='3 and Yong Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' ∗ 1Center for Theoretical Physics and School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Hainan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Haikou 570228,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' China 2Beijing Computational Science Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Beijing 100193,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' China 3Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Beijing Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 100081 Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' China 4Synergetic Innovation Center for Quantum Effects and Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Hunan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Changsha 410081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' China (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2023) An interesting method of fast enantio-specific state transfer is proposed for cyclic three-level systems of chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' We show that the fast population transfer via shortcut to adiabaticity can be accomplished for the cyclic three-level system of a general (chiral) molecule with invariant- based inverse engineering of the coupling strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By choosing appropriate parameters, the two enantiomers, which are initially prepared in their ground states in the three-level systems, will evolve respectively along their enantio-selective shortcut-to-adiabaticity paths to different-energy final states simultaneously, namely achieving the fast enantio-specific state transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' INTRODUCTION Since Pasteur first discovered chiral molecules in 1848, the theoretical and experimental studies of chiral molecules have proliferated in chemistry [1], biotechnolo- gies [2], and pharmaceutics [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Chiral molecules contain two species, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' left- and right-handed ones [4], which are often called enantiomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The two enantiomers are mir- ror images of each other but can be superposed on each other via translations and rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The enantiodiscrim- ination (as well as enantioseparation and enantioconver- sion) [5–8] of chiral molecules remains an enormous chal- lenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The traditional method of enantiodiscrimination is to break the mirror symmetry of the enantiomers by using circularly polarized light [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Some commonly used chiroptical methods of enantiodiscrimination are circular dichroism [10], vibrating circular dichroism [11], optical rotation [9], and Raman optical activity [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' However, these methods rely on the interference between electric- dipole and weak magnetic-dipole (or electric-quadrupole) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Alternatively, enantiodiscrimination methods that only use electric-dipole interactions [13, 14], have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The left- and right-handed chiral molecules can be modeled as cyclic three-level systems, where three electromagnetic (optical or microwave) fields couple respectively to three transitions via electric-dipole interactions [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Due to the intrinsic property of chi- ral molecules, the product of the corresponding three cou- pling strengths (Rabi frequencies) in the cyclic three-level systems can differ in signs for the two enantiomers [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' So the corresponding overall phases in the cyclic three-level systems differ by π with the enantiomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Based on such cyclic three-level systems, one can use different schemes, such as enantio-selective three-wave ∗ yongli@hainanu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='cn mixing [17–21], enantio-selective absorption [22], enantio- selective AC stark effect [23] and enantio-selective two- dimensional spectra [24], to discriminate the left- and right-handed molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Moreover, some more ingenious sources of modern optics physics, such as frequency en- tangled photons [25], quantized photons [26, 27], and cor- related photons in cavities [28], have been introduced to enhance the performance of enantiodiscrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Beyond the enantiodiscrimination, the cyclic three- level systems of chiral molecules have also been used in some more ambitious issues, such as the enantio-specific state transfer (ESST) [15, 29–38], enantioseparation [39– 42], and enantioconversion [16, 43–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The perfect ESST of chiral molecules can be realized by transferring the left- and right-handed chiral molecules from the same-energy initial states to different-energy final states by choosing suitable electromagnetic fields [15, 29–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Recently, the feasibility of ESST based on the cyclic three-level sys- tems has been demonstrated experimentally in gaseous samples by using microwave fields [47–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' After the achievement of the ESST, one can further realize the enantiodiscrimination and spatial enantioseparation for the chiral molecules [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In the original ESST method based on cyclic three- level systems of chiral molecules [15], the ESST was re- alized by using the adiabatic (and also diabatic) passage technique, which makes the ESST process slow and com- plicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' To overcome these defects, several theoretical methods of fast ESST were proposed and developed [29– 38] based on cyclic three-level systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Among them, an ingenious method [31] was proposed to achieve the fast ESST of chiral molecules by using the “shortcut to adi- abaticity” (STA) concept via adding a counterdiabatic field to accelerate the stimulated Raman adiabatic pas- sage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Motivated by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' [31], here we propose to achieve the ESST by a different STA with invariant-based inverse engineering [51–53], instead of the STA with adding the counterdiabatic field [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The invariant-based inverse arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='03341v1 [quant-ph] 9 Jan 2023 2 engineering starts by introducing a Lewis-Riesenfeld in- variant in a time-dependent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The invariant can be used to derive a law that governs the evolution state for the designed Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By means of the invariant- based inverse engineering of the time-dependent Hamil- tonians with designing appropriate control parameters, the left- and right-handed chiral molecules prepared ini- tially in their corresponding ground states would evolve (approximately) along their enantio-selective shortcut- to-adiabaticity paths to different-energy final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' CYCLIC THREE-LEVEL SYSTEMS A general chiral molecule can be modeled as the cyclic three-level system by choosing appropriate three electro- magnetic fields to couple with three electric-dipole tran- sitions [15, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here, we only consider the case that all the three electromagnetic fields couple resonantly with the electric-dipole transitions respectively, as shown sim- ilar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In the basis of {|1⟩, |2⟩, |3⟩}, the Hamil- tonian of the cyclic three-level system can be described in the interaction picture as (ℏ = 1) [31] ˆH(t) = � � 0 Ωx(t) Ωz(t)e−iφ Ωx(t) 0 Ωy(t) Ωz(t)eiφ Ωy(t) 0 � � (1) with |1⟩ = (1, 0, 0)T , |2⟩ = (0, 1, 0)T , |3⟩ = (0, 0, 1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here Ωj(t) (j = x, y, z) are the Rabi frequencies, which can be controlled by varying the amplitudes of the ap- plied electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' φ is the overall phase of the three Rabi frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here we set φ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Without loss of generality, we have assumed Ωj are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Then the Hamiltonian can be expressed as ˆH(t) = Ωx(t) ˆKx + Ωy(t) ˆKy + Ωz(t) ˆKz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (2) Here, ˆKx, ˆKy, and ˆKz are the SU(2) angular-momentum operators [55] ˆKx = � � 0 1 0 1 0 0 0 0 0 � � , ˆKy = � � 0 0 0 0 0 1 0 1 0 � � , ˆKz = � � 0 0 −i 0 0 0 i 0 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (3) They satisfy the commutation relations [ ˆKx, ˆKy] = i ˆKz, [ ˆKy, ˆKz] = i ˆKx, [ ˆKz, ˆKx] = i ˆKy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (4) The fact that Hamiltonian (2) is written as the sum of three SU(2) operators, means it addresses the SU(2) al- gebraic structure [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' For the two enantiomers of chiral molecules, the overall phases in the cyclic three-level systems under consider- ation differ by π [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' For convenience, we specify that the signs before Ωx and Ωz are equal for the two enan- tiomers, while the sign before Ωy is opposite, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' |3〉L |3〉R |2〉L |2〉R |1〉L |1〉R Ωzeiϕ Ωy Ωx Ωzeiϕ Ωy Ωx (a) Left-handed (b) Right-handed FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (a) Left- and (b) right-handed chiral molecules of cyclic three-level systems, where three electromagnetic fields couple resonantly to the three electric-dipole transitions, re- spectively, with Ωx, ±Ωy, and Ωzeiφ the corresponding Rabi frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Therefore, the Hamiltonians of the cyclic three-level systems for the two enantiomers in the basis {|m⟩L} and {|m⟩R} (m = 1, 2, 3) can be described as ˆHL,R(t) = Ωx(t) ˆKL,R x ± Ωy(t) ˆKL,R y + Ωz(t) ˆKL,R z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (5) Here, the indices L and R [which correspond, respec- tively, to the signs + and − in the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (5)], denote the left- and right-handed chiral molecules, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' ˆKQ j (j = x, y, z, Q = L, R) is just ˆKj in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (3) for the two enantiomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In this work, when refer- ring to left- or right-handed chiral molecules, we will add the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' When there is no index, we refer to general molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' INVARIANT DYNAMICS Shortcut to adiabaticity (STA) is a fast route to ac- celerate a slow adiabatic process by controlling the pa- rameters of a system [56], while keeping the same initial and final states as that in the adiabatic passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' A mo- tivation to apply the STA technique is to manipulate the quantum system on timescales shorter than decoherence times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' There are two main STA techniques that have been proposed theoretically and implemented experimen- tally to inversely engineer the time-dependent Hamilto- nian of a quantum system for accelerating slow adiabatic process [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' One is the counterdiabatic driving method with adding an auxiliary field in a reference Hamilto- nian to cancel the nonadiabatic coupling, where the dy- namics follows exactly the adiabatic passage defined by the reference Hamiltonian [52, 57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The other one is the invariant-based inverse engineering method, which is based on the Lewis-Riesenfeld invariant that carries the eigenstates of a system from the initial state to the de- sired final state [52], with keeping the same initial and final states as those in the adiabatic passage, but without following the adiabatic passage at the intermediate time instants [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In what follows, we focus on how to use 3 the latter STA technique to achieve the ESST of chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Commonly a Lewis-Riesenfeld invariant for a Hamilto- nian ˆH(t) is a Hermitian operator ˆI(t) that satisfies [59] dˆI(t) dt ≡ ∂ ˆI(t) ∂t − i[ˆI(t), ˆH(t)] = 0, (6) so that its eigenvalues remain constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Accord- ing to the Lewis-Riesenfeld theory [51, 52, 59], if {|φn(t)⟩} is a set of orthogonal eigenstates of the invariant ˆI(t), the solution to the time-dependent Sch¨ordinger equation can be constructed as |Ψ(t)⟩ = � n cneiαn(t)|φn(t)⟩, with cn being a time-independent coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here αn(t) = � t 0⟨φn(t′)|[i∂t′ − ˆH(t′)]|φn(t′)⟩dt′ is the Lewis-Riesenfeld phase [51, 52, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In general, ˆH(t) does not commute with the invari- ant ˆI(t) at all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' We only require the invariant and the Hamiltonian to commute at the initial and final time instants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=', [ ˆH(0), ˆI(0)] = 0 and [ ˆH(τ), ˆI(τ)] = 0 [51– 53, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The eigenstates of the Hamiltonian and the in- variant coincide at the initial and final time instants but may be different at the intermediate time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' This leaves large freedom to choose how the state evolves in the in- termediate time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' We can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (6) to find the Hamilto- nian (2) that drives such a designed evolution of a given state in the cyclic three-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Moreover, we con- sider, respectively, the evolutions of the left- and right- handed chiral molecules with cyclic three-level structures by invariant-based inverse engineering of the Rabi fre- quencies (equivalently the amplitude of the electromag- netic fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By choosing appropriate Rabi frequencies, the fast ESST can be achieved by transferring the two enantiomers from their ground states to different-energy final states through their corresponding eigenstates of in- variants, following their enantio-selective STA paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Invariant dynamics for the left-handed chiral molecules We first consider the state transfer of the left-handed chiral molecules with the cyclic three-level structures by the invariant-based inverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Since ˆHL(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (5) possesses the SU(2) algebraic structure, the cor- responding invariant ˆIL(t) can be given as [53] ˆIL =Ω0 2 (cos γ sin β · ˆKL x + cos γ cos β · ˆKL y + sin γ · ˆKL z ) =Ω0 2 � � 0 cos γ sin β −i sin γ cos γ sin β 0 cos γ cos β i sin γ cos γ cos β 0 � � L (7) in the basis {|1⟩L, |2⟩L, |3⟩L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here, Ω0 is an arbitrary constant with unit of frequency, and the time-dependent auxiliary parameters γ and β satisfy the equations ˙γ = Ωx cos β − Ωy sin β, ˙β = (Ωx sin β + Ωy cos β) tan γ − Ωz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (8) The eigenstates of the invariant ˆIL(t), which satisfy ˆIL(t)|φn(t)⟩L = λL n|φn(t)⟩L (n = 0, ±), are |φ0⟩L = � � cos γ cos β −i sin γ − cos γ sin β � � L , (9) |φ±⟩L = 1 √ 2 � � sin γ cos β ± i sin β i cos γ − sin γ sin β ± i cos β � � L (10) with the corresponding (time-independent) eigenval- ues λL 0 = 0 and λL ± = ±Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In this case, the Lewis-Riesenfeld phases are αL 0 (t) = 0, and αL ±(t) = ∓ � t 0[ ˙β(t′) sin β(t′) + Ωx(t′) sin β(t′) cos γ(t′) + Ωy(t′) cos β(t′) cos γ(t′) + Ωz(t′) sin γ(t′)]dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here, we take Ωx(t) = Ωz(t) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (8), we have Ωx = Ωz = ˙β sin β + ˙γ cos β tan γ tan γ − sin β , Ωy = ˙β cos β + ˙γ(1 − tan γ sin β) tan γ − sin β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (11) Once the appropriate boundary conditions for γ and β are fixed, one can insert a polynomial function to deter- mine Ωx, Ωy, and Ωz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Our task is to design the Hamilto- nian ˆHL(t) to drive the initial state |1⟩L to the final state |3⟩L (up to a phase factor) along the invariant eigenstate |φ0(t)⟩L in a given time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Therefore, based on the invari- ant eigenstate |φ0(0)⟩L = (1, 0, 0)T L = |1⟩L at the initial instant time and |φ0(τ)⟩L = (0, 0, −1)T L = −|3⟩L at the final instant time τ, the boundary conditions for γ and β can be given as γ(0) = 0, β(0) = 0, γ(τ) = 0, β(τ) = π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (12) On one hand, one needs to impose the boundary con- ditions to make ˆHL(t) and ˆIL(t) commute at t = 0 and t = τ so that they have common eigenstates at these time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' On the other hand, one requires the Rabi fre- quencies to vanish at the initial and final time instants to make the electromagnetic fields turn on and off smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' These requirements further imply the additional bound- ary conditions ˙γ(0) = 0, ˙β(0) = 0, ˙γ(τ) = 0, ˙β(τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (13) There are many interpolating functions consistent with the boundary conditions at the initial and final time in- stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' With these boundary conditions, we can simply choose γ(t) = 0, β(t) = 3π 2τ 2 t2 − π τ 3 t3 + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (14) Here the small value η is set to avoid the infinite values of the Rabi frequencies at the initial time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus 4 the designed Rabi frequencies in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (11) reduce to Ωx = Ωz = 3πt τ 2 � t τ − 1 � , Ωy = 3πt τ 2 � t τ − 1 � cot � 3π 2τ 2 t2 − π τ 3 t3 + η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (15) (a) Ωx (Ωz) Ωy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 t /τ Rabi frequencies (2π /τ) (b) P1 L P3 L P2 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 t /τ population FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (Color online) (a) The designed Rabi frequen- cies for the left-handed chiral molecules with Ωx = Ωz (red solid line) and Ωy (blue dashed line) given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (b) Time evolution of corresponding populations in |1⟩L (red solid line), |2⟩L (black dotted line), and |3⟩L (blue dashed line) for the left-handed chiral molecules with the initial state |1⟩L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2 shows the designed Rabi frequencies for the left- handed chiral molecules and corresponding evolution of the populations in the states |m⟩L (m = 1, 2, 3) for the initial state |Ψ(0)⟩L = |1⟩L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In the ideal condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' the case of η = 0), the left-handed chiral molecules will evolve from the initial state |1⟩L (= |φ0(0)⟩L) to the final target state −|3⟩L (up to a phase factor), along the invariant eigenstate |φ0(t)⟩L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' For the case of small value η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2(b), the initial state |1⟩L ≈ |φ0(0)⟩L, thus the populations in the initial state |1⟩L with P L 1 (0) = 1 are finally transferred approxi- mately to that in the target state |3⟩L with probabil- ity P L 3 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='9991 for the left-handed chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Correspondingly, P L 2 (0) = 0 = P L 3 (0), P L 1 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0005, and P L 2 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Invariant dynamics for the right-handed chiral molecules Then we consider the state transfer of the right-handed chiral molecules with the cyclic three-level structures by the invariant-based inverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Since the Hamiltonian ˆHR(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (5) of the right-handed chi- ral molecules has the same SU(2) algebraic structure as ˆHL(t) of the left-handed ones, similarly the invariant ˆIR(t) can be given in the basis {|1⟩R, |2⟩R, |3⟩R} as the form ˆIR= Ω0 2 (cos ξ sin χ · ˆKR x + cos ξ cos χ · ˆKR y + sin ξ · ˆKR z ) = Ω0 2 � � 0 cos ξ sin χ −i sin ξ cos ξ sin χ 0 cos ξ cos χ i sin ξ cos ξ cos χ 0 � � R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (16) Here the time-dependent auxiliary parameters ξ(t) and χ(t) satisfy the equations ˙ξ = Ωx cos χ + Ωy sin χ, ˙χ = (Ωx sin χ − Ωy cos χ) tan ξ − Ωz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (17) The eigenstates of the invariant ˆIR(t), which satisfy ˆIR(t)|φn(t)⟩R = λR n |φn(t)⟩R (n = 0, ±), are |φ0⟩R = � � cos ξ cos χ −i sin ξ − cos ξ sin χ � � R , (18) |φ±⟩R = 1 √ 2 � � sin ξ cos χ ± i sin χ i cos ξ − sin ξ sin χ ± i cos χ � � R (19) with the corresponding eigenvalues λR 0 = 0 and λR ± = ±Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here the Lewis-Riesenfeld phase is αR 0 (t) = 0, and αR ±(t) = ∓ � t 0[ ˙χ(t′) sin χ(t′) + Ωx(t′) sin χ(t′) cos ξ(t′) − Ωy(t′) cos χ(t′) cos ξ(t′) + Ωz(t′) sin ξ(t′)]dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here we still take Ωx = Ωz for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (17), we have Ωx = Ωz = ˙χ sin χ + ˙ξ cos χ tan ξ tan ξ − sin χ , Ωy = ˙χ cos χ + ˙ξ(1 − tan ξ sin χ) sin χ − tan ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (20) Similar to the case of the left-handed chiral molecules in the above subsection, once the functions χ and ξ are fixed, we can construct Ωx, Ωy, and Ωz and thus the Hamiltonian HR(t) can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here we aim to design the Hamiltonian ˆHR(t) to make the system evolve from the initial state |1⟩R to the finial state |2⟩R (up to a phase factor) along the invariant eigenstate |φ0(t)⟩R in a given time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Therefore, based on the invariant eigenstate |φ0(0)⟩R = (1, 0, 0)T R = |1⟩R at the initial time instant and |φ0(τ)⟩R = (0, −i, 0)T R = −i|2⟩R at the final time instant τ, the boundary conditions for ξ and χ can be given as ξ(0) = 0, χ(0) = 0, ξ(τ) = −π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (21) 5 (a) Ωx (Ωz) Ωy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 t /τ Rabi frequencies (2π /τ) (b) P1 R P2 R P3 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 t /τ population FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (Color online) (a) The designed Rabi frequencies for the right-handed chiral molecules with Ωx = Ωz (red solid line) and Ωy (blue dashed line) given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (b) Time evolution of corresponding populations in |1⟩R (red solid line), |2⟩R (black dotted line), and |3⟩R (blue dashed line) for the right-handed chiral molecules with the initial state |1⟩R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here η′ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Similarly, we set ˆHR(t) and ˆIR(t) commute at the ini- tial and final time instants (so that they have the same eigenstates at these time instants) and make the electro- magnetic fields (equivalently the Rabi frequencies) turn on and off smoothly for the right-handed chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus, the additional boundary conditions for ξ(t) and χ(t) can be given as ˙ξ(0) = 0, ˙χ(0) = 0, ˙ξ(τ) = 0, ˙χ(τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (22) Consistent with these boundary conditions, we can choose χ(t) = 0, ξ(t) = − 3π 2τ 2 t2 + π τ 3 t3 + η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (23) Here the small value η′ is set to avoid the infinite values of the Rabi frequencies at the initial time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus the designed Rabi frequencies in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (20) reduce to Ωx = Ωz = 3πt τ 2 � t τ − 1 � , Ωy = 3πt τ 2 � t τ − 1 � cot � 3π 2τ 2 t2 − π τ 3 t3 − η′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (24) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 3 shows the designed Rabi frequencies of the right- handed chiral molecules and corresponding evolution of the populations in the states |m⟩R (m = 1, 2, 3) for the initial state |Ψ(0)⟩R = |1⟩R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In the ideal condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' the case of η′ = 0), the right-handed chiral molecules will evolve from the initial state |1⟩R (= |φ0(0)⟩R) to the final target state −i|2⟩L (up to a phase factor), along the invariant eigenstate |φ0(t)⟩R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' When we set the small value η′ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 3(b), the initial state |1⟩R ≈ |φ0(0)⟩R, thus the populations in the initial state |1⟩R with P R 1 (0) = 1 are finally transferred approximately to that in the tar- get state |2⟩R with P R 2 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='9991 for the right-handed chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Correspondingly, P R 2 (0) = 0 = P R 3 (0), P R 1 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0005, and P R 3 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Achieving the fast enantio-specific state transfer So far we have designed the desired evolution for the left- and right-handed chiral molecules of the cyclic three- level systems via the STA technique with invariant-based inverse engineering in the above two subsections, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (15) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (24), it can be found that the two groups of designed Rabi frequen- cies for the two enantiomers are exactly the same when η = −η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' This means that the two enantiomers are driven by the same three electromagnetic fields indeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In this case, the left-handed chiral molecule begins with |1⟩L and terminates approximately at −|3⟩L, almost along the in- variant eigenstate |φ0(t)⟩L, while the right-handed chi- ral molecule begins with |1⟩R and terminates approxi- mately at −i|2⟩R, almost along the invariant eigenstate |φ0(t)⟩R simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' As also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 3, the left- and right-handed chiral molecules pre- pared in the same-energy initial states evolves (approx- imately) to the different-energy final states via the dif- ferent enantio-selective STA processes of invariant-based inverse engineering, driven by the same electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus, the fast ESST via enantio-selective STA is achieved (approximately).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In the above ESST method via the enantio-selective STA with invariant-based inverse engineering, the enan- tiomeric excess of the ESST can be defined as [23, 38] ϵ ≡ ���P L 3 (τ) − P R 3 (τ) P L 3 (τ) + P R 3 (τ) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (25) Although the small values η and η′ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' η = −η′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02) have been introduced to avoid the infinite Ωy at the initial time instant, we still obtain a highly efficient ESST with enantiomeric excess ϵ = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='92% at the final time instant (with most of left-chiral molecule staying in |3⟩L and very few of the right-chiral molecule staying in the same-energy state |3⟩R, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In general, the final populations are effected by the small value η (or η′) and are independent of the param- eter τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 4(a), the population of the tar- 6 get state |3⟩L can be further decreased by increasing the small value η, while the population of the other target state |3⟩R would be commonly increased by increasing the small value η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Therefore, it is possible to achieve a better enantiomeric excess with relatively small value η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (15) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (24), decreasing the small amount η (or η′) implies the tradeoff of requiring larger Rabi frequencies and laser intensities [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here we define Ωmax=Max{|Ωx(t)|, |Ωy(t)|, |Ωz(t)|} as the maximum ab- solute value of the Rabi frequencies during the whole evo- lution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 4(b), the maximum ab- solute value of the Rabi frequencies increase dramatically when decreasing the small value η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (a) P3 R P3 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='00 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='04 η population (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='25 5 10 50 100 500 1000 η Ωmax (2π⨯MHz) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (Color online) (a) The corresponding populations in |3⟩L (red solid line) and |3⟩R (blue dashed line) at the final time versus the small value η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The initial states are |1⟩L,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (b) The maximum absolute value of the Rabi frequencies Ωmax versus the small value η with τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In experiments, the typical Rabi frequencies for the transitions of chiral molecules are about 2π×10 MHz [18, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' That means the evolution time can be shortened to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 µs for the experimentally available Rabi fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus, the decoherence effects (typically being about 5 ∼ 6 µs) [17, 47] will become negligable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' This is the advantage of our ESST method since it allows to manipulate the quantum system on the timescales much shorter than the typical decoherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Note that in the previous ESST method via STA [31], an auxiliary counterdiabatic field has been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' It works as a shortcut to adiabaticity for canceling the nonadiabatic coupling and induces perfect population transfer between the states |1⟩L and |3⟩L for the left- handed chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Simultaneously, it also acts oppositely for strengthening the nonadiabatic coupling for the right-handed chiral molecules and the population transfer between the states |1⟩R and |3⟩R is canceled com- pletely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Therefore, under such an ESST process, the left- handed chiral molecule begins with |1⟩L and terminates at −|3⟩L, following a STA path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' But the right-handed chiral molecule is subject to a free evolution, instead of following the STA path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By contrast, in our ESST method via STA, the eigenstates of invariants for the two enantiomers define their corresponding enantio-selective STA paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus, our ESST can be achieved with trans- ferring the two enantiomers from their ground states to different-energy final states along their enantio-selective STA paths simultaneously, by choosing appropriate in- tensities of the three electromagnetic fields (that is, the Rabi frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' CONCLUSION In conclusion, we have proposed the fast ESST method of chiral molecules via the STA technique with invariant- based inverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Based on the cyclic three-level systems, the ESST of chiral molecules can be achieved through enantio-selective STA paths: for the left- and right-handed chiral molecules prepared initially in their ground states, they will evolve (approximately) finally to the different-energy states almost along the eigenstates of the invariants within a short operation time simulta- neously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Hence, our fast ESST method via STA with invariant-based inverse engineering has promising appli- cations in discriminating molecular chirality and control- ling the dynamics of chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the Natural Science Foun- dation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 12074030, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 12274107, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' U1930402), National Science Foundation for Young Scientists of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 12105011), and Bei- jing Institute of Technology Research Fund Program for Young Scholars.' metadata={'source': 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@@ -0,0 +1,2321 @@ +MI-HET-793 +Brane wrapping, AKSZ sigma models, and QP +manifolds +Alex S. Arvanitakisa and David Tennysonb +aTheoretische Natuurkunde, Vrije Universiteit Brussel, and the International Solvay Institutes, Plein- +laan 2, B-1050 Brussels, Belgium +bMitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, +TX, 77843, USA +E-mail: alex.s.arvanitakis@vub.be, dtennyson@tamu.edu +Abstract: We introduce a technique to realise brane wrapping and double dimensional +reduction in the context of AKSZ topological sigma models and also in their target spaces, +which are symplectic Ln-algebroids (i.e. QP-manifolds). Our procedure involves a novel +coisotropic reduction combined with an AKSZ transgression that realises degree-shifting; the +reduced QP-manifold depends on topological data of the ‘wrapped’ cycle. We check our +procedure against the known rules for fluxes under wrapping in the context of M-theory/type +IIA duality, and we also find a new relation between Courant algebroids and Poisson manifolds. +arXiv:2301.02670v1 [hep-th] 6 Jan 2023 + +Contents +1 +Introduction +1 +2 +Wrapping QP manifolds +5 +3 +Example – dim X = 0 +12 +4 +Examples – dim Y = 0 +13 +4.1 +n-brane → (n − 1)-brane +13 +4.2 +From Courant to Poisson +16 +5 +Examples – X = Y +20 +5.1 +M2 on S1 +20 +5.2 +M5 on S1 +23 +5.3 +M5 on X4 +24 +6 +AKSZ sigma models and brane wrapping +26 +6.1 +AKSZ 3-brane to membrane example +28 +7 +Conclusions +29 +A Notation +30 +B QP manifolds +32 +C Coisotropic reduction of graded Poisson algebras +36 +1 +Introduction +In a series of recent papers [1–3] we have been establishing a correspondence between (BPS) +p-branes in String/M-theory on one hand and symplectic Lp+1-algebroids on the other +hand. The latter can be thought of as appropriate generalisations of the (exact) Courant +algebroid that encodes the generalised geometry of type II backgounds with Neveu-Schwarz +3-form H (but without Ramond-Ramond fluxes); a Courant algebroid is precisely a symplectic +Ln-algebroid of degree n = 2 [4]. In this correspondence the exact Courant algebroid (which +is classified by the de Rham cohomology class of H [5]) is associated to the fundamental +string (which couples to H electrically via a Wess-Zumino term). For other branes, such +– 1 – + +as the M2 and M5 branes in M-theory, or even/odd D-branes in type II string theory, the +corresponding algebroids are roughly speaking the ones that are classified by whichever fluxes +couple electrically to the brane in question; such algebroids can be thought of as generalisations +of Courant algebroids to e.g. M-theory scenarios, see [6] for this point of view. +In general, the correspondence is between a “physical” p-brane (an F1, M2, M5. . . ), a +symplectic Ln-algebroid for n = p+1, and a topological AKSZ brane sigma model of dimension +p + 2. This can be schematically summarised in the following diagram: +symplectic Ln-algebroid +topological n-brane +physical (n − 1)-brane +AKSZ +brane phase space +boundary condition +(1.1) +Less tersely, the symplectic Ln-algebroid — that is classified by a certain collection of fluxes +— determines a topological n-brane sigma model via the AKSZ construction [7]. When the +n-brane has an (n−1)-brane boundary, an inflow-type argument with an appropriate boundary +condition produces the WZ term that couples those same fluxes to the (n − 1)-brane [1, 3]1. +The algebroid also determines the corresponding (n − 1)-brane more directly via the brane +phase space construction that yields the Poisson algebra of brane currents on phase space [2] +(i.e. in the hamiltonian formulation of brane dynamics). +This correspondence between branes and algebroids motivates the question: given that +the String/M-theory duality web acts on the branes, how is the duality web realised on +the algebroid side? Heretofore this was only known for dualities that preserve worldvolume +dimension; see [9] for T-duality, and [3] for M-theory/type IIA duality along a transverse +M-theory circle. An example of the latter is the emergence of a D2 brane given an M2 brane +that does not wrap the M-theory circle, whose algebroid avatar is symplectic reduction modulo +the U(1) action. +In this paper we provide an algebroid realisation for the brane wrapping operation. In +the string theory picture, this sends a p-brane to the (p − d)-brane found by wrapping the +original brane around a d-dimensional cycle on target space and then shrinking the volume of +the cycle to zero. (Since both the dimensionality of the brane and that of the target space +are reduced in this way, this is also known as double dimensional reduction.) The most basic +example is M-theory/IIA duality, where M2 branes wrapped around the compactified 11th +dimension give rise to fundamental strings in 10 dimensions [10]. This already poses a puzzle: +the corresponding algebroids are of degree n = p + 1 = 3 (for the M2 brane) and n = 2 (for +the F1); what is the mathematical operation that accounts for this degree shift? +The mystery is resolved in the supergeometric formulation of symplectic Ln-algebroids, +1A slightly different boundary condition for the AKSZ sigma model can produce the entire (n − 1)-brane +lagrangian, including kinetic terms. This was done for the fundamental string by ˇSevera [8]. The other cases +have not yet been considered in the literature. +– 2 – + +defined by the data of a QP manifold (M, ω, Q) where M is a non-negatively graded manifold, +ω a symplectic form of degree n, and Q a nilpotent vector field of degree 1, hamiltonian for ω. +Given a compact manifold X of dimension d — to be identified with the cycle to be ‘wrapped’ +— the odd tangent bundle X ≡ T[1]X possesses an integration measure +� +X : C∞(X) → R of +degree −d, namely the integral of differential forms. Then the mapping space +MX ≡ maps(X → M) +(1.2) +possesses a P-structure of degree (n − d), provided by the AKSZ construction. This is the +correct degree shift; however, this manifold is infinite dimensional, and its structure sheaf is +not non-negatively graded, so it cannot be the sought-after symplectic Ln−d-algebroid. +A ‘brane wrapping’ for QP manifolds. +We introduce a coisotropic reduction of the space MX to a finite-dimensional QP-manifold +that resolves both issues. This resolution is heavily motivated by the intuitive string-theoretic +picture of brane wrapping. We deal with the case where the body of M is a product N × X, +seen as a trivial bundle with fibre X, and we select a map N �→ maps(X → N × X), as in +the figure +maps(X → N × X) +N +maps(X → N × X) +N +maps(X → N × X) +N +Figure 1. The wrapping map specification, for N = R, X = S1. +The idea is that each point n ∈ N is mapped to the cycle of N × X that shrinks to zero size in +the double dimensional reduction procedure. Since maps(X → N × X) is disconnected, with +connected components corresponding to different winding sectors (as they would be called in +physics), the choice of map N �→ maps(X → N × X) includes a choice of winding. On the +string theory side, double dimensional reduction indeed depends on winding: for instance, an +M2 brane wound w times around the M-theory circle yields a fundamental string coupled to +the H-flux wH. Since the algebroids corresponding to these branes via the diagram (1.1) are +defined by the same fluxes, we expect winding dependence in the obtained algebroid, and we +will indeed find it. +In more detail: we start with the data of an NQP — “N” for non-negatively graded — +manifold M with body M and a ‘source’ Q manifold X = T[1]X as above, along with a +– 3 – + +wrapping map w : X → M that defines a degree-zero submanifold N �→ maps(X → M). We +then produce a finite-dimensional, non-negatively graded QP manifold W, whose P-structure +has degree n − d; we will call W the wrapped algebroid, and we will call our procedure +(brane) wrapping. The wrapping of QP manifolds/symplectic Ln algebroids is then a +reduction of MX with respect to a coisotropic submanifold C which may be thought of as the +lift of N �→ maps(X → M) to a graded submanifold of maps(X → M) = MX . The output +QP manifold W depends on the choice of wrapping map w only up to homotopy. +In fact we were able to generalise beyond the case M = N ×X (that was pictorially outlined +above) to the case M = N × Y , with Y and X not necessarily of the same dimension, even; +then the wrapping is a map w : X → N ×Y , and d = dim X controls the degree/dimensionality +shifts as before. This generalisation allows us to accommodate at least one example which +might be of interest outside of string theory, namely the wrapping of a Courant algebroid into +a Poisson manifold discussed in Section 4.2, which has dim Y = 0. When dim X = n + 1 in +addition to dim Y = 0 (so that MX has a degree −1 P structure) our wrapping procedure +agrees with that of [11]. Our approach gives a complementary perspective to the Losev-trick +based ‘wrapping’-style reductions of [12, 13], and to that of [14, 15]. +Brane wrapping and AKSZ sigma models. +Our ‘brane wrapping’ reduction — from a QP manifold M to a QP manifold W — also +induces a reduction of the corresponding AKSZ topological field theories. Essentially, the two +reductions commute, as in the schematic diagam +M +W +MX×S +WS +wrapping +AKSZ +AKSZ +(1.3) +Here MX×S and WS are P-manifolds of degree −1 created by the AKSZ construction for +S of appropriate dimension. The dotted arrow corresponds to a reduction of MX×S with +respect to the coisotropic submanifold CS ≡ maps(S, C), for C the coisotropic submanifold +that appears in the ‘wrapping’ reduction M → W. This ‘dotted’ reduction always exists and +is compatible with the AKSZ/BV master actions if the wrapping reduction does. +We provide the argument for the reduction of AKSZ sigma models in section 6, along with +an example: the reduction of a topological 3-brane sigma model (corresponding to the M2 +brane symplectic L3-algebroid) to a Courant sigma model (corresponding to the fundamental +string symplectic L2-algebroid). This provides an important consistency check: if we were to +derive the corresponding physical brane sigma models, e.g. by introducing boundaries and +using an inflow-type argument as in [1, 8], we would find that the electric Wess-Zumino flux +coupling has the correct winding dependence. +– 4 – + +Structure of the paper. +In section 2, we describe the general procedure for wrapping QP manifolds. We provide the +conditions required of the QP structure on M and define the coisotropic ideal I ⊂ C∞(MX ) +(that defines the coisotropic submanifold C) in general. We show that it is well-defined and +perform the reduction. The next three sections provide a multitude for examples. (If the +reader finds the notation of section 2 too terse, they may find it useful to first work their way +through the examples before coming back to the general procedure.) Section 3 covers the +case where dim X = 0. In this case, we do not get any wrapping and our reduction is very +similar to conventional dimensional reduction [16]. In section 4 we consider examples where +dim X ̸= 0, but the wrapping map w is trivial in homotopy. These provide examples which +are simple but still present some of the main features of the reduction. Among these is the +reduction of a Courant algebroid to a Poisson manifold given in section 4.2. In section 5, we +consider examples relevant for physics and wrap string/M-theory branes on various manifolds. +In section 6 we show how our procedure naturally lifts to a reduction of the AKSZ theory +from MX×S to WS. Section 7 is left for comments and outlook. The appendices cover our +notation (appendix A), some key properties and conventions of QP manifolds (appendix B), +and a review of coisotropic reduction in the graded context (appendix C). +2 +Wrapping QP manifolds +We will describe a process of creating new QP manifolds from old, which effectively generalises +the notion of dimensional reduction, that we describe as ‘wrapping’ QP manifolds. The +nomenclature arises due to the consistency of this process with the AKSZ construction [7] - +that is, one can reduce the AKSZ theory from the original QP manifold to that of the new +manifold. Solutions of this reduced AKSZ theory will look like branes wrapping cycles of the +target space. We will describe the relation to AKSZ sigma models in a later section and will +describe the wrapping procedure here. +We start from the following ingredients. +• An NQP manifold M = N × Y of degree n ≥ 2 where +Y = T ∗[n]T[1]Y +(2.1) +and N is otherwise generic, with underlying commutative manifold2 N. The underlying +commutative manifold for M is M = N × Y , a direct product manifold. The symplectic +form will be written ωM = dϑM, where ϑM is the canonical symplectic potential. The +induced Poisson Bracket on M will be written (·, ·)M. +2By ‘underlying commutative manifold’ we mean the commutative manifold M whose structure sheaf is +the sheaf of degree 0 functions on M, i.e. C∞(M) = C∞ +0 (M). This is well defined since we are working on +non-commutative manifolds with a non-negative grading. We will also refer to this as the manifold in degree 0. +– 5 – + +• The Q-structure of M should be a lift of the de Rham differential of Y , seen as the +vector field dY ≡ ξm∂/∂ym on T[1]Y , with respect to the bundle projection p that +is the composition N × Y +πY +−−→ Y +πT [1]Y +−−−−→ T[1]Y . Explicitly this lift condition means +QMp⋆ = p⋆dY , which partially determines the form of the hamiltonian ΘM in local +coordinates: +ΘM = −ξmqm + +n+1 +� +k=0 +1 +k!θm1...mk(z, y)ξm1...ξmk +(2.2) +where q are the degree n conjugate momenta to y on T ⋆[n]T[1]Y and z are generic +homogeneous coordinates on N. The θk = θk(z, y)ξk can be viewed as (C∞(N)-valued) +differential forms on Y and we demand that they must be dY -closed differential forms. +• A Q manifold X = (T[1]X, d) where X is compact, without boundary, and has dimension +d < n. d is the de Rham differential. +• A choice of ‘wrapping map’ w : X → Y , defined up to homotopy. +We aim to produce a new NQP manifold W from M, X, which describes a brane where +X has been wrapped over Y and both cycles have been shrunk. The resulting QP manifold +should therefore have degree n − d and underlying commutative manifold N. There is a +natural choice of manifold of degree n − d given by the mapping space MX := maps(X → M). +However, this manifold is infinite dimensional. We will see that we can define a coisotropic +reduction of MX that produces a finite dimensional NQP manifold which only depends on +the topology of X and the homotopy class of w. +Properties of the mapping space +The infinite dimensional space MX consists of maps f which are defined by their pullback +action on the coordinates on M. Using generic homogeneous coordinates ZA for M and +coordinates (σα, dσα) for X adapted to d (d(σα) = dσα, ddσα = 0) we have +f∗ZA = ZA(σ, dσ) = ZA +0 (σ) + ZA +1 α(σ)dσα + ... + 1 +d!ZA +d α1...αd(σ)dσα1...dσαd +(2.3) +Defining the components Zk is equivalent to defining the map f. To interpret the Zk we +consider a change of coordinates on M given by ˜ZA = ˜ZA(Z) and note that +f∗ ˜ZA(Z) = ˜ZA(f∗Z) += ˜ZA(Z0) + ZB +1 αdσα ∂ ˜ZA +∂ZB (Z0) ++ 1 +2dσαdσβ +� +ZB +2 αβ +∂ ˜ZA +∂ZB (Z0) + ZB +1 αZC +1 β +∂2 ˜ZA +∂ZB∂ZC (Z0) +� ++ ... +(2.4) +– 6 – + +Therefore, in spite of the index structure, these in general are not vector-bundle-valued +differential forms, with the exception of Z1 which is an f⋆ +0 TM-valued 1-form for the map +f0 = f ◦ s0, where s0 : X → X is the zero section of X = T[1]X. Of the other components, ZA +0 +defines the map f0 : X → M, while the ZA +k for k > 1 transform “affinely” whenever ZA +k′ ̸= 0 +for any 0 < k′ < k.3 Since we may not set ZA +k = 0 consistently in general, this introduces a +subtlety for our reduction procedure which we will discuss later in this section. +The QP structure on the mapping space is induced by that on M through transgression. +The symplectic structure is given by +ωMX = +� +X +1 +2δZA(ωM)AB δZB = +� +k +� +X +1 +2δZA +k (ωM)ABδZB +d−k +(2.5) +which induces a Poisson bracket [·, ·] on MX . This Poisson bracket can be conveniently +expressed in terms of ‘test functions’ as in [2]. Given arbitrary functions ϵ, η on X — which +correspond to differential forms on X since X = T[1]X — they write +�� +X +ZAϵ , +� +X +ZBη +� += (−1)(B+n)ϵ+d +� +X +(ZA, ZB)M ϵη +(2.6) +where in the exponent we have used the shorthand B, ϵ for the degrees of the respective +functions. From (2.5) and (2.6) we can see that if ZA is dual to ZB on M, then ZA +k will +be dual to ZB +d−k on MX . Furthermore, if we are working in Darboux coordinates, so that +components of ωM are constant, then by performing a Hodge decomposition +Ωk(X) = Hk ⊕ dΩk−1 ⊕ d†Ωk+1 +(2.7) +with respect to some arbitrary metric, exact forms ZA +k will be dual to co-exact ZB +d−k and +harmonic forms will be dual to harmonic forms. For convenience we introduce orthogonal +projectors +PH, +Pex , +Pco +(2.8) +onto harmonic, exact, and co-exact forms respectively. +The Q-structure D on MX is defined as the hamiltonian vector field +D = d + QM , +D = [ΘMX , ·] , +(2.9) +where the hamiltonian is +ΘMX = (−1)d +� +X +ΘM + (−1)d+n+1 +� +X +ıdϑM +(2.10) +3Exploiting Batchelor’s theorem to write M as a graded vector bundle only improves this situation in that +some Z0 take values in a vector bundle as well. +– 7 – + +where each term generates the lift of QM and d to MX respectively. Note that implicit in +this formulae is the fact that we have pulled back/transgressed ΘM, ϑM to objects on X; we +have used boldface to highlight this. The signs are such that D = dX + QM. +The coisotrope +We need to perform a coisotropic reduction to obtain a finite dimensional NQP manifold. This +is a generalisation of symplectic reduction for Poisson manifolds which requires a coisotropic +ideal I ⊂ C∞(MX ), i.e. an ideal that satisfies +[I, I] ⊆ I +(2.11) +The description of the quotient manifold is given in two equivalent ways. In one description, we +take the submanifold C ⊂ MX defined by the vanishing of I and quotient by transformations +generated by I. Alternatively, we can describe the structure sheaf of the quotient manifold as +the normaliser N(I) of I, quotiented by I. That is +W = C/[I, ·] +⇔ +C∞(W) = N(I)/I +(2.12) +Such a manifold has a natural Poisson structure induced from that on the mapping space; see +appendix C for a review. Further, provided the ideal is closed with respect to the Q structure, +i.e. DI = [ΘMX , I] ⊆ I, the reduced space has a Q-structure induced from the image of the +Hamiltonian function under the quotient map: +ΘW = Π(ΘMX ) +Π : N(I) → N(I)/I +(2.13) +This closure is precisely the statement that ΘMX ∈ N(I). +We build our ideal I = ⟨IN , IY⟩ in 2 parts, each defining a restriction to some submanifold +of MX = N X × YX . +This factorisation is convenient because Y may be thought of as +‘longitudinal’ to the cycle to be wrapped, while N is ‘transverse’. +On YX , we would like the maps in degree 0 to restrict to the fixed wrapping map +w : X → Y . This restriction is naturally given by the zero locus of the ideal generated by +y − w and its closure under D. Using (2.2) and (2.10), we find +IY = ⟨y − w, ξ + dw⟩ +(2.14) +This is clearly coisotropic in the coordinates on Y. The angled brackets ⟨· · · ⟩ will always +denote the ideal generated by · · · . +On N X , we follow [11] and take the coisotropic submanifold to consist — in the first +instance — of closed maps under the transgressed differential d on N X . In degree zero we +realise this via a choice of degree preserving embedding N �→ N X . By degree counting this +is a map of (ordinary) manifolds N �→ NX, and we choose this to be the map sending each +– 8 – + +n ∈ N to the constant map X → {n} (which is d-closed). Beyond degree zero, we simply set +the coisotropic part of each coordinate in the superfield expansion (2.3) to zero (using the +Hodge decomposition). Therefore we define IN such that +IN ⊃ +� +PcozA +k +� +(2.15) +for all values of k in the expansion (2.3), where zA is a generic coordinate on N. If we +consider the vanishing locus of IN and IY simultaneously, we see that we are restricted to +x0 = const, y0 = w. This gives an embedding N �→ MX . Similarly, a choice of degree +preserving embedding N �→ MX defines our ideal in degree 0. +This alone is not enough as we would like the reduced manifold W to be an N-manifold, +i.e. a graded manifold with non-negative coordinates, such that in degree zero the structure +sheaf is that of an ordinary manifold.4 To remove these, we include harmonic generators of +the maps zA +k for maps such that deg zA − k ≤ 0. The exception to this is the maps x0 for +which we do not include the harmonic (i.e. constant map) representatives. We therefore have +IN = +�� +� +� +PcozA +k , PHzA +k′ | ∀k′ ≥ deg zA +if +deg zA > 0 +PcozA +k , PHzA +k′ | ∀k′ > 0 +if +deg zA = 0 +� +(2.16) +To see that this is coisotropic, we use (2.6) and the surrounding discussion to note that +the co-exact generators are dual to exact generators. Hence, these terms are coisotropic with +respect to all of IN . The harmonic generators could be dual to some other harmonic generator +in IN . However, since we only include harmonic zA +k for 0 ≥ deg zA − k, the dual coordinate +zB +k′ on MX has +deg zB − k′ = n − deg zA − (d − k) += (n − d) − (deg zA − k) +> 0 +(2.17) +so is not included in IN . The total ideal I = ⟨IN , IY⟩ is coisotropic, as required. +Metric and coordinate independence +The construction of the ideal I appears to rely on a choice of metric on X but we claim that +the resulting reduced manifold depends on only the topological data of X. This can be best +seen from the vanishing locus C ⊂ MX . This is given by maps which are either d-closed (for +deg zA = 0 and some values of k, k′) or ones which are also d-exact (in all other cases), as +specified in (2.16). The metric only appears in the specification of the vanishing ideal that +4There are issues not just with negative-graded but also with degree zero ‘formal’ coordinates; see e.g. [17, +section 2]. +– 9 – + +represents C but C does not in itself depend on metric data. (This apparent metric dependence +thus may perhaps be seen as due to ‘gauge-fixing’.) +Our construction also appears to depend on a choice of coordinates on N. To see that this +is well defined, we will show that the submanifold C is invariant under a change of coordinates +˜zA = ˜zA(z). Using formula (2.4), we can write the k-form component of the transgressed ˜z as +˜zA +k ∼ +� +j +CAA1...Aj(z0)zA1 +k1 ...zAj +kj +(2.18) +such that k1 + ... + kj = k and deg zA1 + ... + deg zAj ≤ deg ˜zA +k ; here we emphasise that the +last inequality holds true because M — and thus N — was assumed to be an N-manifold (its +structure sheaf is non-negatively graded). Restricting to C, the coordinates zAi +ki are all closed +under d and hence so is ˜zA +k . Further, if deg ˜zA − k ≤ 0, then at least one of deg zAi − ki ≤ 0. +This means that zAi +ki is exact. A product of closed and exact forms is exact and hence so is ˜zA +k +as required. +Closure under D +The final condition to check is that the ideal I is closed under the Q-structure D = [ΘMX , ·] +on MX . We have already checked that IY is D-closed and so we need only check how D acts +on the generating coordinates of IN . We can use formula (2.6) with Pcoϵ = 0. This choice of +epsilon selects out the harmonic and co-exact generators zA +k , respectively. We have +� +ΘMX , +� +X +zAϵ +� += (−1)d +�� +X +ΘM, +� +X +zAϵ +� ++ (−1)d+n+1 +�� +X +ıdϑM, zAϵ +� += +� +X +(ΘM, zA)M ϵ + +� +X +dzAϵ +(2.19) +The second term vanishes when ϵ is closed. We therefore need to only consider the first +term. We can see whether this term is contained within I by transgressing the function +(ΘM, zA)M to the mapping space and evaluating it over C. If the integral vanishes when +integrated against all closed ϵ then the ideal is closed under D. +Using the form of the +Hamiltonian function we find +(ΘM, zA)M ∝ (ωM)AB � +k +∂θk(z, y) +∂zB +ξk +(2.20) +Transgressing this to the mapping space and restricting to C, we replace y → w, ξ → dw, +z → z for z closed (or exact). Integrating this against ϵ we get +� +ΘMX , +� +X +zAϵ +����� +C +∝ +� +X +� +(ωM)ABw∗ � +k +∂θk +∂zB (z) +� +ϵ +(2.21) +– 10 – + +We require this to vanish for the above ϵ. When ϵ is exact, this indeed vanishes if we +impose dY θk = 0. If ϵ is harmonic, however, we find constraints on the coefficients θk that we +address case-by-case, in general. +The reduction, metric independence, and homotopy invariance +Given the coisotropic reduction I, we consider the reduction given in (2.12). We will consider +the structure sheaf construction of the reduced manifold. The normaliser N(I) of the ideal is +generated by +N(I) ∼ +� +PHx0, PHzA +k , I | 0 < deg zA − k ≤ n − d +� +(2.22) +We can expand the PHzA +k = zA,a +k +ea in some basis {ea} of Hk, so the zA,a +k +are constant +parameters of degree deg zA − k. In the case that the respective cohomology group is 1 +dimensional (e.g. for H0, Hd) we will omit the a index and simply identify e.g. zA +d = zA +d volX. +We see that the structure sheaf C∞(W) = N(I)/I is given therefore generated by +C∞(W) = N(I)/I ∼ {x0, zA,a +k +| 0 < deg zA − k ≤ n − d} +(2.23) +That is, the structure sheaf is given by all smooth functions in the zA,a +k +(and x0). +The Hamiltonian function on W is given by the projection Π : N(I) → N(I)/I of ΘMX . +We will confirm in the examples that the final result is given by +ΘW = Π(ΘMX ) = +� +X +� +k +(−1)kw∗θk(z) +(2.24) +where the z are now the harmonic representatives of the cohomology groups on X. Expanding +the harmonic zA +k in terms of the constant coordinates zA,a +k +, we can perform the integral over +X with the convention that the volume form is on the right of the integrand, so we pull out +constants from the left. Once this is done the final result will no longer be an integral but will +be a function in the zA,a +k +which will involve, in general, a sum over cohomology groups, which +will be discrete in all cases (we only consider wrapping over compact cycles). +Formula (2.24) seems to depend on some metric to choose the harmonic representatives for +the z. However, under a change of metric, the harmonic representatives change by a d-exact +term, and since we have assumed that the forms θk are closed, this shift will not change the +integral. Furthermore, since the forms θk are closed, the evaluation of the integral only depends +on the homotopy class of w : X → Y . Therefore the construction is metric-independent and +homotopy invariant. +Now that we have defined the reduction in complete generality, we will see many examples +of how this works in practice. There are three interesting cases to consider. +1. dim X = 0 – The process effectively shrinks Y to a point. +– 11 – + +2. dim Y = 0 – We produce a QP manifold with the same underlying commutative manifold +but with a different degree. +3. X = Y – We produce a QP manifold which corresponds to a brane wrapping the internal +manifold. +3 +Example – dim X = 0 +We consider first a simple example to show that in the simple case that dim X = 0, our +procedure effectively reduces to dimensional reduction on Y . Consider the ingredients +M = T ∗[n]T[1](N × Y ) +X = pt +(3.1) +Taking N = T ∗[n]T[1]N, Y = T ∗[n]T[1]Y and X = T[1]X = pt, we introduce the Darboux +coordinates +N +Y +coord +xµ +ψµ +χµ +pµ +deg +0 +1 +n − 1 +n +coord +ym +ξm +φm +qm +deg +0 +1 +n − 1 +n +(3.2) +We will take the QP structure to be given by the symplectic form and Hamiltonian function +ωM = dp dx + dq dy − dψ dχ − dξ dφ +(3.3) +ΘM = −ψp − ξq + 1 +n!Fnψn + +1 +(n−1)!Fn−1ψn−1ξ + ... + +1 +d!(n−d)!Fn−dψn−dξd +(3.4) +We have suppressed all indices but they should be read as being contracted in the natural +way. The coefficients Fk can be thought of as elements of Ωk(N) × Ωn−k(Y ). These should be +closed under the differential dY on Y . So for example, Fnψn = Fn(x, y)µ1...µnψµ1...ψµn should +be viewed as a differential n-form on N, but a constant function on Y . In the ansatz above, +we have assumed a trivial connection on the bundle. We can easily reintroduce it by making +the replacement ξ → A = ξ + Aψ, where A is the connection, however it won’t change our +final result so we omit it for simplicity. +The first step in the reduction process is to transgress the QP structure to MX . But +since X is 0 dimensional, we have MX ≃ M. Next, we need to choose a wrapping map +w : X = pt → Y , or equivalently, a (degree preserving) embedding N �→ MX ≃ M. This is +equivalent to choosing some point ˆy ∈ Y and defining the embedding N → (N, ˆy) ⊂ M. This +is described by the ideal +I0 = ⟨ym − ˆym⟩ +(3.5) +We then want to form the closure of this ideal with respect to differential Q on M. We get +IY = ⟨I0, QI0⟩ = ⟨ym − ˆym, ξm⟩ +(3.6) +– 12 – + +It is easy to check from (3.3) that this is indeed coisotropic with respect to the Poisson bracket +on M. In principal, we also need to restrict the maps into N to those that are closed/exact +with respect to d on X. However, since dim X = 0, this is a trivial constraint and so we just +have I = IY. +To perform the coisotropic reduction, we need to go to first find the normaliser N(I) of +I, which can easily be verified to be generated by the coordinates +N(I) ∼ {xµ, ψµ, χµ, pµ, ym − ˆym, ξm} +(3.7) +The structure sheaf of the new QP manifold W is then defined to be the quotient of this by +the ideal I. That is C∞(W) = N(I)/I, which is generated by +N(I)/I ∼ {xµ, ψµ, χµ, pµ} +⇒ +W = T ∗[n]T[1]N +(3.8) +Note that, by construction ΘM ∈ N(I) and so we can find the new Hamiltonian function +through the natural projection Π : N(I) → N(I)/I, which gives +ΘW = Π(ΘM) = −ψp + 1 +n!Fn(x, ˆy)ψn +(3.9) +and the final symplectic form is +ωW = dp dx − dψ dχ +(3.10) +We see that this procedure has produced a new QP manifold with the same degree but +with underlying commutative manifold N. We see that we have effectively collapsed Y to +the point ˆy. In the case where Y is a Lie group, we find the same result as in symplectic +reduction modulo T[1]Y [9]. If we were to choose a different wrapping map w′ : X �→ ˆy′ +that is homotopic to w : X �→ ˆy, then we end up with the same graded manifold where the +Q-structure is evaluated for Fn(x, ˆy′). However, the condition that the Fn is closed on Y says +that it is constant, and hence the Q-structures are the same. This demonstrates the homotopy +invariance of our construction. +4 +Examples – dim Y = 0 +4.1 +n-brane → (n − 1)-brane +Let’s now consider the same example as above, but instead of having dim X = 0, we will +take the dimension of the fibre dim Y = 0 and take X to be non-trivial. We will take the +ingredients +M = T ∗[n]T[1]M +X = S1 +(4.1) +– 13 – + +We will use the homogeneous coordinates coordinates +M +coord +xµ +ψµ +χµ +pµ +deg +0 +1 +n − 1 +n +(4.2) +and use the coordinates σ, dσ on X = T[1]S1. The Hamiltonian function and symplectic form +are given by +ωM = dp dx − dψ dχ +(4.3) +ΘM = −ψp + 1 +n!Fnψn +(4.4) +Since Y = pt in this example, we do not need to impose any constraints on the coefficients Fn. +We need to transgress this structure to the mapping space MX . This is now an infinite +dimensional graded manifold whose points f ∈ MX can be described by their pullback action +on coordinates on M. That is, we have +f∗ZA = ZA(σ, dσ) = ZA +0 (σ) + ZA +1 (σ)dσ . +(4.5) +The transgressed Hamiltonian function is given by +ΘMX = (−1)1 +� +X +ΘM + (−1)n−1+1 +� +X +ıdϑM += − +� +T[1]S1 −ψp + 1 +n!Fn(x)ψn + (−1)n +� +T[1]S1 pdx − 1 +n(ψdχ + (n − 1)χdψ) +(4.6) +The bold-faced letters in the expression correspond to functions pulled back to functions on X +as in (4.5). The Berezin integral over T[1]S1 selects the maximal degree component of the +integrand (i.e. the 1-form components) and integrates it over S1. Our convention is that we +normalise with an overall factor of vol(S1), and so for the flat metric on S1 we have +� +T[1]S1 ... = 1 +2π +� +S1(...)1 +(4.7) +The next step is to define the coisotropic ideal I = ⟨IN , IY⟩. Since Y is trivial, so is +the ideal IY and hence we need only determine IN . Following section 2, we first start by +restricting to all closed maps. That is, we take +IN ⊃ ⟨Pcoxk, Pcoψk, Pcoχk, Pcopk⟩ +(4.8) +To define this ideal we choose some arbitrary metric on S1, and for simplicity we can take the +flat metric. We then also add the harmonic representatives for ZA +k such that deg ZA − k ≤ 0 +– 14 – + +(except for x0). This gives +IN = ⟨Pcox0, Pcoψ0, Pcoχ0, Pcop0, PHx1, PHψ1⟩ +(4.9) +Again, in degree 0, the vanishing locus of this ideal restricts us to maps x0 = const and hence +defines a natural embedding M �→ MX . It is a quick check using (2.6) that this ideal is +coisotropic. Indeed, the Poisson bracket of the co-exact generators with any other generators +will vanish, as they are dual to exact maps. The harmonic x1, ψ1 representatives are dual to +p0, χ0 ∈ H0 respectively and these do not appear in the generating set of IN . +We will also verify that this ideal is closed with respect to the Q-structure (4.6). Using the +test function form of the Poisson bracket (2.6), we can calculate D acting on the generators +by calculating +� +ΘMX , +� +T[1]S1 ZAϵ +� += − +�� +T[1]S1 ΘM, +� +T[1]S1 ZAϵ +� ++ (−1)n +�� +T[1]S1 ıdϑM, +� +T[1]S1 ZAϵ +� +(4.10) +where ϵ is a function on N that is closed under d. Taking ϵ ∈ Hk selects the harmonic +representative Z1−k ∈ H1−k, while taking ϵ to be exact selects the co-exact representative of +ZA +0 . The second term gives us +�� +T[1]S1 ıdϑM, +� +T[1]S1 ZAϵ +� +∝ +� +T[1]S1 dZ ϵ = 1 +2π +� +S1 dZA +0 ϵ0 +(4.11) +Taking ϵ to be closed tells us that ϵ0 is constant. The integrand on the right hand side is +therefore exact and so the integral vanishes. The Poisson bracket is then determined by the +first term alone which is proportional to +� +ΘMX , +� +T[1]S1 ZAϵ +� +∝ +� +T[1]S1(ΘM, ZA)M ϵ +(4.12) +where the function (ΘM, ZA) is transgressed to the mapping space. We can use these results +to confirm that ΘMX lies always in IN as outlined in Section 2. The only non-trivial checks +are for the harmonic generators x1, ψ1, for which we take ϵ = ϵ0 to be constant. We have +(ΘM, x)M = ψ , +(ΘM, ψ)M = 0 +(4.13) +Transgressing these functions and evaluating on C, we take ψ1 to be exact. Hence, both vanish +under the integral (4.12) when ϵ = ϵ0 is constant. This proves that the ideal is closed under D. +Now that we have our coisotropic ideal, we perform the coisotropic reduction. The +– 15 – + +normaliser of I is generated by all the coordinates that are not dual to those in I. +N(I) ∼ {PHx0, PHψ0, PHχ1, PHp1, I} +(4.14) +The structure sheaf for W is then N(I)/I which is generated by +N(I)/I ∼ {x0, ψ0, χ1, p1} +⇒ +W = T ∗[n − 1]T[1]M +(4.15) +(Note in the expression above we are now working in the coordinates zA,a +k +described in section +2: PHx0 = x0 · 1 and PHp1 = p1vol.) Thus we restrict to harmonic functions for x, ψ — so +they retain their original degrees — while we restrict to harmonic 1-forms for χ, p hence they +have their degrees shifted down by 1. We therefore end up with the manifold T ∗[n − 1]T[1]M. +To find the symplectic form, we use the Poisson brackets (2.6) with the ϵ, η appropriate +harmonic representatives. We find that +ωW = −dp1 dx0 − dψ0 dχ1 +(4.16) +To find the form of the Hamiltonian function we project ΘMX under Π : N(I) → N(I)/I. By +restricting all coordinates to the harmonic representatives on which d = 0, we find Π(ı¯d ¯ϑ) = 0. +The term +1 +n!Fnψn gets projected to +1 +n!Fn(x0)ψn +0 which is a function on S1 and hence vanishes +under the Berezin integral. We find that we are left with5 +ΘW = Π(ΘMX ) = ψ0p1 +(4.17) +Making the change of coordinates p1 → −p1 puts the QP manifold in the canonical form for a +(n − 1)-brane. Interestingly, all flux twisting drops out of the Hamiltonian function in this +case. This is what happens in the zero-wrapping sector of wrapped branes where physically +one ends up with a tensionless brane [18]. These are somewhat pathological and hence the +physical interpretation of such reductions is less clear. We will see that one can get more +interesting reductions if one allows X to wrap some part of M. +4.2 +From Courant to Poisson +Using the formulation set out, we can already find interesting relations between different QP +manifolds. Suppose M is a Poisson manifold with Poisson bivector π. There are two distinct +ways to realise this structure as a QP structure. Firstly, we can take the straight cotangent +lift of π to obtain the following QP manifold +W = T ∗[1]M +coord +˜x +˜p +deg +0 +1 +ωW = d˜p d˜x +ΘW = 1 +2π˜p2 +(4.18) +5Our conventions are that we integrate with the volume form on the right of the integrand, and so we pull +constants out from the left. This gives the overall sign. +– 16 – + +A quick calculation shows that (ΘW, ΘW) = 0 if and only if π is Poisson. +Alternatively, we can consider the Lie algebroid structure on T ∗M whose anchor map is +given by the bivector π : T ∗M → TM and whose bracket is given by +[α, β] = Lπ(α)β − ıπ(β)dα +(4.19) +This Lie algebroid can be lifted to a Dirac structure L = (1 + π)T ∗M within the Courant +algebroid TM ⊕ T ∗M. Such a structure can be described by a QP manifold via +M = T ∗[2]T[1]M +coord +x +ψ +χ +p +deg +0 +1 +1 +2 +ωM = dp dx − dψ dχ +ΘM = −πpχ + 1 +2∂πψχ2 +(4.20) +Once again (ΘM, ΘM) = 0 if and only if π is Poisson. We have suppressed indices for +convenience. We want to see how, if at all, these constructions are related. +Let us perform a circle reduction of M as above. We transgress the structure to MX +where X = T[1]S1. As before, we define I = IN by first including all co-exact generators +I ⊃ ⟨Pcox0, Pcoψ0, Pcoχ0, Pcop0⟩ +(4.21) +Then we include harmonic representatives to remove coordinates of 0 or negative degree. We +will slightly relax the construction set out in section 2 by allowing some new coordinates of +degree 0.6 We will define +I = ⟨Pcox0, Pcoψ0, Pcoχ0, Pcop0, PHx1, PHχ1⟩ +(4.22) +As before, this ideal is coisotropic. +We will check the closure of this ideal with respect to the Q-structure D on MX . The +transgressed Hamiltonian function is +ΘMX = − +� +T[1]S1 ΘM + +� +T[1]S1 ıdϑM += − +� +T[1]S1 −π(x)pχ + 1 +2∂π(x)ψχ2 + +� +T[1]S1 p dx − 1 +2(ψdχ + χdψ) +(4.23) +We then act with this on +� +ZAϵ for some test function ϵ that must be harmonic, or exact. As +in (4.12), the only non-trivial constraint to check is for the harmonic representatives. We need +6The construction, as set out previously, would still work in this case but we would end up with a trivial +Q-structure. To result in a QP manifold with non-trivial Q-structure, we will need to perform an intermediate +step before removing the additional degree 0 coordinates. +– 17 – + +to check if the following vanishes +� +T[1]S1(ΘM, ZA)M ϵ +(4.24) +whenever the function (ΘM, ZA) is transgressed and evaluated on C, and if ϵ is harmonic. +Since the only harmonic generators of I are x1, χ1, we calculate +(ΘM, x) = π(x)χ , +(ΘM, χµ) = 1 +2∂π(x)χ2 +(4.25) +We transgress these functions to the mapping space and evaluate on the vanishing locus C +of I. Noting that these are functions of x, χ alone, evaluating them on C means that the +zero-form component must be constant functions on X, while the 1-form component must +be an exact form. Integrating these against a constant function ϵ = ϵ0 selects the 1-form +component, which is exact and hence the integral vanishes as required. +The next step is to perform the coisotropic reduction with respect to this ideal. The +normaliser is generated by all coordinates not dual to those in I. +N(I) ∼ {PHx0, PHχ0, PHψ1, PHp1, I} +(4.26) +and so we obtain the structure sheaf C∞( � +M) = N(I)/I which is generated by +N(I)/I ∼ {x0, ψ1, χ0, p1} +⇒ +� +M = T ∗[1]TM +(4.27) +This time, we restrict to harmonic functions for x, χ so they retain their degree, while we +take harmonic 1-forms for ψ, p and hence their degree is shifted down by 1. The resulting +Hamiltonian function is Π(ΘMX ) and the symplectic form is derived from the Poisson brackets +(2.6) with harmonic representatives for ϵ, η. +Θ � +M = −πp1χ0 − 1 +2∂πψ1χ2 +0 +(4.28) +ω � +M = −dp1dx0 + dψ1dχ0 +(4.29) +We performed the change of coordinates p1 → −p1, χ0 → −χ0 to remove minus signs. +We have arrived at a ‘halfway house’ QP manifold � +M. Interestingly, this is the cotangent +lift of the complete lift of the Poisson structure π on M to the tangent bundle (TM, πc) [19]. +That is, given any Poisson structure (M, π) we define a Poisson structure (TM, πc) by +πc = πµν ∂ +∂xµ +0 +∂ +∂ψν +1 ++ 1 +2ψρ +1∂ρπµν +∂ +∂ψµ +1 +∂ +∂ψν +1 +(4.30) +where x0 are coordinates on M and ψ1 are coordinates along the vector bundle fibres. We can +reduce the QP manifold � +M further by following [19]. Given any (torsionless) connection on +– 18 – + +M, we can define a global vector field on TM given by the geodesic spray +s = ψµ +1 +∂ +∂xµ +0 +− ψµ +1 ψν +1Γρ +µν +∂ +∂ψρ +1 +. +(4.31) +This has a cotangent lift to T ∗[1]TM whose hamiltonian is +S = ψ1p1 − Γψ2 +1χ0 +(4.32) +From this, we define a new Hamiltonian function +Θ′ +� +M = − 1 +2(S, Θ � +M) = 1 +2πp2 +1 + ψ1f(x0, ψ1, χ0, p1) +(4.33) +where f is some function of the coordinates whose precise form is not important. All we will +need is that besides the first term, 1 +2πp2 +1, each term is at least linear in the coordinate ψ1. +Consider the ideal generated by the single coordinate I = ⟨ψ1⟩. This ideal is automatically +closed under the Q-structure since +(Θ′ +� +M, ψ1) = ( 1 +2πp2 +1 + ψ1f(x0, ψ1, χ0, p1), ψ1) += (ψ1f(x0, ψ1, χ0, p1), ψ1) +∝ ψ1(f, ψ1) +∈ I +(4.34) +Performing the coisotropic reduction with respect to this ideal we obtain the structure sheaf +N(I)/I ∼ {x0, p1} +⇒ +W = T ∗[1]M +(4.35) +That is, we reproduce the non-commutative manifold W. Further, the symplectic form and +Hamiltonian function are easily shown to be the following: +ωW = dp1 dx0 , +(4.36) +ΘW = 1 +2πp2 +1 , +(4.37) +We see then that we precisely reproduce the QP manifold associated to the cotangent lift of +the Poisson bivector that we described at the beginning of this section. This construction +provides new relations between the Courant sigma model and the Poisson sigma model that +is different from the WZW-Poisson model [20]. There they view the Poisson model arising +at the boundary of a topological WZW-like theory. Instead, our construction is closer to +dimensional reduction and can be viewed as the geometric counterpart of the Courant sigma +model reduction found in [21]. +– 19 – + +5 +Examples – X = Y +We will now generalise the previous two sections to allow for cases where the source manifold +X wraps the target space fibre Y . In particular, we will be interested in the case where +X = Y . We will see that the reduction procedure requires us to choose some self-wrapping +map w : X → X. The examples we choose are physically motivated and fill our understanding +of how brane dualities in M-theory/IIA arise in the QP setting. In particular, when X = S1, +we will see that our procedure produces the known relations from M-theory/type IIA duality. +We will also see that this procedure reproduces other interesting relations between the M5 +brane and the heterotic string [22, 23]. +5.1 +M2 on S1 +Our first example will be wrapping the M2 brane on an S1. This will be very similar to the +n-brane example in section 4.1, except in this case the wrapping will allow for more interesting +Hamiltonian functions to be produced. +We start with the QP manifold M associated to the M2 brane and a source manifold X: +M = T ∗[3]T[1](N × S1) +X = S1 +(5.1) +Writing N = T ∗[3]T[1]N, Y = T ∗[3]T[1]S1, and X = T[1]S1, we will introduce the coordinates +N +Y +coord +xµ +ψµ +χµ +pµ +deg +0 +1 +2 +3 +coord +y +ξ +φ +q +deg +0 +1 +2 +3 +(5.2) +and use coordinates (σ, dσ) on X. The Hamiltonian function and symplectic form are +ωM = dp dx + dq dy − dψ dχ − dξ dφ +(5.3) +ΘM = −ψp − ξq + 1 +4!F4ψ4 + 1 +3!H3ψ3ξ +(5.4) +We require F4 to be dS1-closed; we will also use the fact that H3ξ is dS1-closed (which is +automatic). +We transgress this to the mapping space MX and choose an ideal whose vanishing locus +describes, in degree 0, some embedding ı : N �→ MX . As explained in section 2, this depends +on the choice of some wrapping map w : S1 → S1. In fact, as stated, the final result only +depends on the homotopy class of w and hence we can take, for some w ∈ Z, +w : S1 −→ S1 +σ �−→ wσ +(5.5) +– 20 – + +To restrict to this wrapping sector of MX , we define a coisotropic ideal I = ⟨IN , IY⟩ with +IY = ⟨y − wσ, ξ + w dσ⟩ +(5.6) +As explained in section 2, this is coisotropic and closed under the Q-structure D = [ΘMX , ·]. +The ideal IN restricts all maps into N to closed maps. That is, we take +IN ⊃ ⟨Pcox0, Pcoψ0, Pcoχ0, Pcop0⟩ +(5.7) +For any coordinate zA +k with deg zA − k ≤ 0, we need to further restrict to exact maps by +including the harmonic representative in the ideal (except for x0). Hence, we have +IN = ⟨Pcox0, Pcoψ0, Pcoχ0, Pcop0, PHx1, PHψ1⟩ +(5.8) +The ideal I = ⟨IN , IY⟩ is clearly coisotropic. +We need to check that IN is closed under D = [ΘMX , ·]. The transgressed Hamiltonian is +ΘMX = − +� +X +ΘM − +� +X +ıdϑM += − +� +X +−ψp − ξq + 1 +4!F4(x, y)ψ4 + 1 +3!H3(x, y)ψ3ξ +− +� +X +pdx + qdy − 1 +3(ψdχ + 2χdψ + ξdφ + 2φdξ) +(5.9) +As in the previous cases, the only non-trivial constraint comes from the Poisson bracket +between the first term and the harmonic generators of IN . We calculate +(ΘM, x) = ψ , +(ΘM, ψ) = 0 +(5.10) +and hence we have +� +ΘMX , +� +X +ψϵ +� +∝ +� +X +(ΘM, ψ)M ϵ = 0 +(5.11) +� +ΘMX , +� +X +xϵ +� +∝ +� +X +(ΘM, x)M ϵ = +� +X +ψϵ +(5.12) +Evaluating this on C, we take ψ1 to be exact and so the integral vanishes when integrated +over a constant ϵ = ϵ0. This shows that the Poisson brackets with the harmonic generators +x1, ψ1 vanish when evaluated on C, i.e. they are in I. +To perform the coisotropic reduction we find the normaliser is generated by +N(I) ∼ {PHx0, PHψ0, PHχ1, PHp1, I} +(5.13) +– 21 – + +and hence the structure sheaf is generated by by +C∞(W) = N(I)/I ∼ {x0, ψ0, χ1, p1} +⇒ +W = T ∗[2]T[1]N +(5.14) +where the coordinates represent harmonic maps. The symplectic form can be derived from +the Poisson brackets on MX , as in section 4.1, and we find +ωW = −dp1 dx0 − dψ0 dχ1 +(5.15) +The Hamiltonian function is given by +ΘW = Π(ΘMX ) = Π +� +− +� +X +ΘM − +� +X +ıdϑM +� +(5.16) +The second term vanishes when evaluated on harmonic maps where d annihilates the maps, +except for the term qdy. We also get a piece −ξq from the first term. We find +Π +�� +X +ξq − qdy +� += +� +X +−dy q − qdy = 0 +(5.17) +where we pick up a minus sign from commuting dy (degree 1) through q (degree 3). This +verifies the statement made in section 2 about this cancellation. We then have +ΘW = Π +�� +T[1]S1 ψp − 1 +4!F4(x, y)ψ4 − 1 +3!H3(x, y)ψ3ξ +� += +� +X +ψ0p1 + − 1 +4!F4(x0, wσ)ψ4 +0 + 1 +3!H3(x0, wσ)ψ3 +0w dσ += 1 +2π +� +X +� +ψ0p1 + w +3!H3(x0, wσ)ψ3 +0 +� +dσ += ψ0p1 + w +3! ˜H3ψ3 +0 +(5.18) +where ˜H3 is the average of H3 over the fibre. +Under the change of coordinates p1 → −p1, we see that we recover the QP manifold +associated to the F1 string with w units of ˜H3 flux, as we would expect from our intuition of +M-theory/IIA duality. Note that in the case that w = 0, the physical interpretation seems to +break down - we find a string which doesn’t couple to the NS 3-form. However, as is noted +in [18], this zero winding case corresponds to a scenario in which the original worldvolume +is “collapsed”. This means that the map from the worldvolume to the target space is not +an embedding. From the IIA perspective, the resulting string is tensionless and thus the M2 +brane must somehow be tensionless. We should discard that case on account of such objects +appear not to exist on physical grounds; nevertheless, the QP procedure is well defined. +– 22 – + +5.2 +M5 on S1 +The next case of interest is wrapping the M5 QP manifold on a circle. The M5 QP manifold +was written down in [1] and our expectation is that we should recover that of the D4 brane +[3]. We start with the following manifolds. +M = T ∗[6]T[1](N × S1) × R[3] +X = S1 +(5.19) +Writing N = T ∗[6]T[1]N × R[3], Y = T ∗[6]T[1]S1 and X = T[1]S1, we introduce the +homogeneous coordinates +N +Y +coord +xµ +ψµ +ζ +χµ +pµ +deg +0 +1 +3 +5 +6 +coord +y +ξ +φ +q +deg +0 +1 +5 +6 +(5.20) +and use coordinates (σ, dσ) for X. We write the symplectic form and Hamiltonian function as +ωM = dp dx + dq dy − dψ dχ − dξ dφ − 1 +2dζ dζ +(5.21) +ΘM = −ψp − ξq + 1 +7!(H7 + A ∧ F6)ψ7 + 1 +6!F6ψ6ξ + 1 +4!(F4 − A ∧ H3)ψ4ζ + 1 +3!H3ψ3ξζ (5.22) +We included, in this example, a non-trivial connection on the fibre bundle N × S1 which we +will assume to be S1 invariant. As previously, we can interpret the coefficients to be elements +of Ωi(N) × Ωj(S1) and we require that they are closed under the dS1 on S1. +We then transgress the structure to the mapping space MX and aim to define a suitable +ideal I = ⟨IN , IY⟩ to perform the coisotropic reduction with respect to. The ideal IY is taken +as in the previous section +IY = ⟨y − wσ, ξ + w dσ⟩ +(5.23) +The ideal IN is also taken as in the previous section, but now with the additional constraints +on the ζ coordinates, restricting them to closed maps. That is, we take +IN = ⟨Pcox0, Pcoψ0, Pcoζ0, Pcoχ0, Pcop0, PHx1, PHψ1⟩ +(5.24) +Since we have only added co-exact generators to the ideal, the proof of coisotropy and closure +under D goes exactly as in the previous case. +Performing the coisotropic reduction, we find the structure sheaf is generated by +C∞(W) = N(I)/I ∼ {x0, ψ0, ζ0, ζ1, χ1, p1} +(5.25) +which gives +W = T ∗[5]T[1]N × R[2] × R[3] +(5.26) +– 23 – + +To find the symplectic form, we use the Poisson brackets on MX given by (2.6) with appropriate +insertions of harmonic test functions ϵ, η, and find +ωW = −dp1 dx0 − dψ0 dχ1 − dζ1 dζ0 +(5.27) +and the Hamiltonian function is given by7 +ΘW = Π(ΘMX ) += Π +�� +T[1]S1 ψp − 1 +7!(H7 + A ∧ F6)ψ7 − 1 +6!F6ψ6ξ − 1 +4!(F4 − A ∧ H3)ψ4ζ − 1 +3!H3ψ3ξζ +� += ψ0p1 + w +6! ˜F6ψ6 +0 − 1 +4!( ˜F4 − A ∧ ˜H3)ψ4 +0ζ1 − w +3! ˜H3ψ3 +0ζ0 +(5.28) +where the tilde denotes the average over the S1 fibre. For w ̸= 0, we perform a canonical +transformation generated by the function − 1 +2wAψζ2 +1 to obtain the Hamiltonian +ΘW = ψ0p1 + +1 +4wF2ψ2 +0ζ2 +1 − w +3! ˜H3ψ3 +0ζ0 − 1 +4! ˜F4ψ4 +0ζ1 + w +6! ˜F6ψ6 +0 +(5.29) +where F2 = dA. Making the change of coordinates p1 → −p1, ζi → −ζi puts the QP manifold +in the canonical form of that associated to the D4 brane [3]. +5.3 +M5 on X4 +The next example will be to wrap the M5 brane over a 4-manifold X4. In [22, 23] it was shown +that one could reproduce the non-critical heterotic string through such a reduction, where the +dimension of the gauge group was related to the cohomology of the wrapping manifold. We +will start with the manifolds +M = T ∗[6]T[1](N × X4) × R[3] +X = X4 +(5.30) +Writing N = T ∗[6]T[1]N ×R[3], Y = T ∗[6]T[1]X4, X = T[1]X4, we introduce the homogeneous +coordinates as in the previous section +N +Y +coord +xµ +ψµ +ζ +χµ +pµ +deg +0 +1 +3 +5 +6 +coord +ym +ξm +φm +qm +deg +0 +1 +5 +6 +(5.31) +7We are using the fact that the ıdϑM term vanishes, apart from the qdy term, which cancels against the ξq +term in ΘM. +– 24 – + +where now α = 1, ..., 4, and we use the DG coordinates (σα, dσα) on X. In these coordinates +the symplectic form and Hamiltonian function take the form +ωM = dp dx + dq dy − dψ dχ − dξ dφ − 1 +2dζ dζ +(5.32) +ΘM = −ψp − ξq + 1 +7!H7ψ7 + 1 +6!H6ψ6ξ + 1 +2 +1 +5!H5ψ5ξ2 + 1 +3! +1 +4!H4ψ4ξ3 + 1 +4! +1 +3!H3ψ3ξ4 ++ 1 +4!F4ψ4ζ + 1 +3!F3ψ3ξζ + 1 +2 +1 +2F2ψ2ξ2ζ + 1 +3!F1ψξ3ζ + 1 +4!F0ξ4ζ +(5.33) +where we have taken a trivial connection on the X4 bundle again. As before, we can view the +coefficients as differential forms on Y valued in Ωk(N) that we take to be dY -closed. +We transgress this structure to MX and define a coisotropic ideal I = ⟨IN , IY⟩. To define +the ideal IY we need to choose some wrapping map w : X4 → X4. Restriction to this winding +sector of MX is given by +IY = ⟨y − w, ξ + dw⟩ +(5.34) +It is easy to verify that this is coisotropic and closed under D. The ideal IN is similar to that +for the circle reduction done in the previous section, except now our transgressed coordinates +are k-forms8 for k = 0, ..., 4. This means that we need to include more co-exact generators +and harmonic generators to remove unwanted coordinates. We take +IN = ⟨Pcoxk, Pcoψk, Pcoζk, Pcoχk, Pcopk, PHxi, PHψi, PHζj | i > 0, j > 2⟩ +(5.35) +We need to check whether this is closed under D. As in previous cases, the only non-trivial +checks come from the harmonic generators. The Q-structure D acting on the harmonic +generators xi, ψi return an element of IN precisely as in previous cases so we need only check +the closure of Dζ3, Dζ4. Once again, this can be done by calculating +(ΘM, ζ)M = 1 +4!F4ψ4 + 1 +3!F3ψ3ξ + 1 +4F2ψ2ξ2 + 1 +3!F1ψξ3 + 1 +4!F0ξ4 +(5.36) +We then transgress this function to MX and evaluate it on the vanishing locus C of I. We +then check whether the following vanishes +� +X +(ΘM, ζ)M ϵ +(5.37) +for suitable harmonic test functions ϵ. To determine the conditions coming from ζ4, we take +ϵ = ϵ0 a constant function. We then get the constraint +� +X +w∗(F0) ϵ0 +!= 0 +(5.38) +8As noted in section 2, the transgressed coordinates zA +k for k ≥ 2 should be viewed as differential forms +evaluated in some affine bundle. Our construction is still well-defined so for simplicity we will ignore this +subtlety here. +– 25 – + +where we are using the fact that F0 is a 4-form on Y which we pull back to X via the wrapping +map. Similarly, the conditions coming from ζ3 are given by choosing an arbitrary harmonic +1-form ϵ = ϵ1 +. +� +X +w∗(F1) ∧ ϵ1 +!= 0 +(5.39) +This puts constraints on the coefficients F0, F1 which can be most easily satisfied if they vanish, +i.e. they act as obstructions to the reduction. Note that in some cases, e.g. for X = K3, there +are no non-trivial harmonic 1-forms and so (5.39) gives no constraints. +Assuming these constraints are satisfied, the coisotropic reduction with respect to I = +⟨IN , IY⟩ gives that the structure sheaf is generated by +C∞(W) = N(I)/I ∼ {x0, ψ0, ζa +2, χ4, p4} +⇒ +W = T ∗[2]T[1]N × H2(X)[1] +(5.40) +We have introduced an index a parameterising a basis {ea} of H2(X4), and have expanded +ζ2 ∈ H2 as ζa +2ea. Using the Poisson brackets on MX , we get the symplectic form and the +Hamiltonian function on W to be +ωW = dp4 dx0 − dψ0 dχ4 − 1 +2κabdζa +2 dζb +2 +(5.41) +ΘW = Π(ΘMX ) = −ψ0p4 + 1 +3! ˜H3ψ3 + 1 +2 ˜Faψ2ζa +2 +(5.42) +where +˜H3 = +� +X +w∗(H3) , +˜Fa = +� +X +w(F2) ∧ ea , +κab = +� +X +ea ∧ eb +(5.43) +We get the canonical form of the QP manifold associated to a heterotic string with abelian +gauge group of dimension b2(X4). The Killing form on the gauge group is also given by the +symmetric form κab on H2(X4). For example, if X4 = T 4, we get an abelian gauge group of +dimension b2(T 4) with Killing form of signature (3, 3). If X4 = K3, then we get a gauge group +of dimension b2(K3) = 22 with Killing form of signature (3, 19). This matches the results +of [22, 23]. The fact that we can only obtain abelian gauge groups arises because we are +assuming that we are reducing on smooth manifolds. Degenerations of X4 to some singular +space should lead to gauge enhancement and non-abelian groups. +6 +AKSZ sigma models and brane wrapping +In previous sections we obtained an NQP manifold W from a coisotropic reduction of the +mapping space MX with respect to a coisotropic submanifold C that is invariant with respect +to the Q-structure D = QM + dX on MX . (In the expression for D we have the lifts of vector +fields on the target and source to the mapping space.) We will now point out that these data +give rise — essentially trivially! — to a reduction of AKSZ sigma models from an AKSZ +model with target M to an AKSZ model with target W. +– 26 – + +We start with the BV manifold of an AKSZ sigma model with target M where the source +takes the form X × S. The N-manifold S is taken to be T[1]S where the (bosonic) manifold S +has dimension dim S = n + 1 − dim X (n being the degree of the target P-structure). Then +the BV master action is the hamiltonian corresponding to the Q-structure on MX×S given by +QBV ≡ QM + dX×S +(6.1) +where again QM denotes the lift to MX×S of the target space M Q-structure of the same +name, and dX×S is the lift of the source X × S de Rham differential again to MX×S. Since +the source is a product we can write dX×S = dX + dS. +The key point that leads to reduction is that we can write +MX×S = (MX )S +(6.2) +which is known as the product-exponential adjunction. Explicitly, this corresponds to inter- +preting a function f ∈ MX×S, which is a function f(x, s) of two arguments, as a function +s → f(•, s) where f(•, s) is a function of x ∈ X for each s ∈ S.9 Since MX is a QP-manifold +and S is an NQ-manifold with an integral measure we can consider the BV structure +on MX×S as arising from an AKSZ construction with source S and target MX . +If C is coisotropic in MX , then the mapping space CS will be a coisotropic submanifold in +(MX )S ∼= MX×S. +The reduced AKSZ sigma model will be given by the coisotropic reduction of MX×S with +respect to CS. We need to confirm that CS is invariant with respect to QBV, so that the BV +master action reduces. We rewrite QBV as +QBV = (QM + dX ) + dS = ˆD + ¯dS +(6.3) +where in the last formula ˆD is the lift from MX to (MX )S of the vector field D on MX , while +¯dS is the lift of dS from S to (MX )S. We denote these lifts explicitly now because it is the +properties of these lifts that guarantee the reduction: if VS is any vector field on S, then the +lift ¯VS always leaves CS invariant (for any submanifold C of MX ); CS is invariant for ˆD if C is +invariant for D. Therefore QBV gives rise to a homological (and hamiltonian) vector field on +the coisotropic reduction of MX×S, which is simply WS. Using the results of appendix C we +find that the new BV master action is given by evaluating the original action on CS. In all +examples we have investigated the result is another topological field theory of AKSZ type. +In summary, the brane wrapping of QP manifolds that we already discussed always leads +to a brane wrapping procedure that takes the BV master action associated to an AKSZ +topological field theory and produces the BV master action of another topological field theory. +9The definition of mapping spaces for graded manifolds is such that this property is true; see e.g. [24]. +– 27 – + +6.1 +AKSZ 3-brane to membrane example +To illustrate, we will treat the reduction of the AKSZ sigma model corresponding to the +wrapping of an M2 algebroid on a circle that we discussed in Section 5.1. This is a reduction +of the 4D topological field theory of Ikeda and Uchino [25] to a (3D) Courant sigma model. +This example thus has X = T[1]S1 and the coisotropic submanifold C ⊂ MX is given by +dX p0 = 0 , +dX xµ +0 = 0 , +PHxµ +1 = Pcoxµ +1 = 0 , +y0 = wσ , +y1 = 0 +dX χ0 = 0 , +dX ψµ +0 = 0 , +PHψµ +1 = Pcoψµ +1 = 0 , +ξ0 = 0 , +ξ1 = −w . +(6.4) +We have used the superfield expansion of Z ¯ +A = {xµ, y, ψµ, ξ, · · · } in form degree (so x(σ, dσ) = +x0(σ) + x1(σ)dσ etc.) +For the original (4D) AKSZ theory degree-counting to work we set S = T[1]S where S +can be any 3-manifold, so that X × S = S1 × S is the four-dimensional worldvolume. Using +the product-exponential adjunction to write MX×S ∼= (MX )S amounts to promoting the +components Z ¯ +A +k of the superfields Z ¯ +A defining a map MX to superfields Z ¯ +A +k that now depend +on the S coordinates {s, ds} as well as the X coordinates ({σ, dσ} in this case). Then the +coisotropic submanifold CS is the locus of functions S → MX such that +dX p0 = 0 , +dX xµ +0 = 0 , +PHxµ +1 = Pcoxµ +1 = 0 , +y0 = wσ , +y1 = 0 +dX χ0 = 0 , +dX ψµ +0 = 0 , +PHψµ +1 = Pcoψµ +1 = 0 , +ξ0 = 0 , +ξ1 = −w . +(6.5) +where all bolded expressions depend on {σ, s, ds}. (The projectors to co-exact/harmonic pieces +refer to the Hodge decomposition with respect to X as above.) +We can explicitly check the claim that CS is invariant with respect to QBV = ˆD + ¯dS. E.g. +ˆD +� +S×X +(y − wσ)ϵ = +� +S×X +(ξ + dσ∂σy)ϵ +mod I(CS) += +� +S×X +(−wdσ + dσw)ϵ = 0 +(6.6) +(We smeared against ϵ ∈ C∞(S ×X) and employed (2.19)). The other differential ¯dS leaves the +ideal invariant independently. This way we may confirm explicitly that SBV lies in N(I(CS)). +It remains to calculate the reduced BV master action, which amounts to calculating +Π(SBV) where Π implements the quotient modulo I(CS). SBV is the hamiltonian for QBV = +D + dS = QM + dX + dS ≡ QM + d which is explicitly given by formula (2.10), which is a +linear combination of +� +X×S ΘM and +� +X×S ιdϑM, for ϑM the transgression of a symplectic +potential on M that satisfies dMϑM = ωM, ωM being given in (5.3). The bolded quantities +are superfields corresponding to MX×S now. We then calculate +Π +� +X×S +ιdϑM = Π +� +X×S +pdx + qdy − χdψ − φdξ += +� +S +( +� +X p1dσ)dSx0 + w( +� +X q0dσ) − ( +� +X χ1dσ)dSψ0 +(6.7) +– 28 – + +Note that terms involving x1, ψ1 will generate dX -exact terms which will vanish under the +� +X integral. Using (5.4), +Π +� +X×S +ΘM = +� +S +−ψ0( +� +X p1dσ) − w( +� +X dσq0) + 0 − w( +� +X +1 +3!H3(ψ0)3dσ) . +(6.8) +We then read off the sign factors from (2.10) to find +ΠSBV = Π +� +− +� +X×S +ΘM + +� +X×S +ιdϑM +� += +� +S +ψ0( +� +X p1dσ) + w( +� +X +1 +3!H3(ψ0)3dσ) + ( +� +X p1dσ)dSx0 − ( +� +X χ1dσ)dSψ0 . +(6.9) +The signs were such that the terms w +� +X q0dσ cancelled. +In the above expression we can identify the integrated expressions ( +� +X p1dσ) , ( +� +X χ1dσ) as +the conjugate momenta superfields (with degrees 2, 1 respectively) that appear in the Courant +sigma model for an exact Courant algebroid structure defined by the 3-form wH3. The result +we calculated via coisotropic reduction of the original (4-dimensional) AKSZ topological sigma +model is identical to the AKSZ sigma model constructed directly from the wrapped QP +manifold W with source manifold S (see (5.14)). +Therefore we have recovered the correct relation between the M-theory fluxes, the M2- +brane winding w, and the type IIA NS-flux wH3 seen by the fundamental strings that arise as +the M-theory circle X = S1 is shrunk to zero, all at the level of the corresponding topological +sigma models. +7 +Conclusions +We defined a reduction procedure of NQP manifolds M → W which encompasses the properties +of wrapped branes. This is consistent with the AKSZ procedure in the sense that the reduction +naturally lifts to a reduction of the AKSZ theory with target M to the AKSZ theory with +target W. We applied this to many examples, including many physically motivated examples +of wrapped branes and we saw that it reproduced the known M-theory/IIA dualities. We also +were able to find a novel relation between the Courant algebroid and the Poisson algebroid +through this reduction. We expect that our work will have many interesting applications to +other topological AKSZ theories. +One can ask how general our procedure is, or whether it is possible to relax some of +the assumptions made in section 2. For example, can we relax the trivial bundle condition +M = N × Y , perhaps by introducing some flat connection similar to [11]? We can also ask +whether we can extend our construction to manifolds X with boundary. We can also relax +the constraint on X = T[1]X, and instead just take X to be some DG manifold with some +invariant measure of degree n + 1. For example, we can try to extend the reduction procedure +– 29 – + +to X = T 1,0[1]X for some complex manifold X with dimC X = n + 1. We could then apply +the reduction to, say, the work of [26]. +In section 4.2, we found an interesting relation between the Courant algebroid and the +Poisson algebroid QP structures. This was based on the embedding of the Poisson differential +dπ into T ⊕ T ∗. There are other interesting differentials that can appear in these Courant +algebroids [27] that are associated to topological theories on G2 and Spin(7) manifolds. One +can try to embed these differentials in the language of QP structures and perform the reduction +to get new topological models associated to these special holonomy manifolds. There are also +similar structures that appear in higher algebroids. That is, one can define the notion of a +Dirac structure for these higher algebroids and define the associated differential [2, 28–33]. +These can be embedded into the Q-structure of the QP manifolds associated to these higher +algebroids. Their reductions may provide further insight into supersymmetric geometries of +string/M-theory. +Acknowledgements +We would like to thank Marco Zambon, Chris Blair, Dan Thompson, and Ondrej Hulik for very +helpful discussions during this project. ASA is supported by the FWO-Vlaanderen through the +project G006119N, as well as by the Vrije Universiteit Brussel through the Strategic Research +Program “High-Energy Physics”. He is also supported by an FWO Senior Postdoctoral +Fellowship (number 1265122N). DT is supported by the NSF grant PHYS-2112859. Part of +the research for this project was performed while DT was supported by the EPSRC New +Horizons Grant “New geometry from string dualities” EP/V049089/1. +A +Notation +Commutative Manifolds +M +Starting/parent commutative manifold which is always a product manifold +of a base and a fibre to be wrapped +N +The commutative manifold which is the base of the trivial fibre bundle M +Y +The fibre of the trivial bundle M. This is the manifold over which we wrap +the branes. +X +The fibre of the brane that is wrapped over Y +– 30 – + +Non-commutative Manifolds +M +Starting/parent QP manifold +N +A submanifold of M which is the natural QP manifold restricted to the +base of the fibration +Y +A submanifold of M which is the natural QP manifold restricted to the +fibre; usually Y = T ⋆[n]T[1]Y +X +The shifted tangent bundle T[1]X; the source of the mapping space MX +W +Final wrapped QP manifold +MX +maps(X → M) +S +A DG manifold with invariant measure of degree n + 1 − dim X +Indices +A, B, C, ... Indices along M, N +µ, ν, ρ, ... +Indices along N +m, n, p, ... +Indices along Y +α, β, γ, ... +Indices along X +r, s, t, ... +Indices correspnding to degree shifted real lines R[nr] +a, b, c, ... +Indices for a basis of differential forms on X +Coordinates +ZA +Homogeneous coordinates on M +zA +Homogeneous coordinates on N +xµ +Degree 0 coordinates on N +ψµ +Degree 1 coordinates on N parameterising the fibre of T[1]N +pµ +Coordinate dual to xµ +χµ +Coordinate dual to ψµ +ym +Degree 0 coordinates on Y parameterising the fibre of T[1]Y +ξm +Degree 1 coordinates on Y +qm +Coordinate dual to yα +φm +Coordinate dual to ξα +(σα, dσα) +Coordinates for the DG manifold (X, d) such that d(σα) = dσα +ζr +Homogeneous coordinates corresponding to degree shifted real lines R[nr] +ZA +Transgressed coordinates of MX +ZA +k +An expansion of the transgressed coordinates ZA into differential k-forms +ZA,a +k +A coordinate labelling the harmonic k-forms, labelled by a, associated to +the transgressed coordinate ZA +– 31 – + +Functions and differential forms +Ωk +The space of differential k-forms +Hk +Harmonic k-forms +ek,a +A basis of harmonic k-form(s) (occasionally the k is dropped) +ΘM +The Hamiltonian function of M (similarly for N, W, ...) +ωM +The symplectic form of M (similarly for N, W, ...) +ϑM +The canonical symplectic potential of M (similarly for N, W, ...) +(·, ·)M +The Poisson bracket for M (similarly for N, W, ...) +[·, ·] +The Poisson bracket on MX +Miscellaneous +w +Wrapping map X → Y +w +Winding number/matrix of a circle/torus over itself +I +The coisotropic ideal within MX +C +The vanishing locus of I within MX +B +QP manifolds +Graded manifolds +A graded manifold M is a supermanifold whose coordinates come equipped with a Z grading.10 +One can always find homogeneous coordinates ZA of definite degree, where deg ZA mod 2 is +the Grassman parity of the coordinate. We will denote by A the degree of ZA and so we have +ZAZB = (−1)ABZBZA +(B.1) +The sheaf of functions on M splits into subsheafs C∞ +n (M) of functions of definite degree. The +degree of a homogeneous function f is measured by the degree counting vector field ε (The +‘Euler vector field’) via +ε(f) = deg(f)f +(B.2) +In local homogeneous coordinates ZA, we have +ε = +� +A +deg(ZA)ZA +∂ +∂ZA +(B.3) +Unless otherwise stated, all derivations are left derivations. Hence, the de Rham d is +df = dZA∂Af +(B.4) +10From [34], the consistency of the Z grading of coordinates comes from the existence of a global degree +counting vector field ε and transition functions which preserve degree. +– 32 – + +and any homogeneous (in degree) vector field X acts as +X(fg) = X(f)g + (−1)XffX(g) +(B.5) +where we have used the shorthand X, f for the degree of the respective components. In local +coordinates we can write X = X(Z)A∂A, and so deg X = deg XA − deg ZA. We also define +deg(df) = deg f + 1 +(B.6) +For this to be consistent with ıAdZB = δBA, where ıA denotes contraction with the vector +field ∂A, we require that the interior product has degree +deg ı = −1 +(B.7) +Poisson and symplectic structures +A graded Poisson structure of degree −n is defined to satisfy +(f, g) = (−1)1+(f+n)(g+n)(g, f) +(B.8) +and the graded Jacobi identity +(f, (g, h)) = ((f, g), h) + (−1)(f+n)(g+n)(g, (f, h)) +(B.9) +for all homogeneous functions f, g, h. It also acts as a left derivation on the right hand +arguments, but a right derivation on the left hand arguments. That is +(f, gh) = (f, g)h + (−1)(f+n)gg(f, h) +(fg, h) = f(g, h) + (−1)(h+n)g(f, h)g +(B.10) +If the Poisson structure is induced from a symplectic structure ω, we have that +ıXf = (−1)fdf +Xf := (f, ·) +(B.11) +In local homogeneous coordinates we can write +ω = 1 +2dZAωABdZB +(B.12) +which implies the symmetry +ωAB = (−1)1+AB+n(A+B)ωBA +(B.13) +– 33 – + +If we define ωAB via ωABωBC = δAC, then (B.11) implies +(f, g) = (−1)f∂R +Af ωAB ∂Bg +(B.14) +where ∂R +A is defined by df = dZA∂Af = ∂R +A dZA. Note that it is not a right derivation by +itself, but the combination (−1)f∂R +Af is a right derivation. This is consistent with (B.10). +Note that this implies +(ZA, ZB) = (−1)AωAB +(B.15) +The symplectic potential is defined such that dϑ = ω, and can be defined canonically +through the Euler vector field ε. We have that11 +nω = Lεω = ıεdω + d(ıεω) = d(ıεω) +(B.16) +where we have used dω = 0. This implies we can take +ϑ = 1 +nıεω = (deg ZA)ZAωABdZB +(B.17) +Transgressed QP structure on MX +Let (X = T[1]X, d) be a DG manifold with homogeneous coordinates σ, dσ. A point f ∈ MX +can be defined by how it pulls back the coordinates on M. We have +f∗ZA = ZA(σ, dσ) = ZA +0 (σ) + ZA +1 α(σ)dσα + ... + 1 +d!ZA +d α1...αd(σ)dσα1...dσαd +(B.18) +We use the shorthand ZA +k = +1 +k!ZA +k α1...αk(σ)dσα1...dσαk, where ZA +k α1...αk(σ) is a function of +degree deg ZA − k. These act as coordinates on MX . Our conventions are always that the +form components come to the right of the function. So, e.g. +ZA +1 = ZA +1 α(σ)dσα = (−1)A−1dσαZA +1 α(σ) +(B.19) +We can always define an evaluation map +ev : MX × X −→ M +(f, σ, dσ) �−→ f(σ, dσ) +(B.20) +We also have the chain map defined by +µ∗ : Ω•(MX × X) −→ Ω•(MX ) +α �−→ +� +X α +(B.21) +11More generally, the Lie derivative on any graded differential form along a vector field X is given by +LX = ıXd + (−1)XdıX. The Euler vector field is degree 0, hence the expression given. +– 34 – + +The combination µ∗ev∗ : Ω•(M) → Ω•(MX ) is called the transgression map. +The QP +structure on the mapping space is defined by +ωMX = µ∗ev∗ωM +ΘMX = (−1)dµ∗ev∗ΘM + (−1)n+d+1ıdµ∗ev∗ϑ +(B.22) +where we use the same symbol d for the lift of the vector field on X to MX . +This can be given more explicitly in the coordinates (B.18). We will use the bold face +notation to denote a function, differential form, or coordinate on M that is pulled back to X +via some function f ∈ MX . That is, we effectively take f = ev∗f. We can then write +ωMX = 1 +2 +� +X +δZA(ωM)AB δZB +(B.23) +Our conventions for integrals is that constants are pulled out from the left. The symplectic +form above gives rise to a Poisson bracket which takes the following form on homogeneous +functionals F, G +[F, G] = +� +X +(−1)F δRF +δZA (ωM)AB δG +δZB +(B.24) +where +δF = +� +X +δZA δF +δZA = +� +X +δRF +δZA δZA +(B.25) +We can define a functional F via some pulled-back function f by +F(ϵ) = +� +X +fϵ +∀ ϵ ∈ C∞(X) +(B.26) +Then the Poisson bracket (B.24) can be expressed nicely as +�� +X +fϵ, +� +X +gη +� += (−1)(f+n)ϵ+d +� +X +(f, g)Mϵη +(B.27) +where (f, g)M = ev∗(f, g)M. +We can use this to calculate the Poisson bracket on two harmonic generators ZA +k , ZB +k′. Let +ek,a be a basis of harmonic k-forms and ˜eb +d−k be a dual basis of harmonic d − k-forms. So +δba = +� +X +ek,a ∧ ˜eb +d−k +(B.28) +Noting that Ω•(X) ≃ C∞(T[1]X) = C∞(X), and by expanding ZA +k = ZA,a +k +ek,a with ZA,a +k +some constant coefficient, we have +ZA,a +k += +� +X +ZA +k ˜ea +d−k = +� +X +ZA˜ea +d−k +(B.29) +– 35 – + +We then see that we get an induced Poisson bracket on the coefficients given by +� +ZA,a +k +, ZB,b +k′ +� +≡ +�� +X +ZA˜ea +d−k, +� +X +ZB˜eb +d−k′ +� += (−1)(A+n)(d−k)+d +� +X +(ZA, ZB)M˜ea +d−k ∧ ˜eb +d−k′ += (−1)(A+n)(d−k)+d(−1)AωAB +� +X +˜ea +d−k ∧ ˜eb +d−k′ += (−1)(A+n)(d−k)+d(−1)AωABκabδk+k′,d +(B.30) +where we have assumed Darboux coordinates, so the ωAB are constant, and where +κab = +� +X +˜ea +d−k ∧ ˜eb +k +(B.31) +We use this to find the symplectic form of the reduced theory. +C +Coisotropic reduction of graded Poisson algebras +Let P be a graded algebra with a graded Poisson bracket [•, •] of degree −P along with a +left derivation V of P, possibly hamiltonian (i.e. given by Poisson brackets, so V = [HV, •] for +HV ∈ P). We will explain how all of these objects behave under coisotropic reduction. The +derivation is as in the ungraded case considered originally by Weinstein and ´Sniatycki [35]. +If I is a (multiplicative, degree-homogeneous) ideal of P, it is a coisotrope if it is a Poisson +subalgebra, i.e. [I, I] ⊆ I. Then the coisotropic reduction of P with respect to I is the +quotient +P ≡ N(I)/I +(C.1) +where N(I) ≡ {f ∈ P|[f, I] ⊆ I} is the Poisson normaliser of I. Then the bracket on P is +defined in terms of the bracket [•, •] via +[Πf, Πg] ¯P ≡ Π[f, g] . +(C.2) +where Πf is the equivalence class f + I. For any P derivation V we define its reduction V via +V(Πf) = ΠV(f) +f ∈ P . +(C.3) +Theorem 1. Given any coisotrope I, the bracket [•, •]P is well-defined. It is moreover a +Poisson bracket of degree −P, and so P is a graded Poisson algebra. +If the derivation V on P preserves the Poisson structure (V[f, g] = [Vf, g] ± [f, Vg]) and +the coisotrope (V(I) ⊆ I) then the reduced derivation V is well-defined. +– 36 – + +Finally if V is furthermore hamiltonian with hamiltonian HV ∈ P (so V = [HV, •]) then +V is hamiltonian with hamiltonian Π(HV). (In this latter case V automatically preserves the +Poisson structure, but the condition V(I) ⊆ I implies [HV, I] ⊆ I.) +If all derivations we are interested in are in fact hamiltonian (which is the case in the +main text) then we just need to check that the ideal I is a coisotrope and that [HV, I] ⊆ I. +Proof. The bracket [•, •]P is well-defined because +[Πf, Πg]P = Π[f + I, g + I] = Π([f, g] + [f, I] + [I, g] + [I, I]) = Π[f, g] +(C.4) +where the last three terms in the second equality vanish because f, g ∈ N(I) and [I, I] ⊆ I. +This new bracket inherits the antisymmetry and Jacobi identity properties from [•, •]. Since +furthermore I is homogeneous in degree, Πf will have a well-defined degree, and so the new +bracket defines a graded Poisson algebra structure. +Similarly since V(f + I) = V(f) + V(I) we have that V is well-defined on P/I when +V(I) ⊆ I. We then need to show that it preserves the subspace N(I)/I = P. Since V +preserves the Poisson bracket we have +[Vf, I] = V[f, I] ± [f, VI] +(C.5) +If f ∈ N(I) this becomes V(I)±[f, VI] which lies in the coisotrope when V(I) ⊂ I. Therefore +V(f) lies in N(I). +Finally if V = [HV, •] then ΠV(Πf) = ΠV(f) = Π[HV, f] = [ΠHV, Πf]ΠP, which completes +the proof. +References +[1] A. S. 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Weinstein, Reduction and quantization for singular momentum mappings, +Letters in mathematical physics 7 (1983) 155. +– 39 – + diff --git a/F9E0T4oBgHgl3EQfzQJ7/content/tmp_files/load_file.txt b/F9E0T4oBgHgl3EQfzQJ7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9983316e6b964c488a7d58a9006c48a047b3d91d --- /dev/null +++ b/F9E0T4oBgHgl3EQfzQJ7/content/tmp_files/load_file.txt @@ -0,0 +1,1098 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf,len=1097 +page_content='MI-HET-793 Brane wrapping, AKSZ sigma models, and QP manifolds Alex S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Arvanitakisa and David Tennysonb aTheoretische Natuurkunde, Vrije Universiteit Brussel, and the International Solvay Institutes, Plein- laan 2, B-1050 Brussels, Belgium bMitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX, 77843, USA E-mail: alex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='arvanitakis@vub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='be, dtennyson@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='edu Abstract: We introduce a technique to realise brane wrapping and double dimensional reduction in the context of AKSZ topological sigma models and also in their target spaces, which are symplectic Ln-algebroids (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' QP-manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Our procedure involves a novel coisotropic reduction combined with an AKSZ transgression that realises degree-shifting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' the reduced QP-manifold depends on topological data of the ‘wrapped’ cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We check our procedure against the known rules for fluxes under wrapping in the context of M-theory/type IIA duality, and we also find a new relation between Courant algebroids and Poisson manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='02670v1 [hep-th] 6 Jan 2023 Contents 1 Introduction 1 2 Wrapping QP manifolds 5 3 Example – dim X = 0 12 4 Examples – dim Y = 0 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1 n-brane → (n − 1)-brane 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2 From Courant to Poisson 16 5 Examples – X = Y 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1 M2 on S1 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2 M5 on S1 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3 M5 on X4 24 6 AKSZ sigma models and brane wrapping 26 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1 AKSZ 3-brane to membrane example 28 7 Conclusions 29 A Notation 30 B QP manifolds 32 C Coisotropic reduction of graded Poisson algebras 36 1 Introduction In a series of recent papers [1–3] we have been establishing a correspondence between (BPS) p-branes in String/M-theory on one hand and symplectic Lp+1-algebroids on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The latter can be thought of as appropriate generalisations of the (exact) Courant algebroid that encodes the generalised geometry of type II backgounds with Neveu-Schwarz 3-form H (but without Ramond-Ramond fluxes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' a Courant algebroid is precisely a symplectic Ln-algebroid of degree n = 2 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In this correspondence the exact Courant algebroid (which is classified by the de Rham cohomology class of H [5]) is associated to the fundamental string (which couples to H electrically via a Wess-Zumino term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' For other branes, such – 1 – as the M2 and M5 branes in M-theory, or even/odd D-branes in type II string theory, the corresponding algebroids are roughly speaking the ones that are classified by whichever fluxes couple electrically to the brane in question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' such algebroids can be thought of as generalisations of Courant algebroids to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' M-theory scenarios, see [6] for this point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In general, the correspondence is between a “physical” p-brane (an F1, M2, M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ), a symplectic Ln-algebroid for n = p+1, and a topological AKSZ brane sigma model of dimension p + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This can be schematically summarised in the following diagram: symplectic Ln-algebroid topological n-brane physical (n − 1)-brane AKSZ brane phase space boundary condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1) Less tersely, the symplectic Ln-algebroid — that is classified by a certain collection of fluxes — determines a topological n-brane sigma model via the AKSZ construction [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' When the n-brane has an (n−1)-brane boundary, an inflow-type argument with an appropriate boundary condition produces the WZ term that couples those same fluxes to the (n − 1)-brane [1, 3]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The algebroid also determines the corresponding (n − 1)-brane more directly via the brane phase space construction that yields the Poisson algebra of brane currents on phase space [2] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' in the hamiltonian formulation of brane dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This correspondence between branes and algebroids motivates the question: given that the String/M-theory duality web acts on the branes, how is the duality web realised on the algebroid side?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Heretofore this was only known for dualities that preserve worldvolume dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' see [9] for T-duality, and [3] for M-theory/type IIA duality along a transverse M-theory circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' An example of the latter is the emergence of a D2 brane given an M2 brane that does not wrap the M-theory circle, whose algebroid avatar is symplectic reduction modulo the U(1) action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In this paper we provide an algebroid realisation for the brane wrapping operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In the string theory picture, this sends a p-brane to the (p − d)-brane found by wrapping the original brane around a d-dimensional cycle on target space and then shrinking the volume of the cycle to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (Since both the dimensionality of the brane and that of the target space are reduced in this way, this is also known as double dimensional reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') The most basic example is M-theory/IIA duality, where M2 branes wrapped around the compactified 11th dimension give rise to fundamental strings in 10 dimensions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This already poses a puzzle: the corresponding algebroids are of degree n = p + 1 = 3 (for the M2 brane) and n = 2 (for the F1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' what is the mathematical operation that accounts for this degree shift?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The mystery is resolved in the supergeometric formulation of symplectic Ln-algebroids, 1A slightly different boundary condition for the AKSZ sigma model can produce the entire (n − 1)-brane lagrangian, including kinetic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This was done for the fundamental string by ˇSevera [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The other cases have not yet been considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 2 – defined by the data of a QP manifold (M, ω, Q) where M is a non-negatively graded manifold, ω a symplectic form of degree n, and Q a nilpotent vector field of degree 1, hamiltonian for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Given a compact manifold X of dimension d — to be identified with the cycle to be ‘wrapped’ — the odd tangent bundle X ≡ T[1]X possesses an integration measure � X : C∞(X) → R of degree −d, namely the integral of differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Then the mapping space MX ≡ maps(X → M) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2) possesses a P-structure of degree (n − d), provided by the AKSZ construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is the correct degree shift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' however, this manifold is infinite dimensional, and its structure sheaf is not non-negatively graded, so it cannot be the sought-after symplectic Ln−d-algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' A ‘brane wrapping’ for QP manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We introduce a coisotropic reduction of the space MX to a finite-dimensional QP-manifold that resolves both issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This resolution is heavily motivated by the intuitive string-theoretic picture of brane wrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We deal with the case where the body of M is a product N × X, seen as a trivial bundle with fibre X, and we select a map N �→ maps(X → N × X), as in the figure maps(X → N × X) N maps(X → N × X) N maps(X → N × X) N Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The wrapping map specification, for N = R, X = S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The idea is that each point n ∈ N is mapped to the cycle of N × X that shrinks to zero size in the double dimensional reduction procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Since maps(X → N × X) is disconnected, with connected components corresponding to different winding sectors (as they would be called in physics), the choice of map N �→ maps(X → N × X) includes a choice of winding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' On the string theory side, double dimensional reduction indeed depends on winding: for instance, an M2 brane wound w times around the M-theory circle yields a fundamental string coupled to the H-flux wH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Since the algebroids corresponding to these branes via the diagram (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1) are defined by the same fluxes, we expect winding dependence in the obtained algebroid, and we will indeed find it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In more detail: we start with the data of an NQP — “N” for non-negatively graded — manifold M with body M and a ‘source’ Q manifold X = T[1]X as above, along with a – 3 – wrapping map w : X → M that defines a degree-zero submanifold N �→ maps(X → M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We then produce a finite-dimensional, non-negatively graded QP manifold W, whose P-structure has degree n − d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' we will call W the wrapped algebroid, and we will call our procedure (brane) wrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The wrapping of QP manifolds/symplectic Ln algebroids is then a reduction of MX with respect to a coisotropic submanifold C which may be thought of as the lift of N �→ maps(X → M) to a graded submanifold of maps(X → M) = MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The output QP manifold W depends on the choice of wrapping map w only up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In fact we were able to generalise beyond the case M = N ×X (that was pictorially outlined above) to the case M = N × Y , with Y and X not necessarily of the same dimension, even;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' then the wrapping is a map w : X → N ×Y , and d = dim X controls the degree/dimensionality shifts as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This generalisation allows us to accommodate at least one example which might be of interest outside of string theory, namely the wrapping of a Courant algebroid into a Poisson manifold discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2, which has dim Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' When dim X = n + 1 in addition to dim Y = 0 (so that MX has a degree −1 P structure) our wrapping procedure agrees with that of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Our approach gives a complementary perspective to the Losev-trick based ‘wrapping’-style reductions of [12, 13], and to that of [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Brane wrapping and AKSZ sigma models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Our ‘brane wrapping’ reduction — from a QP manifold M to a QP manifold W — also induces a reduction of the corresponding AKSZ topological field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Essentially, the two reductions commute, as in the schematic diagam M W MX×S WS wrapping AKSZ AKSZ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3) Here MX×S and WS are P-manifolds of degree −1 created by the AKSZ construction for S of appropriate dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The dotted arrow corresponds to a reduction of MX×S with respect to the coisotropic submanifold CS ≡ maps(S, C), for C the coisotropic submanifold that appears in the ‘wrapping’ reduction M → W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This ‘dotted’ reduction always exists and is compatible with the AKSZ/BV master actions if the wrapping reduction does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We provide the argument for the reduction of AKSZ sigma models in section 6, along with an example: the reduction of a topological 3-brane sigma model (corresponding to the M2 brane symplectic L3-algebroid) to a Courant sigma model (corresponding to the fundamental string symplectic L2-algebroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This provides an important consistency check: if we were to derive the corresponding physical brane sigma models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' by introducing boundaries and using an inflow-type argument as in [1, 8], we would find that the electric Wess-Zumino flux coupling has the correct winding dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 4 – Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In section 2, we describe the general procedure for wrapping QP manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We provide the conditions required of the QP structure on M and define the coisotropic ideal I ⊂ C∞(MX ) (that defines the coisotropic submanifold C) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We show that it is well-defined and perform the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The next three sections provide a multitude for examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (If the reader finds the notation of section 2 too terse, they may find it useful to first work their way through the examples before coming back to the general procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') Section 3 covers the case where dim X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In this case, we do not get any wrapping and our reduction is very similar to conventional dimensional reduction [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In section 4 we consider examples where dim X ̸= 0, but the wrapping map w is trivial in homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' These provide examples which are simple but still present some of the main features of the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Among these is the reduction of a Courant algebroid to a Poisson manifold given in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In section 5, we consider examples relevant for physics and wrap string/M-theory branes on various manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In section 6 we show how our procedure naturally lifts to a reduction of the AKSZ theory from MX×S to WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Section 7 is left for comments and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The appendices cover our notation (appendix A), some key properties and conventions of QP manifolds (appendix B), and a review of coisotropic reduction in the graded context (appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 2 Wrapping QP manifolds We will describe a process of creating new QP manifolds from old, which effectively generalises the notion of dimensional reduction, that we describe as ‘wrapping’ QP manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The nomenclature arises due to the consistency of this process with the AKSZ construction [7] - that is, one can reduce the AKSZ theory from the original QP manifold to that of the new manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Solutions of this reduced AKSZ theory will look like branes wrapping cycles of the target space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will describe the relation to AKSZ sigma models in a later section and will describe the wrapping procedure here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We start from the following ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' An NQP manifold M = N × Y of degree n ≥ 2 where Y = T ∗[n]T[1]Y (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1) and N is otherwise generic, with underlying commutative manifold2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The underlying commutative manifold for M is M = N × Y , a direct product manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The symplectic form will be written ωM = dϑM, where ϑM is the canonical symplectic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The induced Poisson Bracket on M will be written (·, ·)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 2By ‘underlying commutative manifold’ we mean the commutative manifold M whose structure sheaf is the sheaf of degree 0 functions on M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' C∞(M) = C∞ 0 (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is well defined since we are working on non-commutative manifolds with a non-negative grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will also refer to this as the manifold in degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 5 – The Q-structure of M should be a lift of the de Rham differential of Y , seen as the vector field dY ≡ ξm∂/∂ym on T[1]Y , with respect to the bundle projection p that is the composition N × Y πY −−→ Y πT [1]Y −−−−→ T[1]Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Explicitly this lift condition means QMp⋆ = p⋆dY , which partially determines the form of the hamiltonian ΘM in local coordinates: ΘM = −ξmqm + n+1 � k=0 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='θm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='mk(z, y)ξm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='ξmk (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2) where q are the degree n conjugate momenta to y on T ⋆[n]T[1]Y and z are generic homogeneous coordinates on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The θk = θk(z, y)ξk can be viewed as (C∞(N)-valued) differential forms on Y and we demand that they must be dY -closed differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' A Q manifold X = (T[1]X, d) where X is compact, without boundary, and has dimension d < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' d is the de Rham differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' A choice of ‘wrapping map’ w : X → Y , defined up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We aim to produce a new NQP manifold W from M, X, which describes a brane where X has been wrapped over Y and both cycles have been shrunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The resulting QP manifold should therefore have degree n − d and underlying commutative manifold N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' There is a natural choice of manifold of degree n − d given by the mapping space MX := maps(X → M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' However, this manifold is infinite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will see that we can define a coisotropic reduction of MX that produces a finite dimensional NQP manifold which only depends on the topology of X and the homotopy class of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Properties of the mapping space The infinite dimensional space MX consists of maps f which are defined by their pullback action on the coordinates on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Using generic homogeneous coordinates ZA for M and coordinates (σα, dσα) for X adapted to d (d(σα) = dσα, ddσα = 0) we have f∗ZA = ZA(σ, dσ) = ZA 0 (σ) + ZA 1 α(σ)dσα + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' + 1 d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='ZA d α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='αd(σ)dσα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='dσαd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3) Defining the components Zk is equivalent to defining the map f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' To interpret the Zk we consider a change of coordinates on M given by ˜ZA = ˜ZA(Z) and note that f∗ ˜ZA(Z) = ˜ZA(f∗Z) = ˜ZA(Z0) + ZB 1 αdσα ∂ ˜ZA ∂ZB (Z0) + 1 2dσαdσβ � ZB 2 αβ ∂ ˜ZA ∂ZB (Z0) + ZB 1 αZC 1 β ∂2 ˜ZA ∂ZB∂ZC (Z0) � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='4) – 6 – Therefore, in spite of the index structure, these in general are not vector-bundle-valued differential forms, with the exception of Z1 which is an f⋆ 0 TM-valued 1-form for the map f0 = f ◦ s0, where s0 : X → X is the zero section of X = T[1]X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Of the other components, ZA 0 defines the map f0 : X → M, while the ZA k for k > 1 transform “affinely” whenever ZA k′ ̸= 0 for any 0 < k′ < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3 Since we may not set ZA k = 0 consistently in general, this introduces a subtlety for our reduction procedure which we will discuss later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The QP structure on the mapping space is induced by that on M through transgression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The symplectic structure is given by ωMX = � X 1 2δZA(ωM)AB δZB = � k � X 1 2δZA k (ωM)ABδZB d−k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='5) which induces a Poisson bracket [·, ·] on MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This Poisson bracket can be conveniently expressed in terms of ‘test functions’ as in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Given arbitrary functions ϵ, η on X — which correspond to differential forms on X since X = T[1]X — they write �� X ZAϵ , � X ZBη � = (−1)(B+n)ϵ+d � X (ZA, ZB)M ϵη (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) where in the exponent we have used the shorthand B, ϵ for the degrees of the respective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) we can see that if ZA is dual to ZB on M, then ZA k will be dual to ZB d−k on MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Furthermore, if we are working in Darboux coordinates, so that components of ωM are constant, then by performing a Hodge decomposition Ωk(X) = Hk ⊕ dΩk−1 ⊕ d†Ωk+1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='7) with respect to some arbitrary metric, exact forms ZA k will be dual to co-exact ZB d−k and harmonic forms will be dual to harmonic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' For convenience we introduce orthogonal projectors PH, Pex , Pco (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='8) onto harmonic, exact, and co-exact forms respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The Q-structure D on MX is defined as the hamiltonian vector field D = d + QM , D = [ΘMX , ·] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='9) where the hamiltonian is ΘMX = (−1)d � X ΘM + (−1)d+n+1 � X ıdϑM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='10) 3Exploiting Batchelor’s theorem to write M as a graded vector bundle only improves this situation in that some Z0 take values in a vector bundle as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 7 – where each term generates the lift of QM and d to MX respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Note that implicit in this formulae is the fact that we have pulled back/transgressed ΘM, ϑM to objects on X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' we have used boldface to highlight this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The signs are such that D = dX + QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The coisotrope We need to perform a coisotropic reduction to obtain a finite dimensional NQP manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is a generalisation of symplectic reduction for Poisson manifolds which requires a coisotropic ideal I ⊂ C∞(MX ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' an ideal that satisfies [I, I] ⊆ I (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='11) The description of the quotient manifold is given in two equivalent ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In one description, we take the submanifold C ⊂ MX defined by the vanishing of I and quotient by transformations generated by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Alternatively, we can describe the structure sheaf of the quotient manifold as the normaliser N(I) of I, quotiented by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' That is W = C/[I, ·] ⇔ C∞(W) = N(I)/I (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='12) Such a manifold has a natural Poisson structure induced from that on the mapping space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' see appendix C for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Further, provided the ideal is closed with respect to the Q structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' DI = [ΘMX , I] ⊆ I, the reduced space has a Q-structure induced from the image of the Hamiltonian function under the quotient map: ΘW = Π(ΘMX ) Π : N(I) → N(I)/I (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='13) This closure is precisely the statement that ΘMX ∈ N(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We build our ideal I = ⟨IN , IY⟩ in 2 parts, each defining a restriction to some submanifold of MX = N X × YX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This factorisation is convenient because Y may be thought of as ‘longitudinal’ to the cycle to be wrapped, while N is ‘transverse’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' On YX , we would like the maps in degree 0 to restrict to the fixed wrapping map w : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This restriction is naturally given by the zero locus of the ideal generated by y − w and its closure under D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='10), we find IY = ⟨y − w, ξ + dw⟩ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='14) This is clearly coisotropic in the coordinates on Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The angled brackets ⟨· · · ⟩ will always denote the ideal generated by · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' On N X , we follow [11] and take the coisotropic submanifold to consist — in the first instance — of closed maps under the transgressed differential d on N X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In degree zero we realise this via a choice of degree preserving embedding N �→ N X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' By degree counting this is a map of (ordinary) manifolds N �→ NX, and we choose this to be the map sending each – 8 – n ∈ N to the constant map X → {n} (which is d-closed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Beyond degree zero, we simply set the coisotropic part of each coordinate in the superfield expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3) to zero (using the Hodge decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Therefore we define IN such that IN ⊃ � PcozA k � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='15) for all values of k in the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3), where zA is a generic coordinate on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' If we consider the vanishing locus of IN and IY simultaneously, we see that we are restricted to x0 = const, y0 = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This gives an embedding N �→ MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Similarly, a choice of degree preserving embedding N �→ MX defines our ideal in degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This alone is not enough as we would like the reduced manifold W to be an N-manifold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' a graded manifold with non-negative coordinates, such that in degree zero the structure sheaf is that of an ordinary manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='4 To remove these, we include harmonic generators of the maps zA k for maps such that deg zA − k ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The exception to this is the maps x0 for which we do not include the harmonic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' constant map) representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We therefore have IN = �� � � PcozA k , PHzA k′ | ∀k′ ≥ deg zA if deg zA > 0 PcozA k , PHzA k′ | ∀k′ > 0 if deg zA = 0 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='16) To see that this is coisotropic, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) and the surrounding discussion to note that the co-exact generators are dual to exact generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Hence, these terms are coisotropic with respect to all of IN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The harmonic generators could be dual to some other harmonic generator in IN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' However, since we only include harmonic zA k for 0 ≥ deg zA − k, the dual coordinate zB k′ on MX has deg zB − k′ = n − deg zA − (d − k) = (n − d) − (deg zA − k) > 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='17) so is not included in IN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The total ideal I = ⟨IN , IY⟩ is coisotropic, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Metric and coordinate independence The construction of the ideal I appears to rely on a choice of metric on X but we claim that the resulting reduced manifold depends on only the topological data of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This can be best seen from the vanishing locus C ⊂ MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is given by maps which are either d-closed (for deg zA = 0 and some values of k, k′) or ones which are also d-exact (in all other cases), as specified in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The metric only appears in the specification of the vanishing ideal that 4There are issues not just with negative-graded but also with degree zero ‘formal’ coordinates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' [17, section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 9 – represents C but C does not in itself depend on metric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (This apparent metric dependence thus may perhaps be seen as due to ‘gauge-fixing’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') Our construction also appears to depend on a choice of coordinates on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' To see that this is well defined, we will show that the submanifold C is invariant under a change of coordinates ˜zA = ˜zA(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Using formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='4), we can write the k-form component of the transgressed ˜z as ˜zA k ∼ � j CAA1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='Aj(z0)zA1 k1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='zAj kj (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='18) such that k1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' + kj = k and deg zA1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' + deg zAj ≤ deg ˜zA k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' here we emphasise that the last inequality holds true because M — and thus N — was assumed to be an N-manifold (its structure sheaf is non-negatively graded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Restricting to C, the coordinates zAi ki are all closed under d and hence so is ˜zA k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Further, if deg ˜zA − k ≤ 0, then at least one of deg zAi − ki ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This means that zAi ki is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' A product of closed and exact forms is exact and hence so is ˜zA k as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Closure under D The final condition to check is that the ideal I is closed under the Q-structure D = [ΘMX , ·] on MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We have already checked that IY is D-closed and so we need only check how D acts on the generating coordinates of IN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We can use formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) with Pcoϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This choice of epsilon selects out the harmonic and co-exact generators zA k , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We have � ΘMX , � X zAϵ � = (−1)d �� X ΘM, � X zAϵ � + (−1)d+n+1 �� X ıdϑM, zAϵ � = � X (ΘM, zA)M ϵ + � X dzAϵ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='19) The second term vanishes when ϵ is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We therefore need to only consider the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We can see whether this term is contained within I by transgressing the function (ΘM, zA)M to the mapping space and evaluating it over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' If the integral vanishes when integrated against all closed ϵ then the ideal is closed under D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Using the form of the Hamiltonian function we find (ΘM, zA)M ∝ (ωM)AB � k ∂θk(z, y) ∂zB ξk (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='20) Transgressing this to the mapping space and restricting to C, we replace y → w, ξ → dw, z → z for z closed (or exact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Integrating this against ϵ we get � ΘMX , � X zAϵ ����� C ∝ � X � (ωM)ABw∗ � k ∂θk ∂zB (z) � ϵ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='21) – 10 – We require this to vanish for the above ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' When ϵ is exact, this indeed vanishes if we impose dY θk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' If ϵ is harmonic, however, we find constraints on the coefficients θk that we address case-by-case, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The reduction, metric independence, and homotopy invariance Given the coisotropic reduction I, we consider the reduction given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will consider the structure sheaf construction of the reduced manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The normaliser N(I) of the ideal is generated by N(I) ∼ � PHx0, PHzA k , I | 0 < deg zA − k ≤ n − d � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='22) We can expand the PHzA k = zA,a k ea in some basis {ea} of Hk, so the zA,a k are constant parameters of degree deg zA − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In the case that the respective cohomology group is 1 dimensional (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' for H0, Hd) we will omit the a index and simply identify e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' zA d = zA d volX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We see that the structure sheaf C∞(W) = N(I)/I is given therefore generated by C∞(W) = N(I)/I ∼ {x0, zA,a k | 0 < deg zA − k ≤ n − d} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='23) That is, the structure sheaf is given by all smooth functions in the zA,a k (and x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The Hamiltonian function on W is given by the projection Π : N(I) → N(I)/I of ΘMX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will confirm in the examples that the final result is given by ΘW = Π(ΘMX ) = � X � k (−1)kw∗θk(z) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='24) where the z are now the harmonic representatives of the cohomology groups on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Expanding the harmonic zA k in terms of the constant coordinates zA,a k , we can perform the integral over X with the convention that the volume form is on the right of the integrand, so we pull out constants from the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Once this is done the final result will no longer be an integral but will be a function in the zA,a k which will involve, in general, a sum over cohomology groups, which will be discrete in all cases (we only consider wrapping over compact cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='24) seems to depend on some metric to choose the harmonic representatives for the z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' However, under a change of metric, the harmonic representatives change by a d-exact term, and since we have assumed that the forms θk are closed, this shift will not change the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Furthermore, since the forms θk are closed, the evaluation of the integral only depends on the homotopy class of w : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Therefore the construction is metric-independent and homotopy invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Now that we have defined the reduction in complete generality, we will see many examples of how this works in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' There are three interesting cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' dim X = 0 – The process effectively shrinks Y to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 11 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' dim Y = 0 – We produce a QP manifold with the same underlying commutative manifold but with a different degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' X = Y – We produce a QP manifold which corresponds to a brane wrapping the internal manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 3 Example – dim X = 0 We consider first a simple example to show that in the simple case that dim X = 0, our procedure effectively reduces to dimensional reduction on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Consider the ingredients M = T ∗[n]T[1](N × Y ) X = pt (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1) Taking N = T ∗[n]T[1]N, Y = T ∗[n]T[1]Y and X = T[1]X = pt, we introduce the Darboux coordinates N Y coord xµ ψµ χµ pµ deg 0 1 n − 1 n coord ym ξm φm qm deg 0 1 n − 1 n (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2) We will take the QP structure to be given by the symplectic form and Hamiltonian function ωM = dp dx + dq dy − dψ dχ − dξ dφ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3) ΘM = −ψp − ξq + 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='Fnψn + 1 (n−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='Fn−1ψn−1ξ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' + 1 d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (n−d)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='Fn−dψn−dξd (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='4) We have suppressed all indices but they should be read as being contracted in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The coefficients Fk can be thought of as elements of Ωk(N) × Ωn−k(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' These should be closed under the differential dY on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' So for example, Fnψn = Fn(x, y)µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='µnψµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='ψµn should be viewed as a differential n-form on N, but a constant function on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In the ansatz above, we have assumed a trivial connection on the bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We can easily reintroduce it by making the replacement ξ → A = ξ + Aψ, where A is the connection, however it won’t change our final result so we omit it for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The first step in the reduction process is to transgress the QP structure to MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' But since X is 0 dimensional, we have MX ≃ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Next, we need to choose a wrapping map w : X = pt → Y , or equivalently, a (degree preserving) embedding N �→ MX ≃ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is equivalent to choosing some point ˆy ∈ Y and defining the embedding N → (N, ˆy) ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is described by the ideal I0 = ⟨ym − ˆym⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='5) We then want to form the closure of this ideal with respect to differential Q on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We get IY = ⟨I0, QI0⟩ = ⟨ym − ˆym, ξm⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) – 12 – It is easy to check from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3) that this is indeed coisotropic with respect to the Poisson bracket on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In principal, we also need to restrict the maps into N to those that are closed/exact with respect to d on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' However, since dim X = 0, this is a trivial constraint and so we just have I = IY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' To perform the coisotropic reduction, we need to go to first find the normaliser N(I) of I, which can easily be verified to be generated by the coordinates N(I) ∼ {xµ, ψµ, χµ, pµ, ym − ˆym, ξm} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='7) The structure sheaf of the new QP manifold W is then defined to be the quotient of this by the ideal I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' That is C∞(W) = N(I)/I, which is generated by N(I)/I ∼ {xµ, ψµ, χµ, pµ} ⇒ W = T ∗[n]T[1]N (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='8) Note that, by construction ΘM ∈ N(I) and so we can find the new Hamiltonian function through the natural projection Π : N(I) → N(I)/I, which gives ΘW = Π(ΘM) = −ψp + 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='Fn(x, ˆy)ψn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='9) and the final symplectic form is ωW = dp dx − dψ dχ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='10) We see that this procedure has produced a new QP manifold with the same degree but with underlying commutative manifold N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We see that we have effectively collapsed Y to the point ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In the case where Y is a Lie group, we find the same result as in symplectic reduction modulo T[1]Y [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' If we were to choose a different wrapping map w′ : X �→ ˆy′ that is homotopic to w : X �→ ˆy, then we end up with the same graded manifold where the Q-structure is evaluated for Fn(x, ˆy′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' However, the condition that the Fn is closed on Y says that it is constant, and hence the Q-structures are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This demonstrates the homotopy invariance of our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 4 Examples – dim Y = 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1 n-brane → (n − 1)-brane Let’s now consider the same example as above, but instead of having dim X = 0, we will take the dimension of the fibre dim Y = 0 and take X to be non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will take the ingredients M = T ∗[n]T[1]M X = S1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1) – 13 – We will use the homogeneous coordinates coordinates M coord xµ ψµ χµ pµ deg 0 1 n − 1 n (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2) and use the coordinates σ, dσ on X = T[1]S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The Hamiltonian function and symplectic form are given by ωM = dp dx − dψ dχ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3) ΘM = −ψp + 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='Fnψn (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='4) Since Y = pt in this example, we do not need to impose any constraints on the coefficients Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We need to transgress this structure to the mapping space MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is now an infinite dimensional graded manifold whose points f ∈ MX can be described by their pullback action on coordinates on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' That is, we have f∗ZA = ZA(σ, dσ) = ZA 0 (σ) + ZA 1 (σ)dσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='5) The transgressed Hamiltonian function is given by ΘMX = (−1)1 � X ΘM + (−1)n−1+1 � X ıdϑM = − � T[1]S1 −ψp + 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='Fn(x)ψn + (−1)n � T[1]S1 pdx − 1 n(ψdχ + (n − 1)χdψ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) The bold-faced letters in the expression correspond to functions pulled back to functions on X as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The Berezin integral over T[1]S1 selects the maximal degree component of the integrand (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' the 1-form components) and integrates it over S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Our convention is that we normalise with an overall factor of vol(S1), and so for the flat metric on S1 we have � T[1]S1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' = 1 2π � S1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=')1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='7) The next step is to define the coisotropic ideal I = ⟨IN , IY⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Since Y is trivial, so is the ideal IY and hence we need only determine IN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Following section 2, we first start by restricting to all closed maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' That is, we take IN ⊃ ⟨Pcoxk, Pcoψk, Pcoχk, Pcopk⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='8) To define this ideal we choose some arbitrary metric on S1, and for simplicity we can take the flat metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We then also add the harmonic representatives for ZA k such that deg ZA − k ≤ 0 – 14 – (except for x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This gives IN = ⟨Pcox0, Pcoψ0, Pcoχ0, Pcop0, PHx1, PHψ1⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='9) Again, in degree 0, the vanishing locus of this ideal restricts us to maps x0 = const and hence defines a natural embedding M �→ MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' It is a quick check using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) that this ideal is coisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Indeed, the Poisson bracket of the co-exact generators with any other generators will vanish, as they are dual to exact maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The harmonic x1, ψ1 representatives are dual to p0, χ0 ∈ H0 respectively and these do not appear in the generating set of IN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will also verify that this ideal is closed with respect to the Q-structure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Using the test function form of the Poisson bracket (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6), we can calculate D acting on the generators by calculating � ΘMX , � T[1]S1 ZAϵ � = − �� T[1]S1 ΘM, � T[1]S1 ZAϵ � + (−1)n �� T[1]S1 ıdϑM, � T[1]S1 ZAϵ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='10) where ϵ is a function on N that is closed under d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Taking ϵ ∈ Hk selects the harmonic representative Z1−k ∈ H1−k, while taking ϵ to be exact selects the co-exact representative of ZA 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The second term gives us �� T[1]S1 ıdϑM, � T[1]S1 ZAϵ � ∝ � T[1]S1 dZ ϵ = 1 2π � S1 dZA 0 ϵ0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='11) Taking ϵ to be closed tells us that ϵ0 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The integrand on the right hand side is therefore exact and so the integral vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The Poisson bracket is then determined by the first term alone which is proportional to � ΘMX , � T[1]S1 ZAϵ � ∝ � T[1]S1(ΘM, ZA)M ϵ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='12) where the function (ΘM, ZA) is transgressed to the mapping space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We can use these results to confirm that ΘMX lies always in IN as outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The only non-trivial checks are for the harmonic generators x1, ψ1, for which we take ϵ = ϵ0 to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We have (ΘM, x)M = ψ , (ΘM, ψ)M = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='13) Transgressing these functions and evaluating on C, we take ψ1 to be exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Hence, both vanish under the integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='12) when ϵ = ϵ0 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This proves that the ideal is closed under D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Now that we have our coisotropic ideal, we perform the coisotropic reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The – 15 – normaliser of I is generated by all the coordinates that are not dual to those in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' N(I) ∼ {PHx0, PHψ0, PHχ1, PHp1, I} (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='14) The structure sheaf for W is then N(I)/I which is generated by N(I)/I ∼ {x0, ψ0, χ1, p1} ⇒ W = T ∗[n − 1]T[1]M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='15) (Note in the expression above we are now working in the coordinates zA,a k described in section 2: PHx0 = x0 · 1 and PHp1 = p1vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') Thus we restrict to harmonic functions for x, ψ — so they retain their original degrees — while we restrict to harmonic 1-forms for χ, p hence they have their degrees shifted down by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We therefore end up with the manifold T ∗[n − 1]T[1]M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' To find the symplectic form, we use the Poisson brackets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) with the ϵ, η appropriate harmonic representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We find that ωW = −dp1 dx0 − dψ0 dχ1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='16) To find the form of the Hamiltonian function we project ΘMX under Π : N(I) → N(I)/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' By restricting all coordinates to the harmonic representatives on which d = 0, we find Π(ı¯d ¯ϑ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The term 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='Fnψn gets projected to 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='Fn(x0)ψn 0 which is a function on S1 and hence vanishes under the Berezin integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We find that we are left with5 ΘW = Π(ΘMX ) = ψ0p1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='17) Making the change of coordinates p1 → −p1 puts the QP manifold in the canonical form for a (n − 1)-brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Interestingly, all flux twisting drops out of the Hamiltonian function in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is what happens in the zero-wrapping sector of wrapped branes where physically one ends up with a tensionless brane [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' These are somewhat pathological and hence the physical interpretation of such reductions is less clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will see that one can get more interesting reductions if one allows X to wrap some part of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2 From Courant to Poisson Using the formulation set out, we can already find interesting relations between different QP manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Suppose M is a Poisson manifold with Poisson bivector π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' There are two distinct ways to realise this structure as a QP structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Firstly, we can take the straight cotangent lift of π to obtain the following QP manifold W = T ∗[1]M coord ˜x ˜p deg 0 1 ωW = d˜p d˜x ΘW = 1 2π˜p2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='18) 5Our conventions are that we integrate with the volume form on the right of the integrand, and so we pull constants out from the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This gives the overall sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 16 – A quick calculation shows that (ΘW, ΘW) = 0 if and only if π is Poisson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Alternatively, we can consider the Lie algebroid structure on T ∗M whose anchor map is given by the bivector π : T ∗M → TM and whose bracket is given by [α, β] = Lπ(α)β − ıπ(β)dα (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='19) This Lie algebroid can be lifted to a Dirac structure L = (1 + π)T ∗M within the Courant algebroid TM ⊕ T ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Such a structure can be described by a QP manifold via M = T ∗[2]T[1]M coord x ψ χ p deg 0 1 1 2 ωM = dp dx − dψ dχ ΘM = −πpχ + 1 2∂πψχ2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='20) Once again (ΘM, ΘM) = 0 if and only if π is Poisson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We have suppressed indices for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We want to see how, if at all, these constructions are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Let us perform a circle reduction of M as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We transgress the structure to MX where X = T[1]S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' As before, we define I = IN by first including all co-exact generators I ⊃ ⟨Pcox0, Pcoψ0, Pcoχ0, Pcop0⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='21) Then we include harmonic representatives to remove coordinates of 0 or negative degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will slightly relax the construction set out in section 2 by allowing some new coordinates of degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6 We will define I = ⟨Pcox0, Pcoψ0, Pcoχ0, Pcop0, PHx1, PHχ1⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='22) As before, this ideal is coisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will check the closure of this ideal with respect to the Q-structure D on MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The transgressed Hamiltonian function is ΘMX = − � T[1]S1 ΘM + � T[1]S1 ıdϑM = − � T[1]S1 −π(x)pχ + 1 2∂π(x)ψχ2 + � T[1]S1 p dx − 1 2(ψdχ + χdψ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='23) We then act with this on � ZAϵ for some test function ϵ that must be harmonic, or exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' As in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='12), the only non-trivial constraint to check is for the harmonic representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We need 6The construction, as set out previously, would still work in this case but we would end up with a trivial Q-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' To result in a QP manifold with non-trivial Q-structure, we will need to perform an intermediate step before removing the additional degree 0 coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 17 – to check if the following vanishes � T[1]S1(ΘM, ZA)M ϵ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='24) whenever the function (ΘM, ZA) is transgressed and evaluated on C, and if ϵ is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Since the only harmonic generators of I are x1, χ1, we calculate (ΘM, x) = π(x)χ , (ΘM, χµ) = 1 2∂π(x)χ2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='25) We transgress these functions to the mapping space and evaluate on the vanishing locus C of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Noting that these are functions of x, χ alone, evaluating them on C means that the zero-form component must be constant functions on X, while the 1-form component must be an exact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Integrating these against a constant function ϵ = ϵ0 selects the 1-form component, which is exact and hence the integral vanishes as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The next step is to perform the coisotropic reduction with respect to this ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The normaliser is generated by all coordinates not dual to those in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' N(I) ∼ {PHx0, PHχ0, PHψ1, PHp1, I} (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='26) and so we obtain the structure sheaf C∞( � M) = N(I)/I which is generated by N(I)/I ∼ {x0, ψ1, χ0, p1} ⇒ � M = T ∗[1]TM (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='27) This time, we restrict to harmonic functions for x, χ so they retain their degree, while we take harmonic 1-forms for ψ, p and hence their degree is shifted down by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The resulting Hamiltonian function is Π(ΘMX ) and the symplectic form is derived from the Poisson brackets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) with harmonic representatives for ϵ, η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Θ � M = −πp1χ0 − 1 2∂πψ1χ2 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='28) ω � M = −dp1dx0 + dψ1dχ0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='29) We performed the change of coordinates p1 → −p1, χ0 → −χ0 to remove minus signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We have arrived at a ‘halfway house’ QP manifold � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Interestingly, this is the cotangent lift of the complete lift of the Poisson structure π on M to the tangent bundle (TM, πc) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' That is, given any Poisson structure (M, π) we define a Poisson structure (TM, πc) by πc = πµν ∂ ∂xµ 0 ∂ ∂ψν 1 + 1 2ψρ 1∂ρπµν ∂ ∂ψµ 1 ∂ ∂ψν 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='30) where x0 are coordinates on M and ψ1 are coordinates along the vector bundle fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We can reduce the QP manifold � M further by following [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Given any (torsionless) connection on – 18 – M, we can define a global vector field on TM given by the geodesic spray s = ψµ 1 ∂ ∂xµ 0 − ψµ 1 ψν 1Γρ µν ∂ ∂ψρ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='31) This has a cotangent lift to T ∗[1]TM whose hamiltonian is S = ψ1p1 − Γψ2 1χ0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='32) From this, we define a new Hamiltonian function Θ′ � M = − 1 2(S, Θ � M) = 1 2πp2 1 + ψ1f(x0, ψ1, χ0, p1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='33) where f is some function of the coordinates whose precise form is not important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' All we will need is that besides the first term, 1 2πp2 1, each term is at least linear in the coordinate ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Consider the ideal generated by the single coordinate I = ⟨ψ1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This ideal is automatically closed under the Q-structure since (Θ′ � M, ψ1) = ( 1 2πp2 1 + ψ1f(x0, ψ1, χ0, p1), ψ1) = (ψ1f(x0, ψ1, χ0, p1), ψ1) ∝ ψ1(f, ψ1) ∈ I (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='34) Performing the coisotropic reduction with respect to this ideal we obtain the structure sheaf N(I)/I ∼ {x0, p1} ⇒ W = T ∗[1]M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='35) That is, we reproduce the non-commutative manifold W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Further, the symplectic form and Hamiltonian function are easily shown to be the following: ωW = dp1 dx0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='36) ΘW = 1 2πp2 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='37) We see then that we precisely reproduce the QP manifold associated to the cotangent lift of the Poisson bivector that we described at the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This construction provides new relations between the Courant sigma model and the Poisson sigma model that is different from the WZW-Poisson model [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' There they view the Poisson model arising at the boundary of a topological WZW-like theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Instead, our construction is closer to dimensional reduction and can be viewed as the geometric counterpart of the Courant sigma model reduction found in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 19 – 5 Examples – X = Y We will now generalise the previous two sections to allow for cases where the source manifold X wraps the target space fibre Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In particular, we will be interested in the case where X = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will see that the reduction procedure requires us to choose some self-wrapping map w : X → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The examples we choose are physically motivated and fill our understanding of how brane dualities in M-theory/IIA arise in the QP setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In particular, when X = S1, we will see that our procedure produces the known relations from M-theory/type IIA duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will also see that this procedure reproduces other interesting relations between the M5 brane and the heterotic string [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1 M2 on S1 Our first example will be wrapping the M2 brane on an S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This will be very similar to the n-brane example in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1, except in this case the wrapping will allow for more interesting Hamiltonian functions to be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We start with the QP manifold M associated to the M2 brane and a source manifold X: M = T ∗[3]T[1](N × S1) X = S1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1) Writing N = T ∗[3]T[1]N, Y = T ∗[3]T[1]S1, and X = T[1]S1, we will introduce the coordinates N Y coord xµ ψµ χµ pµ deg 0 1 2 3 coord y ξ φ q deg 0 1 2 3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2) and use coordinates (σ, dσ) on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The Hamiltonian function and symplectic form are ωM = dp dx + dq dy − dψ dχ − dξ dφ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3) ΘM = −ψp − ξq + 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F4ψ4 + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H3ψ3ξ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='4) We require F4 to be dS1-closed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' we will also use the fact that H3ξ is dS1-closed (which is automatic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We transgress this to the mapping space MX and choose an ideal whose vanishing locus describes, in degree 0, some embedding ı : N �→ MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' As explained in section 2, this depends on the choice of some wrapping map w : S1 → S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In fact, as stated, the final result only depends on the homotopy class of w and hence we can take, for some w ∈ Z, w : S1 −→ S1 σ �−→ wσ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='5) – 20 – To restrict to this wrapping sector of MX , we define a coisotropic ideal I = ⟨IN , IY⟩ with IY = ⟨y − wσ, ξ + w dσ⟩ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) As explained in section 2, this is coisotropic and closed under the Q-structure D = [ΘMX , ·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The ideal IN restricts all maps into N to closed maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' That is, we take IN ⊃ ⟨Pcox0, Pcoψ0, Pcoχ0, Pcop0⟩ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='7) For any coordinate zA k with deg zA − k ≤ 0, we need to further restrict to exact maps by including the harmonic representative in the ideal (except for x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Hence, we have IN = ⟨Pcox0, Pcoψ0, Pcoχ0, Pcop0, PHx1, PHψ1⟩ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='8) The ideal I = ⟨IN , IY⟩ is clearly coisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We need to check that IN is closed under D = [ΘMX , ·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The transgressed Hamiltonian is ΘMX = − � X ΘM − � X ıdϑM = − � X −ψp − ξq + 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F4(x, y)ψ4 + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H3(x, y)ψ3ξ − � X pdx + qdy − 1 3(ψdχ + 2χdψ + ξdφ + 2φdξ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='9) As in the previous cases, the only non-trivial constraint comes from the Poisson bracket between the first term and the harmonic generators of IN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We calculate (ΘM, x) = ψ , (ΘM, ψ) = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='10) and hence we have � ΘMX , � X ψϵ � ∝ � X (ΘM, ψ)M ϵ = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='11) � ΘMX , � X xϵ � ∝ � X (ΘM, x)M ϵ = � X ψϵ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='12) Evaluating this on C, we take ψ1 to be exact and so the integral vanishes when integrated over a constant ϵ = ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This shows that the Poisson brackets with the harmonic generators x1, ψ1 vanish when evaluated on C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' they are in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' To perform the coisotropic reduction we find the normaliser is generated by N(I) ∼ {PHx0, PHψ0, PHχ1, PHp1, I} (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='13) – 21 – and hence the structure sheaf is generated by by C∞(W) = N(I)/I ∼ {x0, ψ0, χ1, p1} ⇒ W = T ∗[2]T[1]N (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='14) where the coordinates represent harmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The symplectic form can be derived from the Poisson brackets on MX , as in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1, and we find ωW = −dp1 dx0 − dψ0 dχ1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='15) The Hamiltonian function is given by ΘW = Π(ΘMX ) = Π � − � X ΘM − � X ıdϑM � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='16) The second term vanishes when evaluated on harmonic maps where d annihilates the maps, except for the term qdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We also get a piece −ξq from the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We find Π �� X ξq − qdy � = � X −dy q − qdy = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='17) where we pick up a minus sign from commuting dy (degree 1) through q (degree 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This verifies the statement made in section 2 about this cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We then have ΘW = Π �� T[1]S1 ψp − 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F4(x, y)ψ4 − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H3(x, y)ψ3ξ � = � X ψ0p1 + − 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F4(x0, wσ)ψ4 0 + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H3(x0, wσ)ψ3 0w dσ = 1 2π � X � ψ0p1 + w 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H3(x0, wσ)ψ3 0 � dσ = ψ0p1 + w 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ˜H3ψ3 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='18) where ˜H3 is the average of H3 over the fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Under the change of coordinates p1 → −p1, we see that we recover the QP manifold associated to the F1 string with w units of ˜H3 flux, as we would expect from our intuition of M-theory/IIA duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Note that in the case that w = 0, the physical interpretation seems to break down - we find a string which doesn’t couple to the NS 3-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' However, as is noted in [18], this zero winding case corresponds to a scenario in which the original worldvolume is “collapsed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This means that the map from the worldvolume to the target space is not an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' From the IIA perspective, the resulting string is tensionless and thus the M2 brane must somehow be tensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We should discard that case on account of such objects appear not to exist on physical grounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' nevertheless, the QP procedure is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 22 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2 M5 on S1 The next case of interest is wrapping the M5 QP manifold on a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The M5 QP manifold was written down in [1] and our expectation is that we should recover that of the D4 brane [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We start with the following manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' M = T ∗[6]T[1](N × S1) × R[3] X = S1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='19) Writing N = T ∗[6]T[1]N × R[3], Y = T ∗[6]T[1]S1 and X = T[1]S1, we introduce the homogeneous coordinates N Y coord xµ ψµ ζ χµ pµ deg 0 1 3 5 6 coord y ξ φ q deg 0 1 5 6 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='20) and use coordinates (σ, dσ) for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We write the symplectic form and Hamiltonian function as ωM = dp dx + dq dy − dψ dχ − dξ dφ − 1 2dζ dζ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='21) ΘM = −ψp − ξq + 1 7!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (H7 + A ∧ F6)ψ7 + 1 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F6ψ6ξ + 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (F4 − A ∧ H3)ψ4ζ + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H3ψ3ξζ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='22) We included, in this example, a non-trivial connection on the fibre bundle N × S1 which we will assume to be S1 invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' As previously, we can interpret the coefficients to be elements of Ωi(N) × Ωj(S1) and we require that they are closed under the dS1 on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We then transgress the structure to the mapping space MX and aim to define a suitable ideal I = ⟨IN , IY⟩ to perform the coisotropic reduction with respect to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The ideal IY is taken as in the previous section IY = ⟨y − wσ, ξ + w dσ⟩ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='23) The ideal IN is also taken as in the previous section, but now with the additional constraints on the ζ coordinates, restricting them to closed maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' That is, we take IN = ⟨Pcox0, Pcoψ0, Pcoζ0, Pcoχ0, Pcop0, PHx1, PHψ1⟩ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='24) Since we have only added co-exact generators to the ideal, the proof of coisotropy and closure under D goes exactly as in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Performing the coisotropic reduction, we find the structure sheaf is generated by C∞(W) = N(I)/I ∼ {x0, ψ0, ζ0, ζ1, χ1, p1} (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='25) which gives W = T ∗[5]T[1]N × R[2] × R[3] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='26) – 23 – To find the symplectic form, we use the Poisson brackets on MX given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) with appropriate insertions of harmonic test functions ϵ, η, and find ωW = −dp1 dx0 − dψ0 dχ1 − dζ1 dζ0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='27) and the Hamiltonian function is given by7 ΘW = Π(ΘMX ) = Π �� T[1]S1 ψp − 1 7!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (H7 + A ∧ F6)ψ7 − 1 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F6ψ6ξ − 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (F4 − A ∧ H3)ψ4ζ − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H3ψ3ξζ � = ψ0p1 + w 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ˜F6ψ6 0 − 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ( ˜F4 − A ∧ ˜H3)ψ4 0ζ1 − w 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ˜H3ψ3 0ζ0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='28) where the tilde denotes the average over the S1 fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' For w ̸= 0, we perform a canonical transformation generated by the function − 1 2wAψζ2 1 to obtain the Hamiltonian ΘW = ψ0p1 + 1 4wF2ψ2 0ζ2 1 − w 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ˜H3ψ3 0ζ0 − 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ˜F4ψ4 0ζ1 + w 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ˜F6ψ6 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='29) where F2 = dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Making the change of coordinates p1 → −p1, ζi → −ζi puts the QP manifold in the canonical form of that associated to the D4 brane [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3 M5 on X4 The next example will be to wrap the M5 brane over a 4-manifold X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In [22, 23] it was shown that one could reproduce the non-critical heterotic string through such a reduction, where the dimension of the gauge group was related to the cohomology of the wrapping manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will start with the manifolds M = T ∗[6]T[1](N × X4) × R[3] X = X4 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='30) Writing N = T ∗[6]T[1]N ×R[3], Y = T ∗[6]T[1]X4, X = T[1]X4, we introduce the homogeneous coordinates as in the previous section N Y coord xµ ψµ ζ χµ pµ deg 0 1 3 5 6 coord ym ξm φm qm deg 0 1 5 6 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='31) 7We are using the fact that the ıdϑM term vanishes, apart from the qdy term, which cancels against the ξq term in ΘM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 24 – where now α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=', 4, and we use the DG coordinates (σα, dσα) on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In these coordinates the symplectic form and Hamiltonian function take the form ωM = dp dx + dq dy − dψ dχ − dξ dφ − 1 2dζ dζ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='32) ΘM = −ψp − ξq + 1 7!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H7ψ7 + 1 6!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H6ψ6ξ + 1 2 1 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H5ψ5ξ2 + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H4ψ4ξ3 + 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H3ψ3ξ4 + 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F4ψ4ζ + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F3ψ3ξζ + 1 2 1 2F2ψ2ξ2ζ + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F1ψξ3ζ + 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F0ξ4ζ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='33) where we have taken a trivial connection on the X4 bundle again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' As before, we can view the coefficients as differential forms on Y valued in Ωk(N) that we take to be dY -closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We transgress this structure to MX and define a coisotropic ideal I = ⟨IN , IY⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' To define the ideal IY we need to choose some wrapping map w : X4 → X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Restriction to this winding sector of MX is given by IY = ⟨y − w, ξ + dw⟩ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='34) It is easy to verify that this is coisotropic and closed under D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The ideal IN is similar to that for the circle reduction done in the previous section, except now our transgressed coordinates are k-forms8 for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This means that we need to include more co-exact generators and harmonic generators to remove unwanted coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We take IN = ⟨Pcoxk, Pcoψk, Pcoζk, Pcoχk, Pcopk, PHxi, PHψi, PHζj | i > 0, j > 2⟩ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='35) We need to check whether this is closed under D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' As in previous cases, the only non-trivial checks come from the harmonic generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The Q-structure D acting on the harmonic generators xi, ψi return an element of IN precisely as in previous cases so we need only check the closure of Dζ3, Dζ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Once again, this can be done by calculating (ΘM, ζ)M = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F4ψ4 + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F3ψ3ξ + 1 4F2ψ2ξ2 + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F1ψξ3 + 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='F0ξ4 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='36) We then transgress this function to MX and evaluate it on the vanishing locus C of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We then check whether the following vanishes � X (ΘM, ζ)M ϵ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='37) for suitable harmonic test functions ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' To determine the conditions coming from ζ4, we take ϵ = ϵ0 a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We then get the constraint � X w∗(F0) ϵ0 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='= 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='38) 8As noted in section 2, the transgressed coordinates zA k for k ≥ 2 should be viewed as differential forms evaluated in some affine bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Our construction is still well-defined so for simplicity we will ignore this subtlety here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 25 – where we are using the fact that F0 is a 4-form on Y which we pull back to X via the wrapping map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Similarly, the conditions coming from ζ3 are given by choosing an arbitrary harmonic 1-form ϵ = ϵ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' � X w∗(F1) ∧ ϵ1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='= 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='39) This puts constraints on the coefficients F0, F1 which can be most easily satisfied if they vanish, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' they act as obstructions to the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Note that in some cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' for X = K3, there are no non-trivial harmonic 1-forms and so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='39) gives no constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Assuming these constraints are satisfied, the coisotropic reduction with respect to I = ⟨IN , IY⟩ gives that the structure sheaf is generated by C∞(W) = N(I)/I ∼ {x0, ψ0, ζa 2, χ4, p4} ⇒ W = T ∗[2]T[1]N × H2(X)[1] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='40) We have introduced an index a parameterising a basis {ea} of H2(X4), and have expanded ζ2 ∈ H2 as ζa 2ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Using the Poisson brackets on MX , we get the symplectic form and the Hamiltonian function on W to be ωW = dp4 dx0 − dψ0 dχ4 − 1 2κabdζa 2 dζb 2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='41) ΘW = Π(ΘMX ) = −ψ0p4 + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ˜H3ψ3 + 1 2 ˜Faψ2ζa 2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='42) where ˜H3 = � X w∗(H3) , ˜Fa = � X w(F2) ∧ ea , κab = � X ea ∧ eb (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='43) We get the canonical form of the QP manifold associated to a heterotic string with abelian gauge group of dimension b2(X4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The Killing form on the gauge group is also given by the symmetric form κab on H2(X4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' For example, if X4 = T 4, we get an abelian gauge group of dimension b2(T 4) with Killing form of signature (3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' If X4 = K3, then we get a gauge group of dimension b2(K3) = 22 with Killing form of signature (3, 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This matches the results of [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The fact that we can only obtain abelian gauge groups arises because we are assuming that we are reducing on smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Degenerations of X4 to some singular space should lead to gauge enhancement and non-abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 6 AKSZ sigma models and brane wrapping In previous sections we obtained an NQP manifold W from a coisotropic reduction of the mapping space MX with respect to a coisotropic submanifold C that is invariant with respect to the Q-structure D = QM + dX on MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (In the expression for D we have the lifts of vector fields on the target and source to the mapping space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') We will now point out that these data give rise — essentially trivially!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' — to a reduction of AKSZ sigma models from an AKSZ model with target M to an AKSZ model with target W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 26 – We start with the BV manifold of an AKSZ sigma model with target M where the source takes the form X × S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The N-manifold S is taken to be T[1]S where the (bosonic) manifold S has dimension dim S = n + 1 − dim X (n being the degree of the target P-structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Then the BV master action is the hamiltonian corresponding to the Q-structure on MX×S given by QBV ≡ QM + dX×S (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1) where again QM denotes the lift to MX×S of the target space M Q-structure of the same name, and dX×S is the lift of the source X × S de Rham differential again to MX×S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Since the source is a product we can write dX×S = dX + dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The key point that leads to reduction is that we can write MX×S = (MX )S (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2) which is known as the product-exponential adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Explicitly, this corresponds to inter- preting a function f ∈ MX×S, which is a function f(x, s) of two arguments, as a function s → f(•, s) where f(•, s) is a function of x ∈ X for each s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='9 Since MX is a QP-manifold and S is an NQ-manifold with an integral measure we can consider the BV structure on MX×S as arising from an AKSZ construction with source S and target MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' If C is coisotropic in MX , then the mapping space CS will be a coisotropic submanifold in (MX )S ∼= MX×S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The reduced AKSZ sigma model will be given by the coisotropic reduction of MX×S with respect to CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We need to confirm that CS is invariant with respect to QBV, so that the BV master action reduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We rewrite QBV as QBV = (QM + dX ) + dS = ˆD + ¯dS (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3) where in the last formula ˆD is the lift from MX to (MX )S of the vector field D on MX , while ¯dS is the lift of dS from S to (MX )S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We denote these lifts explicitly now because it is the properties of these lifts that guarantee the reduction: if VS is any vector field on S, then the lift ¯VS always leaves CS invariant (for any submanifold C of MX );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' CS is invariant for ˆD if C is invariant for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Therefore QBV gives rise to a homological (and hamiltonian) vector field on the coisotropic reduction of MX×S, which is simply WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Using the results of appendix C we find that the new BV master action is given by evaluating the original action on CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In all examples we have investigated the result is another topological field theory of AKSZ type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In summary, the brane wrapping of QP manifolds that we already discussed always leads to a brane wrapping procedure that takes the BV master action associated to an AKSZ topological field theory and produces the BV master action of another topological field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 9The definition of mapping spaces for graded manifolds is such that this property is true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 27 – 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1 AKSZ 3-brane to membrane example To illustrate, we will treat the reduction of the AKSZ sigma model corresponding to the wrapping of an M2 algebroid on a circle that we discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is a reduction of the 4D topological field theory of Ikeda and Uchino [25] to a (3D) Courant sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This example thus has X = T[1]S1 and the coisotropic submanifold C ⊂ MX is given by dX p0 = 0 , dX xµ 0 = 0 , PHxµ 1 = Pcoxµ 1 = 0 , y0 = wσ , y1 = 0 dX χ0 = 0 , dX ψµ 0 = 0 , PHψµ 1 = Pcoψµ 1 = 0 , ξ0 = 0 , ξ1 = −w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='4) We have used the superfield expansion of Z ¯ A = {xµ, y, ψµ, ξ, · · · } in form degree (so x(σ, dσ) = x0(σ) + x1(σ)dσ etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') For the original (4D) AKSZ theory degree-counting to work we set S = T[1]S where S can be any 3-manifold, so that X × S = S1 × S is the four-dimensional worldvolume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Using the product-exponential adjunction to write MX×S ∼= (MX )S amounts to promoting the components Z ¯ A k of the superfields Z ¯ A defining a map MX to superfields Z ¯ A k that now depend on the S coordinates {s, ds} as well as the X coordinates ({σ, dσ} in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Then the coisotropic submanifold CS is the locus of functions S → MX such that dX p0 = 0 , dX xµ 0 = 0 , PHxµ 1 = Pcoxµ 1 = 0 , y0 = wσ , y1 = 0 dX χ0 = 0 , dX ψµ 0 = 0 , PHψµ 1 = Pcoψµ 1 = 0 , ξ0 = 0 , ξ1 = −w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='5) where all bolded expressions depend on {σ, s, ds}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (The projectors to co-exact/harmonic pieces refer to the Hodge decomposition with respect to X as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') We can explicitly check the claim that CS is invariant with respect to QBV = ˆD + ¯dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ˆD � S×X (y − wσ)ϵ = � S×X (ξ + dσ∂σy)ϵ mod I(CS) = � S×X (−wdσ + dσw)ϵ = 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) (We smeared against ϵ ∈ C∞(S ×X) and employed (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='19)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The other differential ¯dS leaves the ideal invariant independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This way we may confirm explicitly that SBV lies in N(I(CS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' It remains to calculate the reduced BV master action, which amounts to calculating Π(SBV) where Π implements the quotient modulo I(CS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' SBV is the hamiltonian for QBV = D + dS = QM + dX + dS ≡ QM + d which is explicitly given by formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='10), which is a linear combination of � X×S ΘM and � X×S ιdϑM, for ϑM the transgression of a symplectic potential on M that satisfies dMϑM = ωM, ωM being given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The bolded quantities are superfields corresponding to MX×S now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We then calculate Π � X×S ιdϑM = Π � X×S pdx + qdy − χdψ − φdξ = � S ( � X p1dσ)dSx0 + w( � X q0dσ) − ( � X χ1dσ)dSψ0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='7) – 28 – Note that terms involving x1, ψ1 will generate dX -exact terms which will vanish under the � X integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='4), Π � X×S ΘM = � S −ψ0( � X p1dσ) − w( � X dσq0) + 0 − w( � X 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H3(ψ0)3dσ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='8) We then read off the sign factors from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='10) to find ΠSBV = Π � − � X×S ΘM + � X×S ιdϑM � = � S ψ0( � X p1dσ) + w( � X 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='H3(ψ0)3dσ) + ( � X p1dσ)dSx0 − ( � X χ1dσ)dSψ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='9) The signs were such that the terms w � X q0dσ cancelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In the above expression we can identify the integrated expressions ( � X p1dσ) , ( � X χ1dσ) as the conjugate momenta superfields (with degrees 2, 1 respectively) that appear in the Courant sigma model for an exact Courant algebroid structure defined by the 3-form wH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The result we calculated via coisotropic reduction of the original (4-dimensional) AKSZ topological sigma model is identical to the AKSZ sigma model constructed directly from the wrapped QP manifold W with source manifold S (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Therefore we have recovered the correct relation between the M-theory fluxes, the M2- brane winding w, and the type IIA NS-flux wH3 seen by the fundamental strings that arise as the M-theory circle X = S1 is shrunk to zero, all at the level of the corresponding topological sigma models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' 7 Conclusions We defined a reduction procedure of NQP manifolds M → W which encompasses the properties of wrapped branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is consistent with the AKSZ procedure in the sense that the reduction naturally lifts to a reduction of the AKSZ theory with target M to the AKSZ theory with target W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We applied this to many examples, including many physically motivated examples of wrapped branes and we saw that it reproduced the known M-theory/IIA dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We also were able to find a novel relation between the Courant algebroid and the Poisson algebroid through this reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We expect that our work will have many interesting applications to other topological AKSZ theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' One can ask how general our procedure is, or whether it is possible to relax some of the assumptions made in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' For example, can we relax the trivial bundle condition M = N × Y , perhaps by introducing some flat connection similar to [11]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We can also ask whether we can extend our construction to manifolds X with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We can also relax the constraint on X = T[1]X, and instead just take X to be some DG manifold with some invariant measure of degree n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' For example, we can try to extend the reduction procedure – 29 – to X = T 1,0[1]X for some complex manifold X with dimC X = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We could then apply the reduction to, say, the work of [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2, we found an interesting relation between the Courant algebroid and the Poisson algebroid QP structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This was based on the embedding of the Poisson differential dπ into T ⊕ T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' There are other interesting differentials that can appear in these Courant algebroids [27] that are associated to topological theories on G2 and Spin(7) manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' One can try to embed these differentials in the language of QP structures and perform the reduction to get new topological models associated to these special holonomy manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' There are also similar structures that appear in higher algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' That is, one can define the notion of a Dirac structure for these higher algebroids and define the associated differential [2, 28–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' These can be embedded into the Q-structure of the QP manifolds associated to these higher algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Their reductions may provide further insight into supersymmetric geometries of string/M-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Acknowledgements We would like to thank Marco Zambon, Chris Blair, Dan Thompson, and Ondrej Hulik for very helpful discussions during this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ASA is supported by the FWO-Vlaanderen through the project G006119N, as well as by the Vrije Universiteit Brussel through the Strategic Research Program “High-Energy Physics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' He is also supported by an FWO Senior Postdoctoral Fellowship (number 1265122N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' DT is supported by the NSF grant PHYS-2112859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Part of the research for this project was performed while DT was supported by the EPSRC New Horizons Grant “New geometry from string dualities” EP/V049089/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' A Notation Commutative Manifolds M Starting/parent commutative manifold which is always a product manifold of a base and a fibre to be wrapped N The commutative manifold which is the base of the trivial fibre bundle M Y The fibre of the trivial bundle M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is the manifold over which we wrap the branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' X The fibre of the brane that is wrapped over Y – 30 – Non-commutative Manifolds M Starting/parent QP manifold N A submanifold of M which is the natural QP manifold restricted to the base of the fibration Y A submanifold of M which is the natural QP manifold restricted to the fibre;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' usually Y = T ⋆[n]T[1]Y X The shifted tangent bundle T[1]X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' the source of the mapping space MX W Final wrapped QP manifold MX maps(X → M) S A DG manifold with invariant measure of degree n + 1 − dim X Indices A, B, C, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Indices along M, N µ, ν, ρ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Indices along N m, n, p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Indices along Y α, β, γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Indices along X r, s, t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Indices correspnding to degree shifted real lines R[nr] a, b, c, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Indices for a basis of differential forms on X Coordinates ZA Homogeneous coordinates on M zA Homogeneous coordinates on N xµ Degree 0 coordinates on N ψµ Degree 1 coordinates on N parameterising the fibre of T[1]N pµ Coordinate dual to xµ χµ Coordinate dual to ψµ ym Degree 0 coordinates on Y parameterising the fibre of T[1]Y ξm Degree 1 coordinates on Y qm Coordinate dual to yα φm Coordinate dual to ξα (σα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' dσα) Coordinates for the DG manifold (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' d) such that d(σα) = dσα ζr Homogeneous coordinates corresponding to degree shifted real lines R[nr] ZA Transgressed coordinates of MX ZA k An expansion of the transgressed coordinates ZA into differential k-forms ZA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='a k A coordinate labelling the harmonic k-forms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' labelled by a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' associated to the transgressed coordinate ZA – 31 – Functions and differential forms Ωk The space of differential k-forms Hk Harmonic k-forms ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='a A basis of harmonic k-form(s) (occasionally the k is dropped) ΘM The Hamiltonian function of M (similarly for N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') ωM The symplectic form of M (similarly for N, W, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') ϑM The canonical symplectic potential of M (similarly for N, W, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') (·, ·)M The Poisson bracket for M (similarly for N, W, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') [·, ·] The Poisson bracket on MX Miscellaneous w Wrapping map X → Y w Winding number/matrix of a circle/torus over itself I The coisotropic ideal within MX C The vanishing locus of I within MX B QP manifolds Graded manifolds A graded manifold M is a supermanifold whose coordinates come equipped with a Z grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='10 One can always find homogeneous coordinates ZA of definite degree, where deg ZA mod 2 is the Grassman parity of the coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will denote by A the degree of ZA and so we have ZAZB = (−1)ABZBZA (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1) The sheaf of functions on M splits into subsheafs C∞ n (M) of functions of definite degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The degree of a homogeneous function f is measured by the degree counting vector field ε (The ‘Euler vector field’) via ε(f) = deg(f)f (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2) In local homogeneous coordinates ZA, we have ε = � A deg(ZA)ZA ∂ ∂ZA (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3) Unless otherwise stated, all derivations are left derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Hence, the de Rham d is df = dZA∂Af (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='4) 10From [34], the consistency of the Z grading of coordinates comes from the existence of a global degree counting vector field ε and transition functions which preserve degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 32 – and any homogeneous (in degree) vector field X acts as X(fg) = X(f)g + (−1)XffX(g) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='5) where we have used the shorthand X, f for the degree of the respective components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' In local coordinates we can write X = X(Z)A∂A, and so deg X = deg XA − deg ZA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We also define deg(df) = deg f + 1 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='6) For this to be consistent with ıAdZB = δBA, where ıA denotes contraction with the vector field ∂A, we require that the interior product has degree deg ı = −1 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='7) Poisson and symplectic structures A graded Poisson structure of degree −n is defined to satisfy (f, g) = (−1)1+(f+n)(g+n)(g, f) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='8) and the graded Jacobi identity (f, (g, h)) = ((f, g), h) + (−1)(f+n)(g+n)(g, (f, h)) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='9) for all homogeneous functions f, g, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' It also acts as a left derivation on the right hand arguments, but a right derivation on the left hand arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' That is (f, gh) = (f, g)h + (−1)(f+n)gg(f, h) (fg, h) = f(g, h) + (−1)(h+n)g(f, h)g (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='10) If the Poisson structure is induced from a symplectic structure ω, we have that ıXf = (−1)fdf Xf := (f, ·) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='11) In local homogeneous coordinates we can write ω = 1 2dZAωABdZB (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='12) which implies the symmetry ωAB = (−1)1+AB+n(A+B)ωBA (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='13) – 33 – If we define ωAB via ωABωBC = δAC, then (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='11) implies (f, g) = (−1)f∂R Af ωAB ∂Bg (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='14) where ∂R A is defined by df = dZA∂Af = ∂R A dZA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Note that it is not a right derivation by itself, but the combination (−1)f∂R Af is a right derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This is consistent with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Note that this implies (ZA, ZB) = (−1)AωAB (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='15) The symplectic potential is defined such that dϑ = ω, and can be defined canonically through the Euler vector field ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We have that11 nω = Lεω = ıεdω + d(ıεω) = d(ıεω) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='16) where we have used dω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This implies we can take ϑ = 1 nıεω = (deg ZA)ZAωABdZB (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='17) Transgressed QP structure on MX Let (X = T[1]X, d) be a DG manifold with homogeneous coordinates σ, dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' A point f ∈ MX can be defined by how it pulls back the coordinates on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We have f∗ZA = ZA(σ, dσ) = ZA 0 (σ) + ZA 1 α(σ)dσα + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' + 1 d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='ZA d α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='αd(σ)dσα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='dσαd (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='18) We use the shorthand ZA k = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='ZA k α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='αk(σ)dσα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='dσαk, where ZA k α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='αk(σ) is a function of degree deg ZA − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' These act as coordinates on MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Our conventions are always that the form components come to the right of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' So, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' ZA 1 = ZA 1 α(σ)dσα = (−1)A−1dσαZA 1 α(σ) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='19) We can always define an evaluation map ev : MX × X −→ M (f, σ, dσ) �−→ f(σ, dσ) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='20) We also have the chain map defined by µ∗ : Ω•(MX × X) −→ Ω•(MX ) α �−→ � X α (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='21) 11More generally, the Lie derivative on any graded differential form along a vector field X is given by LX = ıXd + (−1)XdıX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The Euler vector field is degree 0, hence the expression given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 34 – The combination µ∗ev∗ : Ω•(M) → Ω•(MX ) is called the transgression map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The QP structure on the mapping space is defined by ωMX = µ∗ev∗ωM ΘMX = (−1)dµ∗ev∗ΘM + (−1)n+d+1ıdµ∗ev∗ϑ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='22) where we use the same symbol d for the lift of the vector field on X to MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This can be given more explicitly in the coordinates (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will use the bold face notation to denote a function, differential form, or coordinate on M that is pulled back to X via some function f ∈ MX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' That is, we effectively take f = ev∗f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We can then write ωMX = 1 2 � X δZA(ωM)AB δZB (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='23) Our conventions for integrals is that constants are pulled out from the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The symplectic form above gives rise to a Poisson bracket which takes the following form on homogeneous functionals F, G [F, G] = � X (−1)F δRF δZA (ωM)AB δG δZB (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='24) where δF = � X δZA δF δZA = � X δRF δZA δZA (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='25) We can define a functional F via some pulled-back function f by F(ϵ) = � X fϵ ∀ ϵ ∈ C∞(X) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='26) Then the Poisson bracket (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='24) can be expressed nicely as �� X fϵ, � X gη � = (−1)(f+n)ϵ+d � X (f, g)Mϵη (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='27) where (f, g)M = ev∗(f, g)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We can use this to calculate the Poisson bracket on two harmonic generators ZA k , ZB k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Let ek,a be a basis of harmonic k-forms and ˜eb d−k be a dual basis of harmonic d − k-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' So δba = � X ek,a ∧ ˜eb d−k (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='28) Noting that Ω•(X) ≃ C∞(T[1]X) = C∞(X), and by expanding ZA k = ZA,a k ek,a with ZA,a k some constant coefficient, we have ZA,a k = � X ZA k ˜ea d−k = � X ZA˜ea d−k (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='29) – 35 – We then see that we get an induced Poisson bracket on the coefficients given by � ZA,a k , ZB,b k′ � ≡ �� X ZA˜ea d−k, � X ZB˜eb d−k′ � = (−1)(A+n)(d−k)+d � X (ZA, ZB)M˜ea d−k ∧ ˜eb d−k′ = (−1)(A+n)(d−k)+d(−1)AωAB � X ˜ea d−k ∧ ˜eb d−k′ = (−1)(A+n)(d−k)+d(−1)AωABκabδk+k′,d (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='30) where we have assumed Darboux coordinates, so the ωAB are constant, and where κab = � X ˜ea d−k ∧ ˜eb k (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='31) We use this to find the symplectic form of the reduced theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' C Coisotropic reduction of graded Poisson algebras Let P be a graded algebra with a graded Poisson bracket [•, •] of degree −P along with a left derivation V of P, possibly hamiltonian (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' given by Poisson brackets, so V = [HV, •] for HV ∈ P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We will explain how all of these objects behave under coisotropic reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The derivation is as in the ungraded case considered originally by Weinstein and ´Sniatycki [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' If I is a (multiplicative, degree-homogeneous) ideal of P, it is a coisotrope if it is a Poisson subalgebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' [I, I] ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Then the coisotropic reduction of P with respect to I is the quotient P ≡ N(I)/I (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='1) where N(I) ≡ {f ∈ P|[f, I] ⊆ I} is the Poisson normaliser of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Then the bracket on P is defined in terms of the bracket [•, •] via [Πf, Πg] ¯P ≡ Π[f, g] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='2) where Πf is the equivalence class f + I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' For any P derivation V we define its reduction V via V(Πf) = ΠV(f) f ∈ P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='3) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Given any coisotrope I, the bracket [•, •]P is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' It is moreover a Poisson bracket of degree −P, and so P is a graded Poisson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' If the derivation V on P preserves the Poisson structure (V[f, g] = [Vf, g] ± [f, Vg]) and the coisotrope (V(I) ⊆ I) then the reduced derivation V is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' – 36 – Finally if V is furthermore hamiltonian with hamiltonian HV ∈ P (so V = [HV, •]) then V is hamiltonian with hamiltonian Π(HV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' (In this latter case V automatically preserves the Poisson structure, but the condition V(I) ⊆ I implies [HV, I] ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=') If all derivations we are interested in are in fact hamiltonian (which is the case in the main text) then we just need to check that the ideal I is a coisotrope and that [HV, I] ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' The bracket [•, •]P is well-defined because [Πf, Πg]P = Π[f + I, g + I] = Π([f, g] + [f, I] + [I, g] + [I, I]) = Π[f, g] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='4) where the last three terms in the second equality vanish because f, g ∈ N(I) and [I, I] ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' This new bracket inherits the antisymmetry and Jacobi identity properties from [•, •].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Since furthermore I is homogeneous in degree, Πf will have a well-defined degree, and so the new bracket defines a graded Poisson algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Similarly since V(f + I) = V(f) + V(I) we have that V is well-defined on P/I when V(I) ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' We then need to show that it preserves the subspace N(I)/I = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Since V preserves the Poisson bracket we have [Vf, I] = V[f, I] ± [f, VI] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content='5) If f ∈ N(I) this becomes V(I)±[f, VI] which lies in the coisotrope when V(I) ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Therefore V(f) lies in N(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfzQJ7/content/2301.02670v1.pdf'} +page_content=' Finally if V = [HV, •] then ΠV(Πf) = ΠV(f) = Π[HV, f] = [ΠHV, Πf]ΠP, which completes the proof.' 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Ferguson +,2 Gabriel Brammer +,4, 5 +Kartheik G. Iyer +,6, ∗ Larry D. Bradley +,2 Pratika Dayal +,7 Rogier A. Windhorst +,8 Adi Zitrin +,9 +Ashish Kumar Meena +,9 Masamune Oguri +,10, 11 Jose M. Diego +,12 Vasily Kokorev +,7 Paola Dimauro +,13 +Angela Adamo +,14 Christopher J. Conselice +,15 Brian Welch +,16, 17, 18 Eros Vanzella +,19 +Tiger Yu-Yang Hsiao +,1 Jinmi Yoon +,2, 20 Xinfeng Xu +,1 Namrata Roy +,1 and Celia R. Mulcahey +1 +1Center for Astrophysical Sciences, Department of Physics and Astronomy, The Johns Hopkins University, 3400 N Charles St. +Baltimore, MD 21218, USA +2Space Telescope Science Institute (STScI), 3700 San Martin Drive, Baltimore, MD 21218, USA +3Association of Universities for Research in Astronomy (AURA), Inc. for the European Space Agency (ESA) +4Cosmic Dawn Center (DAWN), Copenhagen, Denmark +5Niels Bohr Institute, University of Copenhagen, Jagtvej 128, Copenhagen, Denmark +6Columbia Astrophysics Laboratory, Columbia University, 550 West 120th Street, New York, NY 10027, USA +7Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands +8School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287-1404, USA +9Physics Department, Ben-Gurion University of the Negev, P.O. Box 653, Be’er-Sheva 84105, Israel +10Center for Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan +11Department of Physics, Graduate School of Science, Chiba University, 1-33 Yayoi-Cho, Inage-Ku, Chiba 263-8522, Japan +12Instituto de F´ısica de Cantabria (CSIC-UC). Avda. Los Castros s/n. 39005 Santander, Spain +13INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy +14Department of Astronomy, Oskar Klein Centre, Stockholm University, AlbaNova University Centre, SE-106 91 Stockholm, Sweden +15Jodrell Bank Centre for Astrophysics, University of Manchester, Oxford Road, Manchester UK +16Department of Astronomy, University of Maryland, College Park, MD 20742, USA +17Observational Cosmology Lab, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA +18Center for Research and Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771 +19INAF – OAS, Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129 Bologna, Italy +20Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements, USA +Submitted to ApJ +ABSTRACT +We study the spatially resolved stellar populations of 444 galaxies at 0.3 < z < 6.0 in two clusters +(WHL-0137-08 and MACS0647+70) and a blank field, combining imaging data from HST and JWST +to perform spatially resolved spectral energy distribution (SED) modeling using piXedfit. The high +spatial resolution of the imaging data combined with magnification from gravitational lensing in the +cluster fields allows us to resolve some galaxies to sub-kpc scales (for 109 of our galaxies). At redshifts +around cosmic noon and higher (2.5 ≲ z ≲ 6.0), we find mass doubling times to be independent of +radius, inferred from flat specific star formation rate (sSFR) radial profiles and similarities between +the half-mass and half-SFR radii. At lower redshifts (1.5 ≲ z ≲ 2.5), a significant fraction of our +star-forming galaxies show evidence for nuclear starbursts, inferred from centrally elevated sSFR, and +a much smaller half-SFR radius compared to the half-mass radius. At later epochs, we find more +galaxies suppress star formation in their center but are still actively forming stars in the disk. Overall, +these trends point toward a picture of inside-out galaxy growth consistent with theoretical models and +simulations. We also observe a tight relationship between the central mass surface density and global +Corresponding author: Abdurro’uf +fabdurr1@jhu.edu +arXiv:2301.02209v1 [astro-ph.GA] 5 Jan 2023 + +ID2 +stellar mass with ∼ 0.38 dex scatter. Our analysis demonstrates the potential of spatially resolved +SED analysis with JWST data. Future analysis with larger samples will be able to further explore the +assembly of galaxy mass and the growth of their structures. +Keywords: Galaxy evolution (594) — Galaxy formation (595) – Galaxy clusters (584) – Galaxy quench- +ing (2040) +1. INTRODUCTION +Over the last few decades, multiwavelength studies of +galaxies throughout cosmic history reveal that the global +star formation rate density (SFRD) in the universe was +increasing with cosmic time from the reionization epoch +and reached a peak at z ∼ 2 (∼ 3.5 Gyr after the Big +Bang; cosmic noon) after which it declined exponentially +toward the present day (Madau & Dickinson 2014). In +this picture, it is estimated that ∼25% of the present- +day stellar mass density (SMD) was formed before the +peak of the cosmic SFRD, around half of the SMD was +formed during 0.7 < z < 2.0, and another ∼25% was +formed since z = 0.7 (i.e., around the last half of the +universe’s age; Madau & Dickinson 2014). Although the +cosmic SFRD at early cosmic time is still debated due +to the dust obscuration effects (see e.g., Fudamoto et al. +2021; Casey et al. 2018), an emerging picture is that +cosmic SMD increases with cosmic time since the epoch +of reionization, which is believed to take place before +z ∼ 6 (e.g., Treu et al. 2013; McGreer et al. 2015; Dayal +& Ferrara 2018). +Observations also revealed that most of the star for- +mation occurs in galaxies that lie in the so-called star- +forming main sequence (SFMS), which is a tight nearly +linear correlation between the integrated (i.e., global) +star formation rate (SFR) and stellar mass (M∗; Brinch- +mann et al. 2004; Daddi et al. 2007; Elbaz et al. 2007; +Noeske et al. 2007; Whitaker et al. 2012, 2014; Speagle +et al. 2014; Salmon et al. 2015; Tomczak et al. 2016; +Santini et al. 2017; Iyer et al. 2018; Leja et al. 2022). +This relation has been shown to hold at any epoch with +a nearly constant scatter (∼ 0.3 dex; Whitaker et al. +2012; Speagle et al. 2014), suggesting that galaxies grow +in mass over cosmic time in a state of self-regulated semi- +equilibrium (e.g., Bouch´e et al. 2010; Daddi et al. 2010; +Genzel et al. 2010; Tacchella et al. 2016b, 2020). Un- +derstanding this process in detail as well as the mech- +anisms that shut down star formation in galaxies and +move them out of the SFMS onto the “quenched” popu- +lation requires knowledge of not only integrated galaxy +∗ Hubble Fellow +properties but also spatially resolved structures within +galaxies. +The study of spatially resolved properties of galax- +ies with integral field spectroscopy (IFS) and high spa- +tial resolution imaging have been providing important +insights toward a better understanding of galaxy evo- +lution. +Among the important findings is the realiza- +tion that some of the well-known scaling relations ob- +served on global scales are originated from similar re- +lations on kpc scales within galaxies (see a review by +S´anchez 2020). This includes the spatially resolved star- +forming main sequence relation, a relationship between +SFR surface density (ΣSFR) and M∗ surface density +(Σ∗), which is thought to be more fundamental than +the global SFMS (e.g., S´anchez et al. 2013; Wuyts et al. +2013; Cano-D´ıaz et al. 2016; Abdurro’uf & Akiyama +2017; Hsieh et al. 2017; Abdurro’uf & Akiyama 2018; +Lin et al. 2019; Morselli et al. 2020; Abdurro’uf et al. +2022b). This emphasizes the necessity of studying the +spatially resolved properties of galaxies. +Spatially resolved studies of high redshift galaxies +(z ∼ 1 − 4) have hinted on how galaxies assembled +their structures. The emerging picture from these stud- +ies is that galaxies grow their mass in an inside-to- +outside manner (i.e., inside-out growth scenario; e.g., +van Dokkum et al. 2013; Nelson et al. 2012; Morishita +et al. 2015; Nelson et al. 2016) and cease their star for- +mation activities in a similar manner (i.e., inside-out +quenching scenario; e.g., Tacchella et al. 2015; Jung +et al. 2017; Abdurro’uf & Akiyama 2018; Tacchella et al. +2018). Nelson et al. (2016) analyzed the spatially re- +solved distributions (on kpc scales) of Hα emission and +M∗ of 0.7 < z < 1.5 galaxies using the Hubble Space +Telescope (HST)/WFC3 grism data from the 3D-HST +survey (Skelton et al. 2014). They found that the spa- +tial distribution of Hα emission in the galaxies is more +extended than the stellar mass, suggesting that the past +star formation in the galaxies have accumulated stellar +mass in the center and now the star formation progresses +outward to assemble the disk. Tacchella et al. (2015) an- +alyzed the spatial distributions of SFR and M∗ of ∼ 30 +star-forming galaxies at z ∼ 2 using IFS data from the +SINS/zC-SINF survey (F¨orster Schreiber et al. 2018). +They observed that massive galaxies (M∗ ≳ 1011M⊙) + +3 +in their sample have a centrally-suppressed specific SFR +(sSFR) radial profile and a massive central spheroid that +is as dense as the centers of local early-type galaxies. In +contrast to this, less massive galaxies in their sample +have broadly flat sSFR radial profiles. This trend in- +dicates that massive galaxies at this epoch might have +started a quenching process in their central regions and +assembled a mature bulge. +The buildup of the central stellar mass density is +likely correlated with the quenching process in galaxies. +The central stellar mass density within a 1 kpc radius +(Σ∗,1kpc) has been shown to be a good predictor for qui- +escence, where galaxies with high Σ∗,1kpc are tend to +be red and quiescent, whereas galaxies with low Σ∗,1kpc +are tend to be blue and star-forming (e.g., Fang et al. +2013; Tacchella et al. 2015, 2016a; Barro et al. 2017; +Jung et al. 2017; Whitaker et al. 2017). It has also been +shown that Σ∗,1kpc is tightly correlated with the global +M∗, suggesting that M∗ of galaxies grow hand-in-hand +with the central mass density. In this Σ∗,1kpc–M∗ rela- +tion, quiescent galaxies reside in a sequence at the tip +of the overall relation and have a shallower slope than +the relation with star-forming galaxies only, indicating +a formation of a matured bulge in the quiescent galax- +ies (Fang et al. 2013; Tacchella et al. 2015; Barro et al. +2017). +Galaxies also grow their sizes hand-in-hand with the +global M∗, as indicated by the size–mass relation (e.g., +Shen et al. 2003; van der Wel et al. 2014; Suess et al. +2019). Previous studies have shown that star-forming +and quiescent galaxies follow very different size–mass +relations where quiescent galaxies tend to be more com- +pact (i.e., having smaller size) in all M∗ and exhibit +steeper relation than the star-forming galaxies (van der +Wel et al. 2014; Yang et al. 2021). A possible explana- +tion for this trend is that star-forming galaxies build +their mass at all radii by mostly in-situ star forma- +tion, whereas quiescent galaxies grow inside-out through +mergers (e.g., van Dokkum et al. 2015). +Previous studies, some of which are mentioned above, +have used HST for resolving galaxies out to z ∼ 3, +roughly a limit where galaxies can be resolved well by +the telescope, given its spatial resolution and depth. +Furthermore, the wavelength coverage of HST only cov- +ers the rest-frame ultraviolet (UV) and a small portion of +the optical at z ∼ 3, making it difficult to robustly derive +the stellar mass as well as the other stellar population +properties, which typically requires a rest-frame near- +infrared (NIR). Forcing to include NIR imaging from +the ground-based telescopes would need to sacrifice the +spatial resolution of HST (e.g., Jung et al. 2017). With +the advent of the James Webb Space Telescope (JWST) +NIRCam observations (Rigby et al. 2022), with its high +spatial resolution, depth, and its coverage in NIR, now +we can push the analysis of spatially resolved SED of +galaxies to higher redshifts. Some very recent studies +have used JWST/NIRCam imaging to study the inter- +nal structures and morphology of galaxies at z > 3 (e.g., +Ferreira et al. 2022; Chen et al. 2022; Kartaltepe et al. +2022; Gim´enez-Arteaga et al. 2022), and even resolving +a lensed galaxy at z ∼ 11 (Hsiao et al. 2022). +In this paper, we use imaging data from HST/ACS +and JWST/NIRCam to analyze the spatially resolved +SEDs of 0.3 < z < 6.0 galaxies in the sightlines of +WHLJ013719.8–082841 (hereafter WHL0137−08; RA = +01:37:25.0, DEC = −08:27:23, J2000; z = 0.566; Wen +et al. 2012; Wen & Han 2015) and MACSJ0647.7+7015 +(hereafter MACS0647+70; RA = 06:47:50.03, DEC = ++70:14:49.7, J2000; z = 0.591; Ebeling et al. 2007) clus- +ters and examine the spatial distributions of their stellar +populations. Our main goal is to get hints on the as- +sembly of galaxy structures over cosmic time, especially +how galaxies build their stellar masses and quench their +star formation activities. +The high spatial resolution +of JWST/NIRCam combined with magnification from +gravitational lensing in the cluster fields, allow us to re- +solve high-redshift galaxies down to sub-kpc scales. Our +method using piXedfit (Abdurro’uf et al. 2022c) can +simultaneously process imaging data, perform pixel bin- +ning to optimize the signal-to-noise (S/N) ratio of the +spatially resolved SEDs, and perform SED fitting. The +wavelength coverage of HST/ACS and JWST/NIRCam +allow us to get full coverage of the rest-frame UV to +NIR for the majority of our sample, which can give a +strong constraint on model SEDs and break the age– +dust–metallicity degeneracy (see Appendix B). While +IFS observation at z ≳ 2 is lacking, our analysis in +this paper provides a good alternative for the analysis +of spatially resolved SED of high redshift galaxies us- +ing JWST/NIRCam imaging data. Our analysis in this +paper is one of the first robust spatially resolved SED +analyses of hundreds of galaxies using JWST data. Ab- +durro’uf et al. (2021) have demonstrated the capabil- +ities of spatially resolved SED fitting using piXedfit +on local galaxies. In particular, it gives robust SFR on +kpc scales when rest-frame UV to NIR photometry is +available, which is consistent with the SFR derived from +Hα emission maps (dust-corrected based on the Balmer +decrement) from the MaNGA IFS survey (Bundy et al. +2015). +The paper is organized as follows. In Section 2, we +present the data and sample galaxies. We describe the +spatially resolved SED fitting methodology in Section 3 +and present our results in Section 4, which include the + +4 +radial profiles of some key stellar population properties, +comparison between the compactness of the spatial dis- +tributions of SFR and M∗, and Σ∗,1kpc–M∗ relation. In +Section 5, we further discuss our results, focusing on the +evolutionary trends with redshift and what the implica- +tions to the study of galaxy evolution. +Throughout this paper, we assume the Chabrier +(2003) initial mass function (IMF) with a mass range of +0.1 − 100M⊙ and cosmological parameters of Ωm = 0.3, +ΩΛ = 0.7, and H0 = 70 km s−1 Mpc−1. +2. DATA AND SAMPLE +2.1. Observational Data +2.1.1. JWST Observations +We +obtain +JWST/NIRCam +imaging +data +of +WHL0137−08 cluster from Cycle 1 General Observers +(GO) 2282 program (PI Coe) and MACS0647+70 +cluster from GO 1433 program (PI Coe). +The +WHL0137−08 cluster was observed in July 2022, while +the MACS0647+70 cluster was observed in 23 Septem- +ber 2022. The GO 2282 program aims at further inves- +tigating Earendel (Welch et al. 2022a,b) and the Sun- +rise Arc (Vanzella et al. 2022). The JWST/NIRCam +data from this program consist of eight filters (F090W, +F115W, F150W, F200W, F277W, F365W, F410M, and +F444W) spanning a wavelength range of 0.8 − 5.0 µm. +The GO 1433 program is intended to observe the +triply-lensed galaxy MACS0647–JD at z ∼ 11 (Coe +et al. 2013; Hsiao et al. 2022). +This program ob- +tained JWST/NIRCam imaging in six filters (F115W, +F150W, F200W, F277W, F365W, and F444W) span- +ning 1 − 5 µm. The exposure time of each filter in the +two programs is 2104 seconds. It achieves 5σ limiting +AB magnitude of 28.0 to 29.0 in a r = 0.′′2 diameter +circular aperture. +For each filter, we obtained four dithers using IN- +TRAMODULEBOX primary dithers to cover the 4−5′′ +gap between the sort wavelength (SW; λ < 2.4µm) de- +tectors, improve the spatial resolution of final drizzled +images, and minimize the impact of image artifacts and +bad pixels. +We obtained NIRCam imaging over two +2.′26×2.′26 fields separated by 40.′5, covering a total area +of 10.2 arcmin2. +In the observation of WHL0137−08 +cluster, the NIRCam module B was centered at the clus- +ter while the module A covered a nearby field centered +∼ 2.′9 from the cluster center (hereafter called “blank +field”). On the other hand, the MACS0647+70 cluster +was centered at the module A and the module B ob- +serves a blank field nearby to it. +2.1.2. HST Data +We obtain HST imaging data of the WHL0137−08 +cluster from the Reionization Lensing Cluster Survey +(RELICS) HST Treasury program (GO 14096; Coe +et al. 2019). THe RELICS program obtained the first +HST imaging of the WHL0137−08 cluster in 2016 with +three orbits of ACS (F435W, F606W, and F814W) and +two orbits of WFC3/IR (F105W, F125W, F140W, and +F160W) data spanning 0.4 − 1.7 µm. +Two follow-up +HST imaging programs (GO 15842 and GO 16668; PI: +Coe) have thus far obtained an additional 5 orbits of +HST ACS imaging in F814W, 2 orbits in F475W, and +4 orbits with WFC3/IR in F110W. The HST imaging +data only cover the WHL0137−08 cluster field. There- +fore, we do not have HST imaging data for the blank +field. +The HST imaging data of the MACS0647+70 clus- +ter +are +taken +from +multiple +programs. +Overall, +MACS0647+70 has been observed in total of 39 or- +bits of HST imaging in 17 filters. +The cluster was +first observed by programs GO 9722 (PI Ebeling) and +GO 10493, 10793 (PI Gal-Yam) in the ACS F555W +and F814W filters. +Then additional imaging in 15 +filters (WFC3/UVIS, ACS, and WFC3/IR, spanning +0.2 − 1.7 µm ) was obtained by the Cluster Lensing and +Supernova Survey with Hubble (CLASH; Postman et al. +2012; GO 12101, PI Postman). Finally, additional imag- +ing in WFC3/IR F140W was obtained as part of a grism +spectroscopy program (GO 13317, PI Coe). +It is important to note that the nearby blank fields +to the WHL0137−08 and MACS0647+70 clusters that +are observed with NIRCam are not covered in the HST +observations described above. +In this work, we an- +alyze galaxies in three fields: +WHL0137−08 cluster +field, MACS0647+70 cluster field, and the NIRCam +blank field nearby to the WHL0137−08 (hereafter sim- +ply called blank field). We do not analyze galaxies in the +NIRCam blank field of MACS0647+70 because it is ob- +served in less number of NIRCam filters than the blank +field of WHL0137−08 and it does not have F090W ob- +servation, which prevents us from selecting galaxies at +z < 2 in this field as their photometry do not cover the +rest-frame 4000 ˚A break. For the WHL0137−08 and the +blank field, we use 4 HST/ACS filters (F435W, F475W, +F606W, and F814W) and 8 JWST/NIRCam filters, +whereas for the MACS0647+70, we use 7 HST/ACS fil- +ters (F435W, F475W, F555W, F606W, F625W, F775W, +and F814W) and six JWST/NIRCam filters. We do not +use HST/WFC3 IR filters to get high spatial resolution +possible while still get sufficiently wide wavelength cov- +erage with the HST/ACS and JWST/NIRCam. Please +refer to Table 1 for information on limiting magnitudes + +5 +Table 1. HST and JWST Imaging Data Used in the Spatially Resolved SED Fitting +Telescope +Camera +Filter +Wavelength +Deptha +PSF FWHMb +WHL0137−08 +MACS0647+70 +WHL0137−08 +MACS0647+70 +(µm) +(AB mag) +(AB mag) +(arcsec) +(arcsec) +HST +ACS/WFC +F435W +0.37–0.47 +27.7 +28.0 +0.11 +0.11 +HST +ACS/WFC +F475W +0.4–0.55 +28.5 +28.2 +0.11 +0.11 +HST +ACS/WFC +F555W +0.46–0.62 +· · · +28.7 +· · · +0.11 +HST +ACS/WFC +F606W +0.47–0.7 +28.3 +28.3 +0.11 +0.11 +HST +ACS/WFC +F625W +0.54–0.71 +· · · +27.9 +· · · +0.11 +HST +ACS/WFC +F775W +0.68–0.86 +· · · +27.8 +· · · +0.08 +HST +ACS/WFC +F814W +0.7–0.95 +28.7 +28.5 +0.11 +0.11 +JWST +NIRCam +F090W +0.8–1.0 +28.3 +· · · +0.04 +· · · +JWST +NIRCam +F115W +1.0–1.3 +28.4 +28.1 +0.04 +0.04 +JWST +NIRCam +F150W +1.3–1.7 +28.5 +28.3 +0.06 +0.06 +JWST +NIRCam +F200W +1.7–2.2 +28.7 +28.4 +0.06 +0.06 +JWST +NIRCam +F277W +2.4–3.1 +29.1 +28.9 +0.11 +0.11 +JWST +NIRCam +F356W +3.1–4.0 +29.3 +29.0 +0.11 +0.11 +JWST +NIRCam +F410M +3.8–4.3 +28.6 +· · · +0.16 +· · · +JWST +NIRCam +F444W +3.8–5.0 +29.0 +28.8 +0.16 +0.16 +Note— a5σ point source AB magnitude limit measured within a 0.′′2 diameter circular aperture. +bPSF FWHM here +are based on empirical measurement as described in Appendix C. +and the point spread function (PSF) sizes of our HST +and JWST data. +2.2. Sample Galaxies +We use grizli v4 photometric catalogs (will be de- +scribed in Section 3.1) to select our sample galaxies +in the three fields (WHL0137−08, blank field, and +MACS0647+70). +The catalogs provide the aperture +fluxes and photometric redshifts with which we select +our sample. +The sample selection is described in the +following. First, we select galaxies that have integrated +signal-to-noise (S/N) ratio > 5 in all JWST filters that +are available for the fields. This is to ensure that we will +have galaxies with good photometry in at least JWST +filters. This initial cut selects 1322 (out of 2718), 1278 +(out of 3032), and 1331 (out of 2660) galaxies in the +WHL0137−08, blank field, and MACS0647+70, respec- +tively. We do not apply the same S/N criteria on HST +filters because it will exclude many more galaxies as they +have lower S/N than JWST filters. +We further cut the sample galaxies based on their +redshift +to +get +a +sufficient +coverage +of +the +rest- +frame UV–NIR. For galaxies in the WHL0137−08 and +MACS0647+70, which are observed by both JWST and +HST, we select galaxies at 0.3 < z < 6.0, whereas for +galaxies in the blank field, which do not have HST ob- +servations, we select galaxies at 1.3 < z < 6.0. This +redshift cut ensures that the rest-frame 4000 ˚Abreak is +covered. This cut further reduces the sample to be 1258, +581, and 1257 galaxies in the WHL0137−08, blank field, +and MACS0647+70, respectively. After that, we do a +visual inspection to exclude galaxies that appear to be +very small (i.e., unresolved) and in a merger (i.e., one +segmentation region having multiple cores or multiple +galaxies in one segmentation region, despite possible in- +terlopers). This further reduce the sample to be 354, +239, 220 galaxies in the WHL0137−08, blank field, and +MACS0647+70, respectively. +We perform spatially resolved SED analysis on the +galaxies in this initial sample. A detailed description of +the methodology will be given in Section 3. Once the +analysis is done, we inspect the results of all the galaxies +and further exclude galaxies that seem to have bad SED +fitting results based on the χ2 values of the fitting to the +integrated SEDs within the central effective radius (see +Section 3.5) and the average χ2 values of the fitting to +the first 20 spatial bins (see Section 3.4 for definition of +spatial bin). We exclude galaxies that have χ2 > 20 for +the SEDs within the central effective radius and aver- +age χ2 > 40 for the first 20 spatial bins. We note that +χ2 value can be unrealistically high if systematic uncer- +tainties of the photometry are not properly accounted. +Beside this, there is still uncertainty around the zero- + +6 +point calibration of NIRCam photometry in the current +early observations (e.g., Boyer et al. 2022; Finkelstein +et al. 2022). Therefore, we visually inspect SED fitting +results of each galaxy using similar plots as shown in +Figure 4. We find that in most cases, NIRCam fluxes +are well fitted by our models, better than HST/ACS +fluxes. +This might be due the shallower depths (and +lower S/N) of HST compared to JWST. The χ2 values +above are high enough to get sufficient number of galax- +ies and low enough to get good quality of SED fitting re- +sults. This results in our final sample, consisting of 243, +91, and 110 galaxies in the WHL0137−08, blank field, +and MACS0647+70, respectively. Figure 1 shows the +distributions of redshifts and M∗ of our sample galax- +ies. +We note that our sample selection may possibly bias +toward selecting relatively massive, bright, and resolved +galaxies in each redshift. However, due to the lensing +magnification in the cluster fields, we expect to detect +on average lower mass galaxies with better spatial res- +olution than in the standard fields. +The small num- +ber of galaxies and the limited volume sampled might +make our sample to be not representative of the gen- +eral population of galaxies. However, since we do not +make inferences on the average trends or number densi- +ties as function of global properties (e.g., M∗), but in- +stead we show trends in individual galaxies, our results +still provide useful insights on the evolution of galaxy +structures. We also ignore the possible contamination +by the Active Galactic Nucleus (AGN) host galaxies in +our current study because of the lack of diagnostics for +identifying them using our current data. +3. METHODOLOGY +3.1. Data Reduction and Photometric Catalog +We use the grizli pipeline (Brammer et al. 2022) to +process the HST FLT and the JWST pipeline-calibrated +level-2 imaging data. The JWST data were processed +using the calibration pipeline v1.5.3 with CRDS con- +text jwst 0942.pmap, which includes photometric cal- +ibrations based on in-flight data. +The JWST level-2 +imaging data were then scaled with detector-dependent +factors1 based on a NIRCam flux calibration using the +standard star J1743045. Our photometric zeropoints de- +scribed here are similar to those obtained by the JWST +Resolved Stellar Populations ERS program (Boyer et al. +2022; Nardiello et al. 2022) who analyzed the M92 glob- +ular cluster. We also check the consistency of our cali- +bration with the more recent one based on CAL program +1 https://zenodo.org/record/7143382 +Figure 1. +Distributions of M∗ and redshifts of the sam- +ple galaxies analyzed in this work that consist of galaxies in +WHL0137−08 cluster field, a blank field (i.e., a nearby field +of the WHL0137−08 that is observed with NIRCam), and +MACS0647+70 cluster field. +The M∗ here are derived by +summing up M∗ of pixels (in the galaxy’s region) obtained +from our spatially resolved SED analysis. +data jwst 0989.pmap and find out that they are consis- +tent within 3% in all filters analyzed here. +In processing the JWST data, the grizli pipeline ap- +plies a correction to reduce the effect of 1/f noise and +masks “snowballs”2 effect caused by the large cosmic +ray impacts to the NIRCam detectors. Beside this, the +grizli pipeline also corrects for the “wisps”3, which is a +faint, diffuse stray light features that appear at the same +detector locations in NIRCam images and most promi- +nent in the A3, B3, and B4 detectors in the F150W and +F200W images. +The grizli pipeline aligns the HST and JWST imag- +ing data to a common world coordinate system which is +registered based on the GAIA DR3 catalogs (Gaia Col- +laboration et al. 2021). The images are then drizzled to +a common pixel grid using the astrodrizzle (Koeke- +moer et al. 2003; Hoffmann et al. 2021). The 17 HST +filters and 4 JWST NIRCam long-wavelength (LW) fil- +ters (F277W, F356W, F410M, and F444W) are drizzled +2 https://jwst-docs.stsci.edu/data-artifacts-and-features/ +snowballs-artifact +3 https://jwst-docs.stsci.edu/jwst-near-infrared-camera/ +nircam-features-and-caveats/nircam-claws-and-wisps + +50 +0 +WHL0137-08cluster +12 +WHL0137-08blankfield +MACS0647+70cluster +11 +log(M*[M。]) +10 +8 +N=243 +N=91 +N=110 +1 +2 +3 +4 +5 +6 +0 +25 +Photometric Redshift7 +to a spatial sampling of 0.′′04 per pixel while the JWST +short-wavelength (SW) filters (F090W, F115W, F150W, +and F200W) are drizzled to a spatial sampling of 0.′′02 +per pixel. +Source detection is then performed on a weighted sum +of the drizzled NIRCam images in all filters using sep +(Barbary 2016; Bertin & Arnouts 1996). +Fluxes are +then calculated for each source in three circular aper- +tures, 0.′′36, 0.′′5, and 0.′′7. Then photometric redshift +measurement is performed using the 0.′′5 aperture SEDs +employing eazypy(Brammer et al. 2008). This code fits +observed photometry using a set of templates added in +a non-negative linear combination. The processed imag- +ing data along with the photometric catalog are publicly +available4. These data product have also been used in +some recent studies (Welch et al. 2022b; Bradley et al. +2022b; Hsiao et al. 2022; Vanzella et al. 2022; Meena +et al. 2022). +3.2. Analysis of Post-processed Imaging Data +In this work, we combine the post-processed HST +and JWST imaging data (in up to 13 filters) into a +common spatial resolution (i.e., PSF size) and sam- +pling (i.e., pixel size) for extracting the spatially resolved +SEDs of our sample galaxies. These spatially resolved +SEDs are then fitted with models to infer the underlying +properties of the stellar populations. We use piXedfit5 +(Abdurro’uf et al. 2021, 2022c) throughout this analy- +sis. +Basically, this process includes three main tasks: +image processing, pixel binning, and SED fitting. We +will briefly describe these steps in the following. +The image processing is carried out automatically us- +ing piXedfit. +For each galaxy, we first crop stamp +images with a size of 6.′′04 × 6.′′04 (corresponding to +302×302 pixels in NIRCam SW and 151×151 pixels in +NIRCam LW and HST/ACS filters) centered at the +galaxy. +We then perform background subtraction to +each stamp image using photutils (Bradley et al. +2022a). Next, we perform point spread function (PSF) +matching to homogenize the spatial resolution across fil- +ters. We degrade the spatial resolution of the images to +match the resolution of F444W filter, which has the low- +est spatial resolution (see Table 1). For this, we generate +the empirical PSFs of HST/ACS and JWST/NIRcam +filters along with the convolution kernels using photu- +tils package (see Appendix C). The PSF matching is +carried out by convolving the stamp images with the +convolution kernels. After PSF matching, we register +all the stamp images to a common spatial sampling of +4 https://cosmic-spring.github.io/earendel.html +5 https://github.com/aabdurrouf/piXedfit +0.′′04 per pixel. At the end, we have multiband stamp +images with a size of 151×151 pixels for each galaxy in +our sample. +3.3. Constructing Photometric Data Cubes +piXedfit further processes the stamp images to pro- +duce photometric data cubes. First, it defines a galaxy’s +region of interest. For each galaxy, segmentation maps +are first produced in all filters using sep (Barbary 2016) +and those maps are then merged together into a single +map. In the segmentation process, we use same parame- +ters for all filters as follows. We set the detection thresh- +old (thresh), the number of thresholds for deblending +(deblend nthresh), and the minimum contrast ratio for +deblending (deblend cont) to be 2.0, 40, and 0.005, re- +spectively. +In some cases, the merged segmentation map is larger +than expected, as can be inferred from the maps of +multiband fluxes. This can be caused by some factors, +for example, an interference from neighboring objects +that is not separated well by the deblending process. We +visually inspect the merged segmentation map of each +galaxy to find out this issue. To deal with this, we tweak +the deblending parameters to get cleaner segmentation +maps or ignore segmentation map in some filters that +has this deblending issue, then merge them again. +Once the galaxy’s region is defined, then the fluxes of +pixels within the region are calculated. We use PHOT- +FLAM keyword in the header of the grizli imaging data +products to convert the pixel value into flux density in +the units of erg s−1cm−2˚A−1. The data cubes are then +stored into FITS files. Figure 2 shows examples of the +maps of multiband fluxes of galaxies in three fields an- +alyzed in this work. +The color images shown in the +leftmost panels are created using Trilogy6 (Coe et al. +2012). +3.4. Pixel binning +The SEDs of pixels are usually noisy and might not +providing sufficient constraint to the models if fitting is +done to them. Therefore, we perform pixel binning using +piXedfit to optimize the signal-to-noise ratio (S/N) of +the spatially resolved SEDs. Basically, this process bins +neighboring pixels to achieve a certain S/N ratio thresh- +old. +The unique pixel binning scheme in piXedfit, +which takes into account the similarity in SED shape +among pixels, allows for achieving a sufficient S/N ratio +in multiple filters of interest while preserving important +spatial information at pixel level. A detailed description +6 https://github.com/dancoe/trilogy + +8 +Figure 2. Examples of the maps of multiband fluxes produced from the image processing. Example of one galaxy is shown for +each field, from top to bottom: WHL0137−08 cluster (observed in 12 filters), blank field (8 filters), and MACS0647+70 cluster +(13 filters). The ID is based on our grizli v4 public catalog. +of this pixel binning scheme can be seen in Abdurro’uf +et al. (2021). +We assume the following parameters in the pixel bin- +ning process. We refer reader to Abdurro’uf et al. (2021) +for more information about the parameters. We set S/N +thresholds to 5 in all JWST NIRCam filters. +We do +not set S/N threshold to HST filters because the S/N +ratio of pixels in the HST images are low, especially +for galaxies at high redshifts. +Setting a S/N thresh- +old on the HST filters would put a strong constraint in +the pixel binning process which can produce a coarser +binning map and loosing important spatial information +from the original images. +The rest of the binning parameters are as follows. We +set a minimum diameter of 7 pixels, which is larger +than the PSF FWHM size of our data cubes, a re- +duced χ2 limit of 5 in the evaluation of the similarity +of SED shape. We refer to F277W flux in determining +the brightest pixel to be the center of a spatial bin. We +store new data cubes produced from this pixel binning +process into FITS files. The total number of spatial bins +in our sample galaxies is 24999. Figure 3 shows examples +of the pixel binning maps produced from this process. +3.5. Spatially Resolved SED Fitting +Once we have the binned data cubes, we perform SED +fitting to the SEDs of individual spatial bins in our +sample galaxies. Here we use the SED fitting module +in piXedfit. The SED fitting in piXedfit uses fully +Bayesian technique. We refer reader to Abdurro’uf et al. +(2021) for a detailed description of the SED modeling +and fitting methods as well as comprehensive tests of +its capabilities. In Appendix A, we perform SED-fitting +tests using mock SEDs to demonstrate the robustness +of our SED fitting method on the combined HST and +JWST photometry. Moreover, in Appendix B we dis- +cuss how NIRCam photometry can potentially break the + +120 +120 +HST/F435W +HST/F475W +HST/F606W +HST/F814W +JWST/F090W +JWST/F115W +110 - +110 +110 +110 +110 +-20.5 +100: +100 +100 +100 +100 +06 +90 +90 +90 +80 +80 +80 +80. +80 +80 +WHL0137-08 +-21.0 7 +-1cm +70 +70 +70 +70 +6 + 50 +-21.5 + 50 +50 +50 + 50 +s +log(Flux density [erg +40 - +40 - +40 +-22.0 +100 +40 +120 +100 +100 +120, +80 +100 +100 +120 +120 +120 +JWST/F150W +JWST/F200W +JWST/F277W +JWST/F356W +JWST/F410M +JWST/F444W +110 +110 +110 +110 +110 +100 - +100 +100 +-22.5 +100 +100 +90 - +% +80 +80 - +%. +-23.0 +70 +70 +60 +60 +60 + 50 +os +50 + 50 +-23.5 +40 + 40 +40 +40. + 40 +100 +120 +40 +100 +120 +100 +120 +100 +120 +40 +100 +120 +[pixel] +[pixellJWST/F090W +JWST/F115W +JWST/F150W +JWST/F200W +120 +120 +120 - +120 - +20.5 +100 +100 +100 +100 +WHL0137-08 blank field +80 +80 +-21.0 +60 +40 +40 - +-21.5 +ity [erg +20 +20 + +20 - +20 +40 +60 +80 +100 +120 +20 +40 +60 +80 +100 +120 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +120 +JWST/F277W +JWST/F356W +JWST/F410M +JWST/F444W +120 +120 - +120 - +120 +-22.0 +densit +100 +100 +100 +100 - +ID=3240. z=1.77 +80 +-22.5 +60 +60 +60 +40. +40 - +23.0 +20- +40 +60 +80 +100 120 +60 +20 +100 +120 +20 +60 +80 +100 +120 +40 +60 +80 +100 +120 +[pixel] +[pixel] +[pixel] +[pixel]140 +HST/F435W +140 +HST/F475W +HST/F555W +HST/F606W +140 +HST/F625W +HST/F775W +HST/F814W +120 +120 +120 - +120 +20.0.D +100 +MACS0647+70 +80 +80 +-20.5 +60 +60- + 40 +-21.0 +-21.5 +100125150 +75100125 +100125 +150 +¥75100 +125 +7510012515 +JWST/F115W +JWST/F150W +JWST/F200W +JWST/F277W +JWST/F356W +JWST/F444W +120 +120 +-22.0 +120 +100 +100 +100 +. +-22.5 + 60 +60 +60 +40 +40- +20 +125 1509 +Figure 3. Examples of pixel binning results. The pixel binning process achieves a minimum S/N ratio of 5 in all JWST +NIRCam filters. +Table 2. Free Parameters in the SED Modeling and the Assumed Priors. +Parameter +Description +Prior +Sampling/Scale +M∗ +Stellar mass +Uniform: +min= +log(sbest) − 2, +max= log(sbest) + 2a +Logarithmic +Z∗ +Stellar metallicity +Uniform: +min= −2.0 + log(Z⊙), +max= 0.2 + log(Z⊙) +Logarithmic +t +Time since the onset of star formation (agesys) b +Uniform: min= −1.0, max = age +of the universe at the galaxy’s +redshiftc +Logarithmic +τ +Parameter that controls the peak time in the double +power-law SFH modelb +Uniform: min= −1.5, max= 1.14 +Logarithmic +α +Parameter in the double power-law SFH model that con- +trols the slope of the falling star formation episodeb +Uniform: min= −2.0, max= 2.0 +Logarithmic +β +Parameter in the double power-law SFH model that con- +trols the slope of the rising star formation episodeb +Uniform: min= −2.0, max= 2.0 +Logarithmic +ˆτ1 +Dust optical depth of the birth cloud in the Charlot & +Fall (2000) dust attenuation law +Uniform: min= 0.0, max= 4.0 +Linear +ˆτ2 +Dust optical depth of the diffuse ISM in the Charlot & +Fall (2000) dust attenuation law +Uniform: min= 0.0, max= 4.0 +Linear +n +Power law index in the Charlot & Fall (2000) dust atten- +uation law +Uniform: min= −2.2, max= 0.4 +Linear +Note— asbest is the normalization of model SED derived from the initial fitting with the χ2 minimization method (see Section +4.2.1 in Abdurro’uf et al. 2021). +bThe mathematical form of the double power-law SFH is given in Abdurro’uf et al. (2021, +Equation 7 therein). cThe redshift here is z = zeazy − 0.3 with zeazy is the photometric redshift from the grizli catalog which +is derived using eazypy code. Subtraction by 0.3 is made to enlarge the range of t and account for the photometric redshift +uncertainty. + +Bin Index +Bin Index +Bin Index +100 +200 +300 +400 +500 +10 +20 +30 +50 +100 +150 +120 +ID=543 +ID=2430 +ID=3240 +140 +110 +120 +120 +100 +100 +100 +90 +[pixel] +80 +80 +80 +70 +60 +60 +60 +40 +50 +40 +20 +40 +nbins=580 +nbins=32 +20 +nbins=195 +0 - +30. +0 +20 +40 +60 +80 +100 +120140 +30 +40 +50 +60 +70 +80 +90 100 110120 +20 +40 +60 +80 +100 +120 +Bin Index +Bin Index +Bin Index +3 +5 +7 +9 +11 +13 +20 +40 +60 +80 +50 +100 +150 +100 +ID=2808 +ID=4554 +ID=3704 +140 +120 +90 +120 +100 +100 +80 +[pixel] +80 +80 +70 +60 +60 +40 +60 +40 +20 +nbins=13 +nbins=83 +nbins=159 +50 +0 +20 +50 +60 +70 +80 +90 +100 +0 +20 +40 +60 +80100120140 +20 +40 +60 +80 +100 +120 +[pixel] +[pixel] +[pixel]10 +degeneracies among age, dust, and metallicity in the +SED fitting. +In the following, we provide a brief de- +scription of the method and some assumptions applied +in our SED fitting. +We use the Flexible Stellar Population Synthesis code +(FSPS; Conroy et al. 2009; Conroy & Gunn 2010). It +includes the nebular emission modeling that uses the +CLOUDY code (Ferland et al. 1998, 2013). In this work, we +assume the Chabrier (2003) initial mass function (IMF), +Padova isochrones (Girardi et al. 2000; Marigo & Girardi +2007; Marigo et al. 2008), and MILES stellar spectral +library (S´anchez-Bl´azquez et al. 2006; Falc´on-Barroso +et al. 2011). For the star formation history model, we as- +sume an analytic model in the form of double power-law. +It has been shown in Abdurro’uf et al. (2021) that this +SFH form can give robust estimates of the stellar pop- +ulation properties and even SFH of galaxies, as tested +using synthetic SEDs of simulated galaxies in the Illus- +trisTNG simulations. For simulating the effect of dust +attenuation, we use the two-component dust attenuation +law of Charlot & Fall (2000). This dust attenuation law +gives an extra attenuation to stars younger than 10 Myr +that are assumed to be residing in the dense molecular +clouds, in addition to standard attenuation in the dif- +fuse ISM. We model the attenuation due to intergalactic +medium using Inoue et al. (2014) model. Since we do not +have photometry that cover the rest-frames mid-infrared +(MIR) and far-infrared (FIR), we switch off the model- +ing for dust emission and AGN dusty torus emission in +the analysis throughout this work. The SED modeling +has 9 free parameters. We summarize these parameters +along with the assumed priors in Table 2. We assume a +constant ionization parameter (U) of 0.01 in the model- +ing of the nebular emission. +In the current analysis, we rely on photometric red- +shift for all of galaxies in our sample because we do not +have spectroscopic observations at the moment we carry +out this analysis. To get redshift estimates of the galax- +ies, we perform SED fitting with piXedfit in which +redshift is let to be free in the fitting. For this, we fit +integrated SED within the effective radius of the galax- +ies. The effective radius is measured in F444W image +using GALFIT(Peng et al. 2002, ;see Section 4.2). This +is performed to get SEDs with high S/N while reducing +contamination from noisy SEDs of pixels in the outskirt +regions. In this fitting, we apply a prior on redshift in +the form of a Gaussian function centered at the photo- +metric redshift estimated by the eazypy taken from the +grizli catalog (see Section 3.1). We set a width of 0.5 +for this Gaussian prior. This fitting is performed to de- +rive redshift only. We then use this redshift information +for the SED fitting of all spatial bins in the galaxy, in +which we fix the redshift. We apply the Markov Chain +Monte Carlo (MCMC) method in piXedfit˙In the SED +fitting for determining redshifts, we set the number of +walkers to 100 and the number of steps per walker to +1000. For the SED fitting of spatial bins, we use 100 +walkers and less number of steps (600) per walker for +reducing computational time. +We show examples of SED fitting results of two galax- +ies in Figure 4, one galaxy from the WHL0137−08 clus- +ter field (top panel) and the other galaxy from the blank +field (bottom panel). For each galaxy, we show best-fit +SEDs in the top right panel. The observed and best- +fit photometric SEDs are shown with square and circle +symbols, respectively. The SED in black is for the inte- +grated within the effective radius, while those in other +colors are for 5 examples of spatial bins in the galax- +ies. +The corner plot in the lower left side shows the +posterior probability distribution functions (PPDF) of +the model parameters obtained from the fitting on the +integrated SED within the effective radius. Above this +corner plot, we show the PDFs of M∗ and SFR of the +spatial bins which the best-fit SEDs are shown in the top +right panel. The best-fit spectra shown in the plot are +drawn from the MCMC sampler chains, which distribu- +tions reflect the PPDF. Therefore, it is possible to get a +slight shift in wavelength between the best-fit spectra of +the central SED (where z is free in the fitting) and that +of the spatial bins (where z is fixed in the fitting). This +wavelength shift reflects the uncertainty of the estimated +redshift. Finally, in the bottom right panel we show the +maps of stellar population properties, including the M∗ +surface density (Σ∗), SFR surface density (ΣSFR), mass- +weighted age, AV,1 (1.086 × ˆτ1), AV,2 (1.086 × ˆτ2), and +metallicity. +3.6. Lens Modeling +To estimate the magnifications due to the gravita- +tional lensing effect by the clusters, we use the lens mod- +els constructed by our team. For the WHL0137−08 clus- +ter, we use the same lens models that were used for ana- +lyzing the Earendel and the Sunrise Arc in Welch et al. +(2022a), which were made publicly available7. +These +lens models were generated using four independent lens +modeling software packages: Light-Traces-Mass (LTM, +Zitrin et al. 2009, 2015; Broadhurst et al. 2005), Glafic +(Oguri 2010), WSLAP (Diego et al. 2005, 2007), and +Lenstool (Kneib et al. 1993; Jullo et al. 2007; Jullo +& Kneib 2009). +Please see Welch et al. (2022a) for +detailed information about each model. Sample galax- +7 https://relics.stsci.edu/lens models/outgoing/whl0137-08/ + +11 +Figure 4. Examples of SED fits of a galaxy in the WHL0137−08 cluster (top panel) and the blank field (bottom panel). The +SED plots show the best-fit SEDs from the fitting to integrated SED within the effective radius (black color) and 5 examples of +spatial bins (in colors). The corner plots show the posterior probability distributions of the parameters obtained from fitting to +the central integrated SED. Above this corner plots, we show PDFs of M∗ and SFR of the five spatial bins. Finally, the maps +of stellar population properties derived from this analysis are shown in the bottom right corner. + +Bin Index +10 +30 +WHL0137-08 +0 +120 - +Central Re +10-18, +Bin 1 +100 +Bin 3 +Bin 10 +Bin 20 +M +60 +10-19 +Bin 30 +40 +回 +40 +80 +100 +120 +[pixel] +.2 +10-20 +Fx [erg + Normalized PDF + Normalized PDF +10-21. +HST +JWST +10-22+ +7 +8 +9 +10 +11 +-1 +0 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 0.9 1.0 +2.0 +3.0 +4.0 +5.0 +log(M*[Mol) +log(SFR[Moyr-1]) +Wavelength [μm] +120 +120 +1.0 +120 + 9.5 +Corner plot of central Re +110 +110 +110 +0.5 +100 +100 +100 +0.0 +90 +8.5 +06 +06 +10.0 +[pixel] +g0 +80 +0.5 +80 +8.0 +-1.0 +70 +70 +70 +7.5 + 0.15 +60 +1*7)60| +-1.5 + 60 + 50 +7.0 + 50 +-2.0 +50 / +40 - +6.5 +40 +-2.5 +40 +30 +6.0 +30 +3.0 +30 - + 0.00 +40 +60 +80 +100 +120 +40 +60 +80 +100 +120 +40 +60 +80 +100 +120 +[pixel] +[pixel] +[pixel] +120 +120 +120 +3.0 +110 +110 +110 +0.2 +100 +2.5 +100 +100 +-0.4 + [mag] +90 +2.0 +90 +90 + 1.5 +80 +80 +80 +-0.8 +10 +70 +1.02 +70 +1.0 + 60 +Av. +60 +60 +50 + 50 +0.5 +50 +1.4 + 0.5 +40 +40 +40 +1.6 +og(SFR) +og(M-) +Av,2 +"log(age)Tog(ziZo) redshift +30 - +0.0 +30 +0.0 + O +40 +60 +80 +100 +120 +40 +60 +80 +100 +120 +40 +60 +80 +100 +120 +[pixel] +[pixel] +[pixel] Bin Index +WHL0137-08 blank field +50 +100 +150 +Central Re +120 +Bin +100 +Bin 30 +[pixel] +10-18 +Bin 60 +80 +Bin 90 + 60 +Bin 120 +40 - +20 - +25 +50 +[pixel] +Fx [erg + Normalized PDF +Normalized PDF +10-21 +JWST +7 +8 +9 +10 +11 +-4 +-2 +2 +0.3 +0.4 +0.5 +0.60.70.8 0.9 1.0 +2.0 +3.0 +4.0 +5.0 +log(M*[Mol) +log(SFR[M。yr-1]) +Wavelength [μm] +9.5 +3.0 +Corner plot of central Re +120 - +120 +log(2sFR[Moyr-1kpc-2 +120 +100 - +100 +-1 +100 +[pixel] +80 - +8.0 +80 +80 +7.5 + 60 +60 +1*7)60| +60 +7.0 +40. + 40 +40 +6.5 +_5 +20 +20 - +20 +6.0 +0.0 +20 +40 +80 +100120 +20 +40 +60 +80100120 +20 +40 +60 +80 +100120 +[pixel] +[pixel] +[pixel] +3.5 +3.5 +-0.2 +120 +120 +120 +3.0 +3.0 +-0.4 +100 +100 +100 +2.5 +-0.6 ~ +[pixel] +80 +2.0 +80 +80 +-1.0 + 09 +1.5 +1.5 +2 +60 +60 +40 +40 +40 +1.4 +0.5 +0.5 +log(SFR) +log(age)log(ZiZo)"redshit +1.6 +log(m) +20 +20 +20 +0.0 +0.0 +204060 +80100120 +20 +80100120 +2040 +80100120 +[pixel] +[pixel] +[pixel]12 +ies located in the blank field are expected to have only +weak magnifications of µ ≲ 1.1. With the multiple lens +models available for this cluster, we estimate the total +and tangential (i.e., linear) magnifications (µ and µt, re- +spectively) of each galaxy by taking the average values. +By this way, we account for the modeling uncertainties. +Based on the standard deviation values, we find that +the magnifications do not vary a lot among the models. +The median standard deviations of µ and µt are 0.11 +and 0.10 dex, respectively. +The lens models for MACS0647+70 cluster have been +constructed in the past using the HST imaging data. +The first lens model for MACS0647+70, before the +CLASH survey, was provided by Zitrin et al. (2011) us- +ing LTM method. +With the addition of HST imaging +data from CLASH, new lens models were established us- +ing various methods, including Lenstool, LTM, WSLAP, +and LensPerfect (Coe et al. 2008). These lens mod- +els have been used in previous studies in CLASH (e.g., +Coe et al. 2013; Zitrin et al. 2015; Chan et al. 2017). +Now with the addition of JWST NIRCam imaging data, +which add on many new strongly-lensed multiple-image +candidates (thanks to its high spatial resolution and +depth), a new lens model has been established using the +dPIEeNFW method (Zitrin et al. 2015) with some mod- +ifications. A detailed information on this lens model- +ing of MACS0647+70 cluster along with the list of the +multiple-image systems are given in Meena et al. (2022). +This new method has also been implemented to several +clusters using JWST NIRCam data (Pascale et al. 2022; +Roberts-Borsani et al. 2022; Hsiao et al. 2022; Williams +et al. 2022). We used this lens model constructed by +Meena et al. (2022) for galaxies in the MACS0647+70 +field. We correct the M∗ and SFR obtained from SED +fitting for the lensing magnification by dividing them +with µ. +We also correct size or radius measurement +(e.g. half-mass and half-SFR radii; see Section 4.3) by +dividing them with µt. +4. RESULTS +4.1. Integrated Properties +Before analyzing the spatially resolved properties of +our sample galaxies, we first present their integrated +(i.e., global) properties. To bring it into the context of +the global demographics of galaxies, we plot our sample +on the integrated star-forming main sequence (SFMS) +diagram, as shown in the left panel of Figure 5. The +integrated M∗ and SFR of a galaxy are derived by sum- +ming up the values in pixels obtained from the spatially +resolved SED fitting. +Due to our limited sample, we +plot all our galaxies on the SFMS diagram instead of +dividing them into a number of redshift bins and ex- +amine the SFMS relation in each bin. This can cause +the broad distribution as shown in the figure. Different +symbols represent the fields where the galaxies are lo- +cated (WHL0137−08, blank field, and MACS0647+70), +whereas color-coding represent redshift grouping, where +we divide the redshift range into five bins. The dashed +lines show the SFMS relations at the median redshifts of +the five redshift bins, calculated using the prescription +from Speagle et al. (2014). The lines are colored based +on the redshift groups. +We then classify our sample galaxies into star-forming, +green valley, and quiescent groups based on their posi- +tions with respect to the SFMS ridge line at the red- +shift of the galaxies. +We define star-forming, green- +valley, and quiescent galaxies as those having SFR > +SFRMS(z, M∗) − 0.4 dex, SFRMS(z, M∗) − 0.4 ≥ SFR > +SFRMS(z, M∗)−1.0 dex, and SFR ≤ SFRMS(z, M∗)−1.0 +dex, respectively, where SFRMS(z, M∗) is the SFMS +ridge line for exact z and M∗ of the individual galax- +ies. With this selection criteria, we have 219, 108, and +117 total numbers of the star-forming, green-valley, and +quiescent galaxies from the three fields, respectively. We +will use these classified samples throughout the analysis +in this paper to investigate the differences in spatially +resolved properties of galaxies in various evolutionary +stages. The right panel of Figure 5 show the distribu- +tions of these groups on the SFMS diagram. +To get a sense of how the global specific SFR (sSFR≡ +SFR/M∗) evolves with cosmic time in our sample galax- +ies, we plot the sSFR against redshift in Figure 6. We +can see a clear trend of decreasing global sSFR with cos- +mic time and an increasing number of quiescent galaxies +along the way. In our sample, quiescent galaxies start +to emerge from z ∼ 3, ∼ 2 Gyr after the Big Bang. To +compare our global sSFR trend with the similar trend +from previous studies, we plot sSFR(z) inferred from +the SFMS normalization based on Speagle et al. (2014) +prescription. We calculate sSFR(z) for four M∗ of 108.5, +1010, 1011, and 1012 M⊙ and show them in the figure +as black dashed lines. We see an overall agreement be- +tween the evolutionary trend of sSFR in our sample and +that expected based on the evolution of the SFMS nor- +malization from Speagle et al. (2014). +We also show +global sSFR measurements of z ∼ 0 spiral galaxies from +Abdurro’uf & Akiyama (2017) and Abdurro’uf et al. +(2022a) (personal communication) who performed spa- +tially resolved SED fitting using piXedfit. +Previous studies have classified passive galaxies using +various methods. +One of the methods is by compar- +ing the Hubble time (tH) with the mass doubling time +(i.e., inverse of sSFR). Basically, this method defines qui- +escent galaxies as those having sSFR < 1/tH. The red + +13 +Figure 5. Left panel: Integrated (i.e., global) M∗ and SFR of our sample galaxies. Different symbols represent different fields +where the galaxies are located, whereas the color-coding represents redshift grouping. The dashed lines show the global SFMS +relations at the median redshifts of the 5 redshift groups calculated using the prescription from Speagle et al. (2014). The lines +are colored based on the redshift groups. Right panel: Distribution the star-forming, green-valley, and quiescent galaxies in our +sample that are classified as having SFR > SFRMS(z, M∗) − 0.4 dex, SFRMS(z, M∗) − 0.4 ≥ SFR > SFRMS(z, M∗) − 1.0 dex, +and SFR ≤ SFRMS(z, M∗) − 1.0 dex, respectively. SFRMS(z, M∗) is the SFMS ridge line that is calculated for exact z and M∗ +of the individual galaxies. +dashed line in Figure 6 represents 1/tH. To compare our +quiescent classification with this method, we plot 1/tH +in Figure 6, which is shown with red dashed line. We +can see that our quiescent galaxies lie below this line, +indicating that our classification method is consistent +with that based on tH. +4.2. Radial Profiles of the Stellar Population +Properties +As we have shown the global properties of the sam- +ple galaxies and classified them into star-forming, green- +valley, and quiescent groups, now we will analyze their +spatially resolved properties. +We start by presenting +the radial profiles of the stellar population properties to +get a sense of how the properties vary radially within +the galaxies. To derive the radial profiles, first, we per- +form 2D single-component S´ersic fitting using GALFIT +(Peng et al. 2002) on F444W stamp image of each galaxy +to get their ellipticities, position angles, and central co- +ordinates. We then use this information to define ellip- +tical annuli in the radial profile calculation. The radial +profiles are derived from the 2D maps of properties ob- +tained from the spatially resolved SED fitting by aver- +aging values of pixels within the annuli. Since galaxies +have a wide range of size, we normalize the radius by the +half-mass radius (Re), which is the radius that covers +half of the integrated M∗. We use radial increment (δr) +Figure 6. Evolution of the integrated specific SFR (sSFR) +with redshift. The black dashed lines represent sSFR evolu- +tion of SFMS galaxies with M∗ = 108.5, 1010, 1011, and 1012 +M⊙ (decreasing normalization) as inferred from the normal- +ization of the SFMS, calculated using the prescription from +Speagle et al. (2014). The red dashed line represents 1/tH +where tH is the Hubble time. Almost all our quiescent galax- +ies lie below this line, indicating that their mass doubling +timescale is longer than tH. +of 0.3Re. Thanks to the gravitational lensing effect, we + +20 +0 +4 +3 +log(SFR[Moyr-1]) +2 +0 +Star-forming +-3 +Green valley +Quiescent +8 +9 +10 +11 +12 +0 +25 +log(M*[Mo])z2 +0 +1 +3 +4 +5 +6 +7 +8 +WHL0137-08cluster +Star-forming +WHL0137-08blankfield +-6 +Green valley +MACS0647+70cluster +Quiescent +-7 +log(sSFR[yr-1]) +8 +.9 +-10 +11 +-12 +1/Hubbletime +-13 +SFMSSpeagle etal.(2014) +z=0Abdurro'uf&Akivama(2o17） +log(M*[Mo])=8.5,10.0,11.0,12.0 +¥z=0Abdurro'ufetal.(2022a) +-14 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +log(1 + z)6.00 +C +WHL0137-08cluster +★ +WHL0137-08blankfield +3 +MACS0647+70cluster +4.86 +2 +log(SFR[Moyr-1 ]) +1 +3.72 +Redshift +0 +2.58 +-2 +1.44 +-3 +Lines:SFMSSpeagleetal.(2014) +0.30 +8 +9 +10 +11 +12 +log(M*[Mo])14 +can resolve many of our galaxies down to sub-kpc scales +(109 galaxies in our sample have delensed Re < 1 kpc). +Figure 7 shows the radial profiles of the stellar mass +surface density (Σ∗). We divide the sample into 5 bins +of redshift and 4 bins of M∗ to see how the radial pro- +files vary with global M∗ and cosmic time. Moreover, +we indicate the star-forming, green valley, and quiescent +galaxies with different colors, in a similar way as in Fig- +ure 6. For groups that contain at least 5 galaxies, we +show average radial profiles with tick line. Some inter- +esting trends from Figure 7 is that at each redshift bin, +more massive galaxies tend to have higher Σ∗(r) normal- +ization than less massive galaxies, indicating that the +excess in mass happens across the entire radius. More- +over, we also see that quiescent galaxies tend to have +higher Σ∗(r) normalization than the star-forming and +green-valley galaxies in all redshift. This is especially +clear in the most massive groups. It is also interesting +to see that Σ∗(r) profiles have negative gradient (i.e., de- +creasing mass with increasing radius) in all redshift and +mass bins, although the profiles seem to be shallower at +higher redshifts. +To see how galaxies quench their star formation, +specifically where in the galaxies the suppression of star +formation first happens and how it progresses over cos- +mic time, next we analyze the radial profiles of sSFR. +The radial profiles of sSFR are shown in Figure 8. As +we can see from this figure, the sSFR radial profiles +of the majority of our sample galaxies at z ≳ 2.5 are +broadly flat, while they show more diversity in shape at +lower redshifts. At 0.8 ≲ z ≲ 2.5, star-forming galax- +ies in our sample tend to have a flat or centrally-peaked +sSFR(r), while quiescent galaxies tend to have centrally- +suppressed sSFR(r). On the other hand, green-valley +galaxies in our sample seem to have broadly flat ra- +dial profile up to z ∼ 1.0, except in the most massive +group, where some of them show a sSFR suppression +in their central regions. +At lower redshifts, the ma- +jority of our sample galaxies have centrally-suppressed +sSFR(r). It is also interesting to see that the majority +of star-forming galaxies at 0.8 ≲ z ≲ 2.5 (in which the +cosmic noon epoch is covered), have a centrally-peaked +sSFR(r). This central elevation of sSFR is not observed +at higher redshifts, instead they have roughly flat radial +profiles. +Next, we analyze the radial profiles of the stellar popu- +lation age to see how this quantity varies radially within +our sample galaxies and investigate the underlying stel- +lar population properties causing the diversity in the +sSFR radial profiles. From our spatially resolved SED +fitting, we obtain maps of the mass-weighted ages, which +is the average age of stars in a stellar population as +weighted by the stellar mass formed over the course of +the star formation history. The age radial profiles are +shown in Figure 9. As can be seen from this figure, there +is a trend of increasing overall age of the stellar popu- +lations in galaxies over cosmic time, as indicated by the +increasing normalization of the radial profiles with de- +creasing redshift. The star-forming galaxies that have +a centrally-peaked sSFR(r) at 0.8 ≲ z ≲ 2.5 (possibly +around the cosmic noon epoch) as shown in Figure 8 +are likely in a phase of rapid star formation in their +centers (i.e., a nuclear starburst; e.g., Dekel & Burk- +ert 2014; Zolotov et al. 2015; Tacchella et al. 2016b), as +indicated by the young stellar populations (age ≲ 100 +Myr) in their central regions. At this epoch, green-valley +and quiescent galaxies tend to have radially decreasing +age profiles (i.e., negative gradient). At 0.3 < z < 0.8, +low-mass galaxies (log(M∗/M⊙) < 9.5) in all stages of +star formation have radially decreasing age radial pro- +files (i.e., negative gradient). A similar trend is still hold +for star-forming and green-valley galaxies in higher mass +group (9.5 < log(M∗/M⊙) < 10.5). On the other hand, +quiescent galaxies at this epoch tend to have overall flat +and old stellar populations across their entire radial ex- +tents, with higher normalization (i.e., older) than that +of star-forming and quiescent galaxies. +4.3. Compactness of the Spatial Distributions of +Stellar Mass and SFR +The centrally-peaked sSFR(r) of star-forming galax- +ies at around the cosmic noon epoch indicates that they +are likely undergoing a nuclear starburst that builds the +bulge component. +The centrally-suppressed sSFR(r) +profiles which start to emerge in quiescent galaxies at +around the same epoch can be caused by the cessation +of star formation in the center and/or a matured bulge +that has been formed in these galaxies. This trend pro- +vides a hint on how galaxies quench their star formation, +which seems to progress in an inside-to-outside manner +(i.e., quenching starts from the center and then prop- +agates outward). +At the same time, this trends may +indicate that galaxies build their central regions first, +forming a mature bulge, and then subsequently assem- +ble their disk through star formation (i.e., inside-out +growth). To further investigate this, next we compare +the compactness of the spatial distributions of M∗ and +SFR by means of the half-mass and half-SFR radii. +We compare the half-mass radius and the half-SFR ra- +dius in Figure 10. The half-SFR radius is a radius (mea- +sured along the elliptical semi-major axis) that covers +half of the total SFR. Similar as in Figure 6, the star- +forming, green-valley, and quiescent galaxies are shown +in blue, green, and red colors, respectively. +To com- + +15 +Figure 7. Radial profiles of the stellar mass surface density (Σ∗). The sample galaxies are divided into 4 bins of global M∗ +and 5 bins of redshift. At each group, we further classify the galaxies into star-forming, green-valley, and quiescent groups and +indicate them with different colors. For sub-groups that contain at least 5 galaxies, we show average radial profiles with tick +line. At each redshift bin, more massive galaxies tend to have higher Σ∗(r) normalization than less massive galaxies, indicating +that the excess in mass happens across the galaxy region. Quiescent galaxies tend to have higher Σ∗(r) normalization than +star-forming in all redshifts. This is especially clear in high M∗ bins. +pare the distributions of our star-forming and quies- +cent galaxies on this diagram, we plot the density con- +tours. +As can be seen from this figure, star-forming +galaxies broadly follow the one-to-one line, whereas qui- +escent galaxies are excess above the line. This means +that in quiescent galaxies, the spatial distribution of +SFR is more extended than that of stellar mass, indi- +cating that star formation is on-going in the disk and +less active in the central region. +It is also possible +that a massive bulge might has been formed in the cen- +ters, making a more compact stellar mass distribution. +On the other hand, star-forming galaxies are equally +distributed. Some star-forming galaxies have spatially +more compact star formation distribution than the stel- +lar mass (i.e., below the one-to-one line), which indicates +that active star-formation happens at their centers. On +the other hand, in the star-forming galaxies that have +extended star formation (i.e., above the one-to-one line), +the bulge might has been built and active star formation +is now progressing outward and building the disk. +It has been known that galaxy size correlates with +global M∗ for galaxies out to at least z ∼ 3 (i.e., the +size–mass relation; e.g., Shen et al. 2003; van der Wel +et al. 2014; Morishita et al. 2014; Yang et al. 2021). +However, most of the previous studies rely on galaxy +half-light radii as a measure of galaxy size. Since mass- +to-light ratios are not constant across a galaxy’s region, +but instead has a gradient, the half-light radii are not +a direct probe of the underlying stellar mass profiles. +Therefore, it is expected that the half-mass and half- + +7.5 11.0 +Star-forming +1011 +Green Valley +1010 +Quiescent +0.8 +Z*[M。kpc-2] +10° +>N +108 +0.3 +107 +106 +105 +1011 +0.8 11.0 +10-7 +8'0>z>0 +sSFR[yr-1]] +10-9 +10-11 +10-13 +Star-forming +GreenValley +Quiescent +10-15 +10-7 +0.8 z>9 +sSFR[yr-1] +10-9 +10-11 +1.5 +10-13 +10-15 +10-7 +2.5 11.0 +Mass-weighted Age [Gyr] +10- +8'0>z>0 +100 +10 +Star-forming +GreenValley +Quiescent +101 +Mass-weighted Age [Gyr] +0.8 1 +implies a more compact mass distribution than star for- +mation. Re,SFR/Re<< 1 may indicate an on-going nu- +clear starburst, while Re,SFR/Re>> 1 indicates that +a massive bulge has been built in the galaxies. From +Figure 13, we can see that from an early epoch up to +z ∼ 3.5, the two ratios are close to unity, indicating a +similar mass doubling time across the galaxy’s region. +In our sample, we only have star-forming and green val- +ley at this epoch. Quiescent galaxies emerge from z ∼ 3 +in our sample and we start to see more dispersion in the +two ratios at this later epoch. At 1.5 ≲ z ≲ 2.5 we see a +large fraction of our star-forming galaxies have a com- +pact star formation with Re,SFR/Re as low as ∼ −0.5 +dex and centrally-peaked sSFR with sSFRin/sSFRout of +up to ∼ 3 dex. +In contrast to this, the majority of +green valley and quiescent galaxies at this epoch have +Re,SFR/Re> 1 and sSFRin/sSFRout< 1. At later epoch +(z ≲ 1.5), the majority of quiescent, green valley, and +star-forming galaxies have extended SFR distributions +and centrally-suppressed sSFR. The toy model has con- +stant ratios of ∼ 1 from the early epoch up to z ∼ 1.5 +after which its sSFR declines and SFR distribution be- +comes more extended. There is an indication that its +sSFRin/sSFRout actually increases a little bit above 1 +at z ∼ 1.5. +The trends observed at z ∼ 0 indicate that local spiral +galaxies have overall extended SFR distributions and +centrally-suppressed sSFR radial profiles. These trends +agree with the scenarios inferred from our results in the +current work and provide a nice extension to our results +toward low redshift, complementing the general picture. +Overall, the above trend agrees with the inside-out +growth and quenching scenarios. At early cosmic time, +galaxies get steady gas accretion for star formation but +they yet to form bulge and the star formation is likely +distributed evenly across their regions. At z ∼ 2, which +coincides with the peak epoch of the cosmic SFRD and +perhaps the cosmic gas accretion (Madau & Dickinson +2014), star-forming galaxies in our sample may experi- +ence gas compaction event that later build bulge in their +centers. After that, quenching might has been started +in their central regions, but star formation is still active +in the disk that further build the disk. In addtion to in- +situ star formation, minor mergers can also contribute +to the buildup of stellar mass in the disk and grow the +galaxy size. +5.2. The buildup of the Central Stellar Mass Density +Over Cosmic Time +As we have seen in Section 4.4, our sample galax- +ies exhibit a tight relationship between the global M∗ +and the stellar mass surface density at the central 1 kpc +(Σ∗,1kpc). Σ∗,1kpc is a good indicator for quiescent galax- +ies because they form a rather distinct sequence at the +tip of the overall M∗–Σ∗,1kpc relation and has a shal- +lower slope. +Here we discuss the evolution of Σ∗,1kpc +with redshift to see how the central bulge is built over +cosmic time in our sample galaxies. As Σ∗,1kpc develops +over time, it is also interesting to analyze how sSFR at +the central 1 kpc evolves following the development of +Σ∗,1kpc. The evolution of these two quantities are shown +in Figure 14. As we can see from this figure, Σ∗,1kpc +tend to increase with cosmic time, whereas sSFR1kpc de- +clines with cosmic time. The quiescent galaxies tend to +have higher Σ∗,1kpc and lower sSFR1kpc in all redshifts. +sSFR1kpc of quiescent and green-valley galaxies tend +to be declined more rapidly than that of star-forming +galaxies. Interestingly, the overall sSFR1kpc of our star- +forming galaxies does not decline much from z = 6 up to +z ∼ 1.5. There is an indication that sSFR1kpc of some +star-forming galaxies even increases at 1.5 ≲ z ≲ 2.5. +The black profiles show expected evolution of Milky +Way analogs based on our toy model derived in Sec- +tion 5.1. Its Σ∗,1kpc increases by ∼ 2 magnitude over +0.5 ≲ z ≲ 4.5, while its sSFR1kpc decreases significantly +(∼ 3.7 magnitude) over the same period. This implies +that the central SFR within 1 kpc also decreases with +time. +At 1.5 ≲ z ≲ 2.5, the sSFR1kpc of this model +seems to be constant. +In addition to our toy model, +we also compare our observational trend with the pre- +dictions from the VELA zoom-in cosmological hydrody- +namical simulations (Ceverino et al. 2014; Zolotov et al. +2015) that was analyzed by Tacchella et al. (2016a). + +21 +Figure 13. Evolution of the ratio between the half-SFR and half-mass radii (top panel) and the ratio between the sSFR +inside and outise of the half-mass radius (bottom panel). The overall symbols and color-coding are the same as those in the +right panel of Figure 5. The profiles shown with black diamond symbols represent an expectation from a toy empirical model +for the evolution of the Milky Way analogs. For comparison, we also show the trends observed in local spiral galaxies from +Abdurro’uf & Akiyama (2017) and Abdurro’uf et al. (2022a). At z ≳ 3.5, the two ratios are close to unity, indicating a similar +mass doubling time across the galaxy’s region. At 1.5 ≲ z ≲ 2.5, many star-forming galaxies have a compact star formation +(Re,SFR/Re as low as ∼ −0.5 dex and sSFRin/sSFRout up to ∼ 3 dex). The majority of the green valley and quiescent galaxies +at this epoch have Re,SFR/Re> 1 and sSFRin/sSFRout< 1. At later epoch (z ≲ 1.5), the majority of quiescent, green valley, +and star-forming galaxies have extended SFR distributions and centrally-suppressed sSFR. Overall these trends point toward +the inside-out growth and quenching scenario. +These predictions, which are shown in red profiles, are +obtained by averaging the evolutionary trends of rel- +atively massive galaxies in the simulations that have +log(M∗/M⊙) = 10.2 at z = 2. +This simulation was +run over 1 < z < 7. +The Σ∗,1kpc and sSFR1kpc pre- +dicted from the cosmological simulation at z ∼ 6 are in +good agreement with our observations. The evolution +of the Σ∗,1kpc and sSFR1kpc from the simulation seem +to be consistent with the evolution of the progenitors +of local massive quiescent galaxies, as implied from our +observations. However, there is an excess of Σ∗,1kpc in +z ∼ 0.5 quiescent galaxies compared to the simulation. +Those galaxies are likely members of the WHL0137−08 +or MACS0647+70 clusters. In cluster, galaxies could ac- +crete more mass through minor mergers which can result +in denser stellar mass in their central regions. +The trends at z ∼ 0 from Abdurro’uf & Akiyama +(2017) and Abdurro’uf et al. (2022a) provide a good ex- +tension for our results in the current work. They overall +agree with the picture of increasing Σ∗,1kpc and decreas- +ing sSFR1kpc with cosmic time. However, we also see a +lower Σ∗,1kpc in the local spiral galaxies than in the qui- +escent galaxies at z ∼ 0.5 that are possibly the cluster +members. If we ignore these galaxies and assume that +Σ∗,1kpc trend from the VELA simulation will evolve to +have similar value as those of the observed Σ∗,1kpc of +local galaxies (which is likely given its shallow slope at +z ∼ 1), we see a possible saturation of central mass den- +sity in galaxies. +Next, we compare between the effects of Σ∗,1kpc +and the global M∗ on sSFRin/sSFRout ˙We show the +Σ∗,1kpc–sSFRin/sSFRout and M∗–sSFRin/sSFRout rela- + +Z2 +0 +1 +3 +4 +5 +6 +７78 +( ) +1.0 +WHL0137-08cluster +WHL0137-08blankfield +0.8 +MACS0647+70cluster +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +3 +2 +0 +++Toymodel:MilkyWaylike +4 +AZ=0Abdurro'uf&Akiyama(2017) +¥ z=0 Abdurro'uf et al. (2022a) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +log(1 + z)22 +Figure 14. +The builtup of the stellar mass density at the central 1 kpc (Σ∗,1kpc) over cosmic time (top panel) and the +evolution of sSFR at the central 1 kpc (sSFR1kpc; bottom panel). The profiles shown with black diamond symbols represent an +expectation from a toy empirical model for the evolution of the Milky Way analogs, whereas the red pentagon symbols show +a prediction from zoom-in cosmological hydrodynamical simulations by Tacchella et al. (2016a) for the evolution of massive +galaxies (log(M∗(z = 2)/M⊙) > 10.2). +tions in Figure 15. +We can see from this figure that +sSFRin/sSFRout is correlated with both Σ∗,1kpc and the +global M∗ such that increasing Σ∗,1kpc and M∗ cor- +respond to a steeper sSFR decline in the central re- +gions. However, the M∗–sSFRin/sSFRout relation seems +to be broader and less significant compared to M∗– +sSFRin/sSFRout. The majority of those that have cen- +tral sSFR suppression are the quiescent and green-valley +galaxies, whereas a significant fraction of star-foroming +galaxies have broadly flat or centrally-peaked sSFR ra- +dial profile (i.e., negative gradient). This further suggest +that Σ∗,1kpc is a good predictor for quiescent galaxies. +6. SUMMARY AND CONCLUSIONS +We perform spatially resolved SED fitting on 444 +galaxies at 0.3 < z < 6.0 in two clusters (WHL0137−08 +and MACS0647+70) and a blank field using imaging +data from JWST and HST in up to 13 bands. We use +piXedfit throuhgout the analysis. This software can si- +multaneously perform image processing, pixel binning, +and spatially resolved SED fitting. By using the maps of +spatially resolved stellar population properties (on kpc +scales) obtained from this analysis, we investigate how +galaxies grow their structures and quench their star for- +mation activities across cosmic time. Overall, our key +results are summarized in the following: +1. The normalization of the stellar mass surface den- +sity radial profiles (Σ∗,1kpc(r)) increases with in- +creasing cosmic time and global M∗. +At each +redshift, quiescent galaxies tend to have higher +Σ∗,1kpc across the entire radius than green-valley +and star-forming galaxies. The sSFR radial pro- +files (sSFR(r)) show more variations across red- +shift and global M∗. The sSFR(r) are broadly flat +at 2.5 ≲ z ≲ 6.0 in all galaxies. At 0.8 ≲ z ≲ 2.5, +less massive (log(M∗/M⊙) < 11.0) star-forming +galaxies have flat or centrally-peaked sSFR(r), +whereas the majority of quiescent galaxies have +centrally-suppressed sSFR(r). +At lower redshift +(z < 0.8), almost all galaxies (regardless of M∗ and + +N2 +0 +1 +3 +4 +5 +6 +78 +12 +WHL0137-08cluster +WHL0137-08blankfield +11 +MACS0647+70cluster +10 +9 +8 +6 +-6 +-8 +-10 +12 +Toymodel:MilkyWaylike +-14 +z=0Abdurro'ufAkiyama(2017) +¥ z=0 Abdurro'uf et al. (2022a) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +log(1 + z)23 +Figure 15. The sSFRin/sSFRout ratio as a function of the central mass density (left panel) and global M∗(right panel). The +overall symbols in this figure is the same as those in Figure 10. Galaxies that are massive and have high Σ∗,1kpc tend to have +centrally-suppressed sSFR profiles. This trend is predominantly observed in quiescent galaxies. +star formation stage) have centrally-suppressed +sSFR(r). The radial profiles of stellar ages show +that those galaxies with centrally-peaked sSFR(r) +have very young stellar populations in their cen- +tral regions, indicating an on-going nuclear star- +burst. We also see an increasing normalization of +age radial profiles and flattening of their slopes +with increasing cosmic time. +2. The majority of quiescent galaxies have larger +half-SFR radius than half-mass radius, indicat- +ing that they have extended spatial distribution of +SFR and compact distribution of stellar mass. In +contrast, some star-forming galaxies, especially at +high redshifts, have half-SFR radius being roughly +similar or smaller to half-mass radius, whereas +those at low redshifts have half-SFR radius be- +ing larger than half-mass radius. The half-mass +radius of the star-forming galaxies is on average +larger than the quiescent galaxies in all global M∗. +3. We observe a tight correlation between the global +M∗ and stellar mass density at the central 1 kpc +(Σ∗,1kpc) with 0.38 dex, indicating that galaxies +grow their central mass density hand-in-hand with +their global M∗. The quiescent galaxies reside in +a sequence at the tip of the overall relation and +have a shallower slope. This trend indicates that +Σ∗,1kpc is a good predictor of quenching, where +passive galaxies tend to have higher Σ∗,1kpc and +global M∗. The shallower slope of M∗–Σ∗,1kpc in +quiescent galaxies suggest that their central mass +density has reached a saturation point (i.e., a ma- +ture bulge has been formed). +4. We investigate the evolution of the Re,SFR/Re and +sSFRin/sSFRout ratios with redshift to try to un- +derstand how galaxies grow their structures and +quench their star formations over cosmic time. We +find that the ratios are close to unity from the early +epoch up to z ∼ 3.5 and the ratios start to deviate +from unity since then. At 1.5 ≲ z ≲ 2.5, a fraction +of our star-forming sample has a low Re,SFR/Re +and high sSFRin/sSFRout, indicating that they +may be experiencing a nuclear starburst. At the +later epoch, most of our sample galaxies, especially +quiescent and green-valley have a high Re,SFR/Re +and low sSFRin/sSFRout, suggesting that massive +bulges might have been formed in these galaxies +and the star formation has been quenched in their +central regions. +5. We also investigate the evolution of Σ∗,1kpc and +sSFR at the central 1 kpc (sSFR1kpc). +In gen- +eral, we see an increasing Σ∗,1kpc and decreasing +sSFR1kpc with cosmic time, indicating the buildup +of the central bulge component and the quench- +ing process in the central region of the galax- +ies. We also find that quiescent galaxies tend to +have higher Σ∗,1kpc and lower sSFR1kpc than star- +forming galaxies in all redshifts. +6. Finally, we find a relationship between Σ∗,1kpc and +sSFRin/sSFRout with a negative slope, indicating +that galaxies that have more massive Σ∗,1kpc tend +to have steeper central suppression in their sSFR + +WHL0137-08 cluster +4 +WHL0137-08blankfield +MACS0647+70 cluster +2 +4 +6 +7 +8 +9 +10 +11 +7 +8 +9 +10 +11 +12 +log(Z*, 1kpc[Mokpc-2 ]) +log(M*[Mol)24 +radial profiles. The quiescent galaxies tend to have +higher Σ∗,1kpc and sSFRin/sSFRout< 1, suggest- +ing that the formation of bulge might happen si- +multaneously with the quenching of star formation +in the central regions. +Our study in this paper demonstrates the great po- +tentials of spatially resolved SED analysis using JWST +imaging data. It is interesting to extend this study with +larger sample galaxies taken from various surveys to bet- +ter understand the buildup of stellar mass in galaxies +and the growth of their structures over cosmic time. +More comprehensive comparisons with zoom-in cosmo- +logical simulations would help to better understand the +underlying physics. We will pursue this in our future +works. +ACKNOWLEDGEMENTS +This work is based on observations made with +the NASA/ESA/CSA James Webb Space Telescope +(JWST). The data were obtained from the Mikulski +Archive for Space Telescopes (MAST) at the Space Tele- +scope Science Institute (STScI), which is operated by +the Association of Universities for Research in Astron- +omy (AURA), Inc., under NASA contract NAS 5-03127 +for JWST. These observations are associated with pro- +gram JWST GO 2282, JWST GO 1433, and HST GO +14096, 15842, 16668, 9722, 10493, 10793, and 12101. +A and TH are funded by a grant for JWST-GO- +01433 provided by STScI under NASA contract NAS5- +03127. +The CosmicDawn Center is funded by the +Danish National Research Foundation (DNRF) un- +der grant #140. +PD acknowledges support from +the NWO grant 016.VIDI.189.162 (“ODIN”) and from +the European Commission’s and University of Gronin- +gen’s CO-FUND Rosalind Franklin program. +RAW +acknowledges support from NASA JWST Interdisci- +plinary Scientist grants NAG5-12460, NNX14AN10G +and 80NSSC18K0200 from GSFC. AZ and AKM ac- +knowledge support by grant 2020750 from the United +States-Israel Binational Science Foundation (BSF) and +grant 2109066 from the United States National Sci- +ence Foundation (NSF), and by the Ministry of Sci- +ence & Technology, Israel. MO acknowledges support +from JSPS KAKENHI Grant Numbers JP22H01260, +JP20H05856, JP20H00181, JP22K21349. AA acknowl- +edges support from the Swedish Research Council +(Vetenskapsr˚adet project grants 2021-05559). +EV ac- +knowledges financial support through grants PRIN- +MIUR 2017WSCC32, 2020SKSTHZ and the INAF GO +Grant 2022 (P.I. E. Vanzella). +Facilities: HST, JWST +Software: +astropy (Astropy Collaboration et al. +2013, 2018, 2022), piXedfit (Abdurro’uf et al. 2021, +2022c), SExtractor (Bertin & Arnouts 1996), sep +(Barbary 2016), photutils (Bradley et al. 2022a), gri- +zli (Brammer et al. 2022), eazypy (Brammer et al. +2008). +APPENDIX +A. ROBUSTNESS OF THE SED FITTING METHOD: FITTING TESTS WITH MOCK SEDS +To test the robustness of our SED fitting method on this new set of photometric data, we perform SED-fitting tests +using semiempirical mock SEDs, following a similar procedure as performed in Abdurro’uf et al. (2022a, Appendix A +therein). We draw the parameter values for our mock SEDs from the measured parameters of real galaxies. In this +case, we use the measured parameters obtained from our fitting to the SEDs within the central effective radius that +were used for determining our photometric redshifts (see Section 3.5). Here, we use 290 galaxies selected randomly +from 354 galaxies in the WHL0137−08 (before further exclusion). We prefer to use the parameters of real galaxies for +generating mock SEDs, instead of drawing them randomly because we can not be sure that the combinations of those +random parameters are physically realistic. Using the set of parameters of 290 galaxies, we generate mock SEDs using +the same modeling setup as in our main analysis. We generate two sets of mock SEDs. The first one with 12 filters +of JWST and HST, the same set of filters as available for the WHL0137−08 cluster. The second set with 8 JWST +filters, excluding HST filters. We then inject Gaussian noises assuming S/N of 20 in all filters. After that, we fit the +mock SEDs using the same method as we used in the main analysis of this paper. For simplicity, here we fix redshift. +We present the results in Figure 16, which shows the comparisons between the best-fit parameters derived from SED +fitting and the true values form the mock SEDs. Histograms in the insets show ratios between the best-fit parameters +and the true values. Results of the SED fitting with two sets of photometry are shown with different symbols and the +histograms are shown with different colors. The data points are color-coded based on their redshifts. + +25 +Figure 16. Comparisons between the best-fit parameters derived from SED-fitting tests using mock SEDs and the ground +truth. In this test, two sets of photometry are used, one with combined 12 filters of HST and JWST and the other with 8 filters +of JWST only. The results of those two SED fitting tests are shown with different symbols (circle and squares). Histograms in +the insets show ratios between the best-fit parameters and the true values from the mock SEDs. The color-coding of the data +points represents redshift. +Overall, our SED fitting can recover the true parameters reasonably well. Stellar mass and mass-weighted age are +recovered very well in both 8-band and 12-band SED fitting with small offset (≲ 0.06 dex) and small standard deviation +(≲ 0.3 dex). It is interesting to see that mass-weighed age is well recovered here. This may be due to the fact that +our photometry covers the Balmer break and we have good photometry in rest-frame NIR from JWST, which also +provides good constraint for M∗ (see Appendix B). The stellar metallicity (Z) and dust attenuation in the diffuse +ISM (AV,2) are also recovered well with an offset of ≲ 0.07 dex and a standard deviation of ≲ 0.5 dex although they +look to be more scattered due to its small dynamical range. The SFR is more difficult to be recovered for passive +galaxies (log(SFR) ≲ −1) than for star-forming ones. For whole sample, the SED fitting with 12 bands gives better +SFR estimates (a small offset of 0.07 dex and scatter of 0.64 dex) than with only JWST bands (offset of 0.16 dex and +scatter of 0.77 dex). +B. THE AGE–DUST–METALLICITY DEGENERACIES +The addition of the JWST NIRCam data extends the wavelength coverage up to roughly the rest-frame NIR for +our sample galaxies. This sufficiently wide wavelength coverage has a potential to break the well-known age–dust– +metallicity degeneracy in SED fitting, which is very important for the analysis in this paper. In particular, it is crucial +to be able to determine if the reddening observed in the central region of some galaxies in our sample is due to aging +(i.e., quiescence) or the dust attenuation. To check if our data set provides a sufficient constraint for resolving this +degeneracy, we examine model color-color diagrams among the NIRCam filters. We base our analysis on the observed- +frame, instead of the rest-frame, and explore different sets of filters for different redshifts. To generate model SEDs, +we use overall similar setting as that used in the main analysis of this paper and assume a double power-law SFH with +α = 3.0, β = 0.5, and τ = 1.5 Gyr, and M∗ = 1010.5M⊙. We then generate model SEDs in grids of age, AV,2, and +metallicity. For simplicity, we assume AV,1 = 1.5 × AV,2. + +4.5 +12Bands +12 +8 Bands +4.0 +11 +3.5 +log(M*, fit[Mo]) +3.0 +Redshift +10 +2.5 +2.0 +R +9 +μ=-0.06g=0.29 +u=-0.040=0.29 +1.5 +12Bands +8Bands +8 +50 +1.0 +0 +0.5 +log(fit/true) +8 +9 +10 +11 +12 +log(M*, true[Mo])3 +4.5 +12Bands +8 Bands +2 +4.0 +1 +3.5 +log(SFRrit[ Moyr +0 +3.0 +Redshift +-1 +2.5 +2.0 R +-2 +μ= 0.07 +g= 0.64 +μ=0.16 +0=0.77 +1.5 +-3 +12 Bands +50 +8 Bands +25 +1.0 +-4 +0 +2 +0.5 +log(fit/true) +-5 +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +log(SFRtrue[M。yr-11)4.5 +12 Bands +1.0 +8 Bands +[Gyr]) +4.0 +0.5 +3.5 +fit +0.0 +3.0 +-0.5 +Redshift +log(mass-weighted +2.5 +-1.0 +2.00 +R +-1.5 +μ= - 0.03 g= 0.29 +u=-0.02g=0.29 +1.5 +-2.0 +12Bands +100 +8 Bands +50 +1.0 +-2.5 +0 +0.5 +log(fit/true) +-3.0 +3.0-2.5-2.0-1.5-1.0-0.50.0 +0.5 +1.0 +log(mass-weighted age - true [Gyr])4.5 +12Bands +8 Bands +0.0 +4.0 +3.5 +-0.5 +(Z/Z)60l +3.0 +Redshift +2.5 +1.0 +2.0 +R +0.07 +0=0.34 +0.05 +=0.32 +1.5 +12Bands +-1.5 +8Bands +50 +1.0 +0 +0.5 +log(fit/true) +-2.0 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +log(Ztrue/Zo)3.5 +4.5 +12Bands +8 Bands +3.0 +4.0 +3.5 +2.5 +[mag] +3.0 +Redshift +2.0 +2.5 +fit +2 1.5 +Av, +2.00 +R +0.030=0.51 +1.0 +0.05 =0.53 +1.5 +12Bands +8Band +50 +0.5 +1.0 +0 +log(fit/true) +0.5 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Ay.2.true L +[mag]26 +Figure 17. The patterns of model stellar populations on the observed-frame color-color diagrams involving JWST NIRCam +filters. We plot the patterns of models at z = 2.5 and z = 3.5 on the color-color diagrams to check how good the NIRCam +photometry in resolving the degeneracies among age, dust attenuation, and metallicity in high redshifts. +Different colors +represent different age, whereas an incerasing marker size represents an increasing Z (for square symbol) and increasing AV,2 +(for circle symbol). The rightmost panel shows examples of the model SEDs of old less-dusty (brown color), young metal-rich +(purple), and young dusty (green) galaxies at z = 3.5. The reddening effects by aging and dust are almost orthogonal with +each other (i.e., distinguishable) on the F090W−F200W vs. F200W−F444W diagram. The reddening effects by aging and +dust attenuation are distinguishable on the F090W−F277W vs. +F277W−F444W for z = 2.5 and the F150W−F356W vs. +F356W−F444W for z = 3.5. +Figure 17 shows the color-color diagrams of models at z = 2.5 and 3.5. Different colors represent different ages, +whereas increasing marker size represents increasing metallicity (for square symbol in the first column) and AV,2 (for +circle symbol in the second column). For the two plots in the first column, we fix AV,2 = 0.0, whereas for the two plots +in the second column, we fix Z = Z⊙. F090W−F200W vs. F200W−F444W diagram at the two redshifts seems to be +able to distinguish the reddening effect by age and metallicity in such a way that the both effects are almost orthogonal +with each other. An increasing Z at a fixed age corresponds to reddening in F200W−F444W and roughly constant +F090W−F200W (i.e., vertical shift on the diagram). On the other hand, an increasing age tends to make reddening +in the two colors (i.e., a diagonal shift on the diagram) for galaxies with low Z, while it corresponds to a roughly +horizontal shift for galaxies with high Z. In the second column, we show a relative effect of the dust attenuation and +aging on F090W−F277W vs. F277W−F444W (for z = 2.5) and F150W−F356W vs. F356W−F444W (for z = 3.5) +diagrams. The two effects are distinguishable in these two diagrams with dust attenuation seem to make a diagonal +shift, whereas an aging effect is roughly orthogonal to it. For an illustration, in the rightmost panel we show examples +of the model SEDs of an old less-dusty (brown color; age1.5 Gyr, log(Z/Z⊙) = −0.5, AV,2 = 0.1 mag), young metal- +rich (purple; age0.1 Gyr, log(Z/Z⊙) = 0.2, AV,2 = 0.1 mag), and young dusty (green; age0.1 Gyr, log(Z/Z⊙) = −0.5, +AV,2 = 2.0 mag) galaxies z = 3.5. We normalize the SEDs by dividing them with the F200W flux. We also show the +transmission curves of the filters that are used in the color-color diagrams for z = 3.5 in the left two columns. As we +can see from this figure, the three model SEDs are distinguishable with NIRCam photometry. The reddening due to +the dust attenuation is easily recognisable in the rest-frame UV to NIR, while that due to the metallicity is not easily +recognisable in the rest-frame UV colors but it is detectable in the rest-frame around the Balmer break (∼ 4000 ˚A) +and NIR. +C. CONSTRUCTION OF EMPIRICAL POINT SPREAD FUNCTIONS AND CONVOLUTION KERNELS +We generate the empirical PSFs of the HST ACS and JWST NIRcam filters in the WHL0137−08 and MACS0647+70 +clusters by stacking images of the bright isolated stars in those fields. For this, we use some functions in photutils + +0.6 +1.0- +z=2.5 +z=2.5 +Av.2 = 2.0 mag +z=3.5 +old & less dust +0.4 +100.2 +1.5 +young & metal rich +0.8 +F444W +0.2 +young & dusty +100.0 +1.0 +0.0 +101 +1 +F200W +0.2 +0.5 +10-0.4 +0.3/ +0.4 - +0.05 +0.7 +1.0 +0.2 +[arbitrary unit] +0.6 +0.2 +5 +2.0 Gyr +Z = 10-0.8Zo +0.5 +-0.8 1 +0.0 1 +-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 +0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 +4.0 +F090W - F200W +F090W-F277W +0.4 +z=3.5 +F6'0 + z=3.5 +Av,2 = 2.0 mag +100 +1.5 +0.8 +0.2 +100.2 +F444W +1.0 +0.0 +0.5 +- +F200W +0.5 +10-0.4 +0.0 +-0.2 +0.02 +0.5 +0.3 +0.4 - +0.1 +1.0 +0.2 +1.5 Gyr +F090W +F150W F200W +F356W +F444W +Z = 10-0.8Zo +0.2 +10-1 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +1 +2 +3 +4 +5 +F090W - F200W +F150W- F356W +Wavelength [μm]27 +package (Bradley et al. 2022a). We show the encircled energy of the empirical PSFs in the left column of Figure 18. +After generating the PSFs, we then construct the convolution kernels that can be used for PSF matching. As described +in Section 3.2, we perform PSF matching to homogenize the PSF sizes of our imaging data to match the F444W PSF, +which is the largest among the filters used in our work. We also use photutils for generating the kernels. To check +the reliability of our kernels and PSF matching, we convolve the PSF images of the filters other than F444W with the +kernels and compare the encircled energy of the convolved-PSFs to that of F444W filter. We show this comparison in +the right column of Figure 18. As we can see from this figure, there is an overall good agreement among the encircle +energy after the PSF matching. 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West 120th Street, New York, NY 10027, USA 7Kapteyn Astronomical Institute, University of Groningen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Box 800, 9700 AV Groningen, The Netherlands 8School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287-1404, USA 9Physics Department, Ben-Gurion University of the Negev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Box 653, Be’er-Sheva 84105, Israel 10Center for Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan 11Department of Physics, Graduate School of Science, Chiba University, 1-33 Yayoi-Cho, Inage-Ku, Chiba 263-8522, Japan 12Instituto de F´ısica de Cantabria (CSIC-UC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Los Castros s/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 39005 Santander,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Spain 13INAF - Osservatorio Astronomico di Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' via di Frascati 33,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Sweden 15Jodrell Bank Centre for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' University of Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Oxford Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Manchester UK 16Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' College Park,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' NASA/GSFC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Greenbelt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' MD 20771 19INAF – OAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Osservatorio di Astrofisica e Scienza dello Spazio di Bologna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' via Gobetti 93/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' I-40129 Bologna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Italy 20Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' USA Submitted to ApJ ABSTRACT We study the spatially resolved stellar populations of 444 galaxies at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 < z < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 in two clusters (WHL-0137-08 and MACS0647+70) and a blank field, combining imaging data from HST and JWST to perform spatially resolved spectral energy distribution (SED) modeling using piXedfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The high spatial resolution of the imaging data combined with magnification from gravitational lensing in the cluster fields allows us to resolve some galaxies to sub-kpc scales (for 109 of our galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' At redshifts around cosmic noon and higher (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 ≲ z ≲ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0), we find mass doubling times to be independent of radius, inferred from flat specific star formation rate (sSFR) radial profiles and similarities between the half-mass and half-SFR radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' At lower redshifts (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 ≲ z ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5), a significant fraction of our star-forming galaxies show evidence for nuclear starbursts, inferred from centrally elevated sSFR, and a much smaller half-SFR radius compared to the half-mass radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' At later epochs, we find more galaxies suppress star formation in their center but are still actively forming stars in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Overall, these trends point toward a picture of inside-out galaxy growth consistent with theoretical models and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We also observe a tight relationship between the central mass surface density and global Corresponding author: Abdurro’uf fabdurr1@jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='02209v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='GA] 5 Jan 2023 ID2 stellar mass with ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='38 dex scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Our analysis demonstrates the potential of spatially resolved SED analysis with JWST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Future analysis with larger samples will be able to further explore the assembly of galaxy mass and the growth of their structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Keywords: Galaxy evolution (594) — Galaxy formation (595) – Galaxy clusters (584) – Galaxy quench- ing (2040) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' INTRODUCTION Over the last few decades, multiwavelength studies of galaxies throughout cosmic history reveal that the global star formation rate density (SFRD) in the universe was increasing with cosmic time from the reionization epoch and reached a peak at z ∼ 2 (∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 Gyr after the Big Bang;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' cosmic noon) after which it declined exponentially toward the present day (Madau & Dickinson 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In this picture, it is estimated that ∼25% of the present- day stellar mass density (SMD) was formed before the peak of the cosmic SFRD, around half of the SMD was formed during 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0, and another ∼25% was formed since z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', around the last half of the universe’s age;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Madau & Dickinson 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Although the cosmic SFRD at early cosmic time is still debated due to the dust obscuration effects (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Fudamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2018), an emerging picture is that cosmic SMD increases with cosmic time since the epoch of reionization, which is believed to take place before z ∼ 6 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Treu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' McGreer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Dayal & Ferrara 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Observations also revealed that most of the star for- mation occurs in galaxies that lie in the so-called star- forming main sequence (SFMS), which is a tight nearly linear correlation between the integrated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', global) star formation rate (SFR) and stellar mass (M∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Brinch- mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Elbaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Noeske et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Whitaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2012, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Salmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Tomczak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Santini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Leja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This relation has been shown to hold at any epoch with a nearly constant scatter (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 dex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Whitaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2014), suggesting that galaxies grow in mass over cosmic time in a state of self-regulated semi- equilibrium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Bouch´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Tacchella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2016b, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Un- derstanding this process in detail as well as the mech- anisms that shut down star formation in galaxies and move them out of the SFMS onto the “quenched” popu- lation requires knowledge of not only integrated galaxy ∗ Hubble Fellow properties but also spatially resolved structures within galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The study of spatially resolved properties of galax- ies with integral field spectroscopy (IFS) and high spa- tial resolution imaging have been providing important insights toward a better understanding of galaxy evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Among the important findings is the realiza- tion that some of the well-known scaling relations ob- served on global scales are originated from similar re- lations on kpc scales within galaxies (see a review by S´anchez 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This includes the spatially resolved star- forming main sequence relation, a relationship between SFR surface density (ΣSFR) and M∗ surface density (Σ∗), which is thought to be more fundamental than the global SFMS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', S´anchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Wuyts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Cano-D´ıaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Abdurro’uf & Akiyama 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Hsieh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Abdurro’uf & Akiyama 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Morselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This emphasizes the necessity of studying the spatially resolved properties of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Spatially resolved studies of high redshift galaxies (z ∼ 1 − 4) have hinted on how galaxies assembled their structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The emerging picture from these stud- ies is that galaxies grow their mass in an inside-to- outside manner (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', inside-out growth scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', van Dokkum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Morishita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2016) and cease their star for- mation activities in a similar manner (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', inside-out quenching scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Tacchella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Abdurro’uf & Akiyama 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Tacchella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2016) analyzed the spatially re- solved distributions (on kpc scales) of Hα emission and M∗ of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7 < z < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 galaxies using the Hubble Space Telescope (HST)/WFC3 grism data from the 3D-HST survey (Skelton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' They found that the spa- tial distribution of Hα emission in the galaxies is more extended than the stellar mass, suggesting that the past star formation in the galaxies have accumulated stellar mass in the center and now the star formation progresses outward to assemble the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Tacchella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2015) an- alyzed the spatial distributions of SFR and M∗ of ∼ 30 star-forming galaxies at z ∼ 2 using IFS data from the SINS/zC-SINF survey (F¨orster Schreiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' They observed that massive galaxies (M∗ ≳ 1011M⊙) 3 in their sample have a centrally-suppressed specific SFR (sSFR) radial profile and a massive central spheroid that is as dense as the centers of local early-type galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In contrast to this, less massive galaxies in their sample have broadly flat sSFR radial profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This trend in- dicates that massive galaxies at this epoch might have started a quenching process in their central regions and assembled a mature bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The buildup of the central stellar mass density is likely correlated with the quenching process in galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The central stellar mass density within a 1 kpc radius (Σ∗,1kpc) has been shown to be a good predictor for qui- escence, where galaxies with high Σ∗,1kpc are tend to be red and quiescent, whereas galaxies with low Σ∗,1kpc are tend to be blue and star-forming (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Tacchella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2015, 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Barro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Whitaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' It has also been shown that Σ∗,1kpc is tightly correlated with the global M∗, suggesting that M∗ of galaxies grow hand-in-hand with the central mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In this Σ∗,1kpc–M∗ rela- tion, quiescent galaxies reside in a sequence at the tip of the overall relation and have a shallower slope than the relation with star-forming galaxies only, indicating a formation of a matured bulge in the quiescent galax- ies (Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Tacchella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Barro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Galaxies also grow their sizes hand-in-hand with the global M∗, as indicated by the size–mass relation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' van der Wel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Suess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Previous studies have shown that star-forming and quiescent galaxies follow very different size–mass relations where quiescent galaxies tend to be more com- pact (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', having smaller size) in all M∗ and exhibit steeper relation than the star-forming galaxies (van der Wel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' A possible explana- tion for this trend is that star-forming galaxies build their mass at all radii by mostly in-situ star forma- tion, whereas quiescent galaxies grow inside-out through mergers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', van Dokkum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Previous studies, some of which are mentioned above, have used HST for resolving galaxies out to z ∼ 3, roughly a limit where galaxies can be resolved well by the telescope, given its spatial resolution and depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Furthermore, the wavelength coverage of HST only cov- ers the rest-frame ultraviolet (UV) and a small portion of the optical at z ∼ 3, making it difficult to robustly derive the stellar mass as well as the other stellar population properties, which typically requires a rest-frame near- infrared (NIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Forcing to include NIR imaging from the ground-based telescopes would need to sacrifice the spatial resolution of HST (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' With the advent of the James Webb Space Telescope (JWST) NIRCam observations (Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022), with its high spatial resolution, depth, and its coverage in NIR, now we can push the analysis of spatially resolved SED of galaxies to higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Some very recent studies have used JWST/NIRCam imaging to study the inter- nal structures and morphology of galaxies at z > 3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Ferreira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Kartaltepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Gim´enez-Arteaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022), and even resolving a lensed galaxy at z ∼ 11 (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In this paper, we use imaging data from HST/ACS and JWST/NIRCam to analyze the spatially resolved SEDs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 < z < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 galaxies in the sightlines of WHLJ013719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8–082841 (hereafter WHL0137−08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' RA = 01:37:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0, DEC = −08:27:23, J2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='566;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Wen & Han 2015) and MACSJ0647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7+7015 (hereafter MACS0647+70;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' RA = 06:47:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='03, DEC = +70:14:49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7, J2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='591;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Ebeling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2007) clus- ters and examine the spatial distributions of their stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Our main goal is to get hints on the as- sembly of galaxy structures over cosmic time, especially how galaxies build their stellar masses and quench their star formation activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The high spatial resolution of JWST/NIRCam combined with magnification from gravitational lensing in the cluster fields, allow us to re- solve high-redshift galaxies down to sub-kpc scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Our method using piXedfit (Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022c) can simultaneously process imaging data, perform pixel bin- ning to optimize the signal-to-noise (S/N) ratio of the spatially resolved SEDs, and perform SED fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The wavelength coverage of HST/ACS and JWST/NIRCam allow us to get full coverage of the rest-frame UV to NIR for the majority of our sample, which can give a strong constraint on model SEDs and break the age– dust–metallicity degeneracy (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' While IFS observation at z ≳ 2 is lacking, our analysis in this paper provides a good alternative for the analysis of spatially resolved SED of high redshift galaxies us- ing JWST/NIRCam imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Our analysis in this paper is one of the first robust spatially resolved SED analyses of hundreds of galaxies using JWST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Ab- durro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2021) have demonstrated the capabil- ities of spatially resolved SED fitting using piXedfit on local galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In particular, it gives robust SFR on kpc scales when rest-frame UV to NIR photometry is available, which is consistent with the SFR derived from Hα emission maps (dust-corrected based on the Balmer decrement) from the MaNGA IFS survey (Bundy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In Section 2, we present the data and sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We describe the spatially resolved SED fitting methodology in Section 3 and present our results in Section 4, which include the 4 radial profiles of some key stellar population properties, comparison between the compactness of the spatial dis- tributions of SFR and M∗, and Σ∗,1kpc–M∗ relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In Section 5, we further discuss our results, focusing on the evolutionary trends with redshift and what the implica- tions to the study of galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Throughout this paper, we assume the Chabrier (2003) initial mass function (IMF) with a mass range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1 − 100M⊙ and cosmological parameters of Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7, and H0 = 70 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' DATA AND SAMPLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Observational Data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' JWST Observations We obtain JWST/NIRCam imaging data of WHL0137−08 cluster from Cycle 1 General Observers (GO) 2282 program (PI Coe) and MACS0647+70 cluster from GO 1433 program (PI Coe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The WHL0137−08 cluster was observed in July 2022, while the MACS0647+70 cluster was observed in 23 Septem- ber 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The GO 2282 program aims at further inves- tigating Earendel (Welch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022a,b) and the Sun- rise Arc (Vanzella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The JWST/NIRCam data from this program consist of eight filters (F090W, F115W, F150W, F200W, F277W, F365W, F410M, and F444W) spanning a wavelength range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The GO 1433 program is intended to observe the triply-lensed galaxy MACS0647–JD at z ∼ 11 (Coe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This program ob- tained JWST/NIRCam imaging in six filters (F115W, F150W, F200W, F277W, F365W, and F444W) span- ning 1 − 5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The exposure time of each filter in the two programs is 2104 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' It achieves 5σ limiting AB magnitude of 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 to 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 in a r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′′2 diameter circular aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For each filter, we obtained four dithers using IN- TRAMODULEBOX primary dithers to cover the 4−5′′ gap between the sort wavelength (SW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' λ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4µm) de- tectors, improve the spatial resolution of final drizzled images, and minimize the impact of image artifacts and bad pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We obtained NIRCam imaging over two 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′26×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′26 fields separated by 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′5, covering a total area of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2 arcmin2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In the observation of WHL0137−08 cluster, the NIRCam module B was centered at the clus- ter while the module A covered a nearby field centered ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′9 from the cluster center (hereafter called “blank field”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' On the other hand, the MACS0647+70 cluster was centered at the module A and the module B ob- serves a blank field nearby to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' HST Data We obtain HST imaging data of the WHL0137−08 cluster from the Reionization Lensing Cluster Survey (RELICS) HST Treasury program (GO 14096;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Coe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' THe RELICS program obtained the first HST imaging of the WHL0137−08 cluster in 2016 with three orbits of ACS (F435W, F606W, and F814W) and two orbits of WFC3/IR (F105W, F125W, F140W, and F160W) data spanning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Two follow-up HST imaging programs (GO 15842 and GO 16668;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' PI: Coe) have thus far obtained an additional 5 orbits of HST ACS imaging in F814W, 2 orbits in F475W, and 4 orbits with WFC3/IR in F110W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The HST imaging data only cover the WHL0137−08 cluster field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' There- fore, we do not have HST imaging data for the blank field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The HST imaging data of the MACS0647+70 clus- ter are taken from multiple programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Overall, MACS0647+70 has been observed in total of 39 or- bits of HST imaging in 17 filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The cluster was first observed by programs GO 9722 (PI Ebeling) and GO 10493, 10793 (PI Gal-Yam) in the ACS F555W and F814W filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Then additional imaging in 15 filters (WFC3/UVIS, ACS, and WFC3/IR, spanning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7 µm ) was obtained by the Cluster Lensing and Supernova Survey with Hubble (CLASH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Postman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' GO 12101, PI Postman).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Finally, additional imag- ing in WFC3/IR F140W was obtained as part of a grism spectroscopy program (GO 13317, PI Coe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' It is important to note that the nearby blank fields to the WHL0137−08 and MACS0647+70 clusters that are observed with NIRCam are not covered in the HST observations described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In this work, we an- alyze galaxies in three fields: WHL0137−08 cluster field, MACS0647+70 cluster field, and the NIRCam blank field nearby to the WHL0137−08 (hereafter sim- ply called blank field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We do not analyze galaxies in the NIRCam blank field of MACS0647+70 because it is ob- served in less number of NIRCam filters than the blank field of WHL0137−08 and it does not have F090W ob- servation, which prevents us from selecting galaxies at z < 2 in this field as their photometry do not cover the rest-frame 4000 ˚A break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For the WHL0137−08 and the blank field, we use 4 HST/ACS filters (F435W, F475W, F606W, and F814W) and 8 JWST/NIRCam filters, whereas for the MACS0647+70, we use 7 HST/ACS fil- ters (F435W, F475W, F555W, F606W, F625W, F775W, and F814W) and six JWST/NIRCam filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We do not use HST/WFC3 IR filters to get high spatial resolution possible while still get sufficiently wide wavelength cov- erage with the HST/ACS and JWST/NIRCam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Please refer to Table 1 for information on limiting magnitudes 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' HST and JWST Imaging Data Used in the Spatially Resolved SED Fitting Telescope Camera Filter Wavelength Deptha PSF FWHMb WHL0137−08 MACS0647+70 WHL0137−08 MACS0647+70 (µm) (AB mag) (AB mag) (arcsec) (arcsec) HST ACS/WFC F435W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='37–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='47 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 HST ACS/WFC F475W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='55 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 HST ACS/WFC F555W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='46–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='62 · · 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 HST ACS/WFC F606W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='47–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 HST ACS/WFC F625W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='54–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='71 · · 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='9 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 HST ACS/WFC F775W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='68–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='86 · · 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='08 HST ACS/WFC F814W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='95 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 JWST NIRCam F090W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='04 · · JWST NIRCam F115W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='04 JWST NIRCam F150W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='06 JWST NIRCam F200W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='06 JWST NIRCam F277W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 JWST NIRCam F356W 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 JWST NIRCam F410M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='6 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='16 · · JWST NIRCam F444W 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='16 Note— a5σ point source AB magnitude limit measured within a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′′2 diameter circular aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' bPSF FWHM here are based on empirical measurement as described in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' and the point spread function (PSF) sizes of our HST and JWST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Sample Galaxies We use grizli v4 photometric catalogs (will be de- scribed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1) to select our sample galaxies in the three fields (WHL0137−08, blank field, and MACS0647+70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The catalogs provide the aperture fluxes and photometric redshifts with which we select our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The sample selection is described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' First, we select galaxies that have integrated signal-to-noise (S/N) ratio > 5 in all JWST filters that are available for the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This is to ensure that we will have galaxies with good photometry in at least JWST filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This initial cut selects 1322 (out of 2718), 1278 (out of 3032), and 1331 (out of 2660) galaxies in the WHL0137−08, blank field, and MACS0647+70, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We do not apply the same S/N criteria on HST filters because it will exclude many more galaxies as they have lower S/N than JWST filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We further cut the sample galaxies based on their redshift to get a sufficient coverage of the rest- frame UV–NIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For galaxies in the WHL0137−08 and MACS0647+70, which are observed by both JWST and HST, we select galaxies at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 < z < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0, whereas for galaxies in the blank field, which do not have HST ob- servations, we select galaxies at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 < z < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This redshift cut ensures that the rest-frame 4000 ˚Abreak is covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This cut further reduces the sample to be 1258, 581, and 1257 galaxies in the WHL0137−08, blank field, and MACS0647+70, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' After that, we do a visual inspection to exclude galaxies that appear to be very small (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', unresolved) and in a merger (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', one segmentation region having multiple cores or multiple galaxies in one segmentation region, despite possible in- terlopers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This further reduce the sample to be 354, 239, 220 galaxies in the WHL0137−08, blank field, and MACS0647+70, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We perform spatially resolved SED analysis on the galaxies in this initial sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' A detailed description of the methodology will be given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Once the analysis is done, we inspect the results of all the galaxies and further exclude galaxies that seem to have bad SED fitting results based on the χ2 values of the fitting to the integrated SEDs within the central effective radius (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5) and the average χ2 values of the fitting to the first 20 spatial bins (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 for definition of spatial bin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We exclude galaxies that have χ2 > 20 for the SEDs within the central effective radius and aver- age χ2 > 40 for the first 20 spatial bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We note that χ2 value can be unrealistically high if systematic uncer- tainties of the photometry are not properly accounted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Beside this, there is still uncertainty around the zero- 6 point calibration of NIRCam photometry in the current early observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Boyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Therefore, we visually inspect SED fitting results of each galaxy using similar plots as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We find that in most cases, NIRCam fluxes are well fitted by our models, better than HST/ACS fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This might be due the shallower depths (and lower S/N) of HST compared to JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The χ2 values above are high enough to get sufficient number of galax- ies and low enough to get good quality of SED fitting re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This results in our final sample, consisting of 243, 91, and 110 galaxies in the WHL0137−08, blank field, and MACS0647+70, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Figure 1 shows the distributions of redshifts and M∗ of our sample galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We note that our sample selection may possibly bias toward selecting relatively massive, bright, and resolved galaxies in each redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' However, due to the lensing magnification in the cluster fields, we expect to detect on average lower mass galaxies with better spatial res- olution than in the standard fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The small num- ber of galaxies and the limited volume sampled might make our sample to be not representative of the gen- eral population of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' However, since we do not make inferences on the average trends or number densi- ties as function of global properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', M∗), but in- stead we show trends in individual galaxies, our results still provide useful insights on the evolution of galaxy structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We also ignore the possible contamination by the Active Galactic Nucleus (AGN) host galaxies in our current study because of the lack of diagnostics for identifying them using our current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' METHODOLOGY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Data Reduction and Photometric Catalog We use the grizli pipeline (Brammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022) to process the HST FLT and the JWST pipeline-calibrated level-2 imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The JWST data were processed using the calibration pipeline v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 with CRDS con- text jwst 0942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='pmap, which includes photometric cal- ibrations based on in-flight data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The JWST level-2 imaging data were then scaled with detector-dependent factors1 based on a NIRCam flux calibration using the standard star J1743045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Our photometric zeropoints de- scribed here are similar to those obtained by the JWST Resolved Stellar Populations ERS program (Boyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Nardiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022) who analyzed the M92 glob- ular cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We also check the consistency of our cali- bration with the more recent one based on CAL program 1 https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='org/record/7143382 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Distributions of M∗ and redshifts of the sam- ple galaxies analyzed in this work that consist of galaxies in WHL0137−08 cluster field, a blank field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', a nearby field of the WHL0137−08 that is observed with NIRCam), and MACS0647+70 cluster field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The M∗ here are derived by summing up M∗ of pixels (in the galaxy’s region) obtained from our spatially resolved SED analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' data jwst 0989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='pmap and find out that they are consis- tent within 3% in all filters analyzed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In processing the JWST data, the grizli pipeline ap- plies a correction to reduce the effect of 1/f noise and masks “snowballs”2 effect caused by the large cosmic ray impacts to the NIRCam detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Beside this, the grizli pipeline also corrects for the “wisps”3, which is a faint, diffuse stray light features that appear at the same detector locations in NIRCam images and most promi- nent in the A3, B3, and B4 detectors in the F150W and F200W images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The grizli pipeline aligns the HST and JWST imag- ing data to a common world coordinate system which is registered based on the GAIA DR3 catalogs (Gaia Col- laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The images are then drizzled to a common pixel grid using the astrodrizzle (Koeke- moer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The 17 HST filters and 4 JWST NIRCam long-wavelength (LW) fil- ters (F277W, F356W, F410M, and F444W) are drizzled 2 https://jwst-docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='edu/data-artifacts-and-features/ snowballs-artifact 3 https://jwst-docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='edu/jwst-near-infrared-camera/ nircam-features-and-caveats/nircam-claws-and-wisps 50 0 WHL0137-08cluster 12 WHL0137-08blankfield MACS0647+70cluster 11 log(M*[M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=']) 10 8 N=243 N=91 N=110 1 2 3 4 5 6 0 25 Photometric Redshift7 to a spatial sampling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′′04 per pixel while the JWST short-wavelength (SW) filters (F090W, F115W, F150W, and F200W) are drizzled to a spatial sampling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′′02 per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Source detection is then performed on a weighted sum of the drizzled NIRCam images in all filters using sep (Barbary 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Bertin & Arnouts 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Fluxes are then calculated for each source in three circular aper- tures, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′′36, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′′5, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′′7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Then photometric redshift measurement is performed using the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′′5 aperture SEDs employing eazypy(Brammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This code fits observed photometry using a set of templates added in a non-negative linear combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The processed imag- ing data along with the photometric catalog are publicly available4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' These data product have also been used in some recent studies (Welch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Vanzella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Meena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Analysis of Post-processed Imaging Data In this work, we combine the post-processed HST and JWST imaging data (in up to 13 filters) into a common spatial resolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', PSF size) and sam- pling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', pixel size) for extracting the spatially resolved SEDs of our sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' These spatially resolved SEDs are then fitted with models to infer the underlying properties of the stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We use piXedfit5 (Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2021, 2022c) throughout this analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Basically, this process includes three main tasks: image processing, pixel binning, and SED fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We will briefly describe these steps in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The image processing is carried out automatically us- ing piXedfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For each galaxy, we first crop stamp images with a size of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′′04 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′′04 (corresponding to 302×302 pixels in NIRCam SW and 151×151 pixels in NIRCam LW and HST/ACS filters) centered at the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We then perform background subtraction to each stamp image using photutils (Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Next, we perform point spread function (PSF) matching to homogenize the spatial resolution across fil- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We degrade the spatial resolution of the images to match the resolution of F444W filter, which has the low- est spatial resolution (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For this, we generate the empirical PSFs of HST/ACS and JWST/NIRcam filters along with the convolution kernels using photu- tils package (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The PSF matching is carried out by convolving the stamp images with the convolution kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' After PSF matching, we register all the stamp images to a common spatial sampling of 4 https://cosmic-spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='io/earendel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='html 5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='com/aabdurrouf/piXedfit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='′′04 per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' At the end, we have multiband stamp images with a size of 151×151 pixels for each galaxy in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Constructing Photometric Data Cubes piXedfit further processes the stamp images to pro- duce photometric data cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' First, it defines a galaxy’s region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For each galaxy, segmentation maps are first produced in all filters using sep (Barbary 2016) and those maps are then merged together into a single map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In the segmentation process, we use same parame- ters for all filters as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We set the detection thresh- old (thresh), the number of thresholds for deblending (deblend nthresh), and the minimum contrast ratio for deblending (deblend cont) to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0, 40, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='005, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In some cases, the merged segmentation map is larger than expected, as can be inferred from the maps of multiband fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This can be caused by some factors, for example, an interference from neighboring objects that is not separated well by the deblending process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We visually inspect the merged segmentation map of each galaxy to find out this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' To deal with this, we tweak the deblending parameters to get cleaner segmentation maps or ignore segmentation map in some filters that has this deblending issue, then merge them again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Once the galaxy’s region is defined, then the fluxes of pixels within the region are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We use PHOT- FLAM keyword in the header of the grizli imaging data products to convert the pixel value into flux density in the units of erg s−1cm−2˚A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The data cubes are then stored into FITS files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Figure 2 shows examples of the maps of multiband fluxes of galaxies in three fields an- alyzed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The color images shown in the leftmost panels are created using Trilogy6 (Coe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Pixel binning The SEDs of pixels are usually noisy and might not providing sufficient constraint to the models if fitting is done to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Therefore, we perform pixel binning using piXedfit to optimize the signal-to-noise ratio (S/N) of the spatially resolved SEDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Basically, this process bins neighboring pixels to achieve a certain S/N ratio thresh- old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The unique pixel binning scheme in piXedfit, which takes into account the similarity in SED shape among pixels, allows for achieving a sufficient S/N ratio in multiple filters of interest while preserving important spatial information at pixel level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' A detailed description 6 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='com/dancoe/trilogy 8 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Examples of the maps of multiband fluxes produced from the image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Example of one galaxy is shown for each field, from top to bottom: WHL0137−08 cluster (observed in 12 filters), blank field (8 filters), and MACS0647+70 cluster (13 filters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The ID is based on our grizli v4 public catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' of this pixel binning scheme can be seen in Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We assume the following parameters in the pixel bin- ning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We refer reader to Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2021) for more information about the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We set S/N thresholds to 5 in all JWST NIRCam filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We do not set S/N threshold to HST filters because the S/N ratio of pixels in the HST images are low, especially for galaxies at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Setting a S/N thresh- old on the HST filters would put a strong constraint in the pixel binning process which can produce a coarser binning map and loosing important spatial information from the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The rest of the binning parameters are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We set a minimum diameter of 7 pixels, which is larger than the PSF FWHM size of our data cubes, a re- duced χ2 limit of 5 in the evaluation of the similarity of SED shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We refer to F277W flux in determining the brightest pixel to be the center of a spatial bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We store new data cubes produced from this pixel binning process into FITS files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The total number of spatial bins in our sample galaxies is 24999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Figure 3 shows examples of the pixel binning maps produced from this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Spatially Resolved SED Fitting Once we have the binned data cubes, we perform SED fitting to the SEDs of individual spatial bins in our sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Here we use the SED fitting module in piXedfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The SED fitting in piXedfit uses fully Bayesian technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We refer reader to Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2021) for a detailed description of the SED modeling and fitting methods as well as comprehensive tests of its capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In Appendix A, we perform SED-fitting tests using mock SEDs to demonstrate the robustness of our SED fitting method on the combined HST and JWST photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Moreover, in Appendix B we dis- cuss how NIRCam photometry can potentially break the 120 120 HST/F435W HST/F475W HST/F606W HST/F814W JWST/F090W JWST/F115W 110 - 110 110 110 110 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 100: 100 100 100 100 06 90 90 90 80 80 80 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 80 80 WHL0137-08 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 7 1cm 70 70 70 70 6 50 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 50 50 50 50 s log(Flux density [erg 40 - 40 - 40 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 100 40 120 100 100 120, 80 100 100 120 120 120 JWST/F150W JWST/F200W JWST/F277W JWST/F356W JWST/F410M JWST/F444W 110 110 110 110 110 100 - 100 100 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 100 100 90 - % 80 80 - %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 70 70 60 60 60 50 os 50 50 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 40 40 40 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 40 100 120 40 100 120 100 120 100 120 40 100 120 [pixel] [pixellJWST/F090W JWST/F115W JWST/F150W JWST/F200W 120 120 120 - 120 - 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 100 100 100 100 WHL0137-08 blank field 80 80 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 60 40 40 - 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 ity [erg 20 20 + 20 - 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 20 40 60 80 100 120 JWST/F277W JWST/F356W JWST/F410M JWST/F444W 120 120 - 120 - 120 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 densit 100 100 100 100 - ID=3240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='77 80 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 60 60 60 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 40 - 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 20- 40 60 80 100 120 60 20 100 120 20 60 80 100 120 40 60 80 100 120 [pixel] [pixel] [pixel] [pixel]140 HST/F435W 140 HST/F475W HST/F555W HST/F606W 140 HST/F625W HST/F775W HST/F814W 120 120 120 - 120 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='D 100 MACS0647+70 80 80 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 60 60- 40 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 100125150 75100125 100125 150 ¥75100 125 7510012515 JWST/F115W JWST/F150W JWST/F200W JWST/F277W JWST/F356W JWST/F444W 120 120 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 120 100 100 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 60 60 60 40 40- 20 125 1509 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Examples of pixel binning results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The pixel binning process achieves a minimum S/N ratio of 5 in all JWST NIRCam filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Free Parameters in the SED Modeling and the Assumed Priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Parameter Description Prior Sampling/Scale M∗ Stellar mass Uniform: min= log(sbest) − 2, max= log(sbest) + 2a Logarithmic Z∗ Stellar metallicity Uniform: min= −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 + log(Z⊙), max= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2 + log(Z⊙) Logarithmic t Time since the onset of star formation (agesys) b Uniform: min= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0, max = age of the universe at the galaxy’s redshiftc Logarithmic τ Parameter that controls the peak time in the double power-law SFH modelb Uniform: min= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5, max= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='14 Logarithmic α Parameter in the double power-law SFH model that con- trols the slope of the falling star formation episodeb Uniform: min= −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0, max= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 Logarithmic β Parameter in the double power-law SFH model that con- trols the slope of the rising star formation episodeb Uniform: min= −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0, max= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 Logarithmic ˆτ1 Dust optical depth of the birth cloud in the Charlot & Fall (2000) dust attenuation law Uniform: min= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0, max= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 Linear ˆτ2 Dust optical depth of the diffuse ISM in the Charlot & Fall (2000) dust attenuation law Uniform: min= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0, max= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 Linear n Power law index in the Charlot & Fall (2000) dust atten- uation law Uniform: min= −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2, max= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 Linear Note— asbest is the normalization of model SED derived from the initial fitting with the χ2 minimization method (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1 in Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' bThe mathematical form of the double power-law SFH is given in Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2021, Equation 7 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' cThe redshift here is z = zeazy − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 with zeazy is the photometric redshift from the grizli catalog which is derived using eazypy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Subtraction by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 is made to enlarge the range of t and account for the photometric redshift uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Bin Index Bin Index Bin Index 100 200 300 400 500 10 20 30 50 100 150 120 ID=543 ID=2430 ID=3240 140 110 120 120 100 100 100 90 [pixel] 80 80 80 70 60 60 60 40 50 40 20 40 nbins=580 nbins=32 20 nbins=195 0 - 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 0 20 40 60 80 100 120140 30 40 50 60 70 80 90 100 110120 20 40 60 80 100 120 Bin Index Bin Index Bin Index 3 5 7 9 11 13 20 40 60 80 50 100 150 100 ID=2808 ID=4554 ID=3704 140 120 90 120 100 100 80 [pixel] 80 80 70 60 60 40 60 40 20 nbins=13 nbins=83 nbins=159 50 0 20 50 60 70 80 90 100 0 20 40 60 80100120140 20 40 60 80 100 120 [pixel] [pixel] [pixel]10 degeneracies among age,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' dust,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' and metallicity in the SED fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In the following, we provide a brief de- scription of the method and some assumptions applied in our SED fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We use the Flexible Stellar Population Synthesis code (FSPS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Conroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Conroy & Gunn 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' It includes the nebular emission modeling that uses the CLOUDY code (Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 1998, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In this work, we assume the Chabrier (2003) initial mass function (IMF), Padova isochrones (Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Marigo & Girardi 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Marigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2008), and MILES stellar spectral library (S´anchez-Bl´azquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Falc´on-Barroso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For the star formation history model, we as- sume an analytic model in the form of double power-law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' It has been shown in Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2021) that this SFH form can give robust estimates of the stellar pop- ulation properties and even SFH of galaxies, as tested using synthetic SEDs of simulated galaxies in the Illus- trisTNG simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For simulating the effect of dust attenuation, we use the two-component dust attenuation law of Charlot & Fall (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This dust attenuation law gives an extra attenuation to stars younger than 10 Myr that are assumed to be residing in the dense molecular clouds, in addition to standard attenuation in the dif- fuse ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We model the attenuation due to intergalactic medium using Inoue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2014) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Since we do not have photometry that cover the rest-frames mid-infrared (MIR) and far-infrared (FIR), we switch off the model- ing for dust emission and AGN dusty torus emission in the analysis throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The SED modeling has 9 free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We summarize these parameters along with the assumed priors in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We assume a constant ionization parameter (U) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='01 in the model- ing of the nebular emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In the current analysis, we rely on photometric red- shift for all of galaxies in our sample because we do not have spectroscopic observations at the moment we carry out this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' To get redshift estimates of the galax- ies, we perform SED fitting with piXedfit in which redshift is let to be free in the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For this, we fit integrated SED within the effective radius of the galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The effective radius is measured in F444W image using GALFIT(Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2002, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This is performed to get SEDs with high S/N while reducing contamination from noisy SEDs of pixels in the outskirt regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In this fitting, we apply a prior on redshift in the form of a Gaussian function centered at the photo- metric redshift estimated by the eazypy taken from the grizli catalog (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We set a width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 for this Gaussian prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This fitting is performed to de- rive redshift only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We then use this redshift information for the SED fitting of all spatial bins in the galaxy, in which we fix the redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We apply the Markov Chain Monte Carlo (MCMC) method in piXedfit˙In the SED fitting for determining redshifts, we set the number of walkers to 100 and the number of steps per walker to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For the SED fitting of spatial bins, we use 100 walkers and less number of steps (600) per walker for reducing computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We show examples of SED fitting results of two galax- ies in Figure 4, one galaxy from the WHL0137−08 clus- ter field (top panel) and the other galaxy from the blank field (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For each galaxy, we show best-fit SEDs in the top right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The observed and best- fit photometric SEDs are shown with square and circle symbols, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The SED in black is for the inte- grated within the effective radius, while those in other colors are for 5 examples of spatial bins in the galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The corner plot in the lower left side shows the posterior probability distribution functions (PPDF) of the model parameters obtained from the fitting on the integrated SED within the effective radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Above this corner plot, we show the PDFs of M∗ and SFR of the spatial bins which the best-fit SEDs are shown in the top right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The best-fit spectra shown in the plot are drawn from the MCMC sampler chains, which distribu- tions reflect the PPDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Therefore, it is possible to get a slight shift in wavelength between the best-fit spectra of the central SED (where z is free in the fitting) and that of the spatial bins (where z is fixed in the fitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This wavelength shift reflects the uncertainty of the estimated redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Finally, in the bottom right panel we show the maps of stellar population properties, including the M∗ surface density (Σ∗), SFR surface density (ΣSFR), mass- weighted age, AV,1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='086 × ˆτ1), AV,2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='086 × ˆτ2), and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Lens Modeling To estimate the magnifications due to the gravita- tional lensing effect by the clusters, we use the lens mod- els constructed by our team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For the WHL0137−08 clus- ter, we use the same lens models that were used for ana- lyzing the Earendel and the Sunrise Arc in Welch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2022a), which were made publicly available7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' These lens models were generated using four independent lens modeling software packages: Light-Traces-Mass (LTM, Zitrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2009, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Broadhurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2005), Glafic (Oguri 2010), WSLAP (Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2005, 2007), and Lenstool (Kneib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Jullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Jullo & Kneib 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Please see Welch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2022a) for detailed information about each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Sample galax- 7 https://relics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='edu/lens models/outgoing/whl0137-08/ 11 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Examples of SED fits of a galaxy in the WHL0137−08 cluster (top panel) and the blank field (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The SED plots show the best-fit SEDs from the fitting to integrated SED within the effective radius (black color) and 5 examples of spatial bins (in colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The corner plots show the posterior probability distributions of the parameters obtained from fitting to the central integrated SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Above this corner plots, we show PDFs of M∗ and SFR of the five spatial bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Finally, the maps of stellar population properties derived from this analysis are shown in the bottom right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Bin Index 10 30 WHL0137-08 0 120 - Central Re 10-18, Bin 1 100 Bin 3 Bin 10 Bin 20 M 60 10-19 Bin 30 40 回 40 80 100 120 [pixel] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2 10-20 Fx [erg Normalized PDF Normalized PDF 10-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' HST JWST 10-22+ 7 8 9 10 11 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 0.' metadata={'source': 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+page_content='0 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 30 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='00 40 60 80 100 120 40 60 80 100 120 40 60 80 100 120 [pixel] [pixel] [pixel] 120 120 120 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 110 110 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 100 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 [mag] 90 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 40 40 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='6 og(SFR) og(M-) Av,2 "log(age)Tog(ziZo) redshift 30 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 O 40 60 80 100 120 40 60 80 100 120 40 60 80 100 120 [pixel] [pixel] [pixel] Bin Index WHL0137-08 blank field 50 100 150 Central Re 120 Bin 100 Bin 30 [pixel] 10-18 Bin 60 80 Bin 90 60 Bin 120 40 - 20 - 25 50 [pixel] Fx [erg Normalized PDF 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 2 60 60 40 40 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 log(SFR) log(age)log(ZiZo)"redshit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='6 log(m) 20 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 204060 80100120 20 80100120 2040 80100120 [pixel] [pixel] [pixel]12 ies located in the blank field are expected to have only weak magnifications of µ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' With the multiple lens models available for this cluster, we estimate the total and tangential (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', linear) magnifications (µ and µt, re- spectively) of each galaxy by taking the average values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' By this way, we account for the modeling uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Based on the standard deviation values, we find that the magnifications do not vary a lot among the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The median standard deviations of µ and µt are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='11 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='10 dex, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The lens models for MACS0647+70 cluster have been constructed in the past using the HST imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The first lens model for MACS0647+70, before the CLASH survey, was provided by Zitrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2011) us- ing LTM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' With the addition of HST imaging data from CLASH, new lens models were established us- ing various methods, including Lenstool, LTM, WSLAP, and LensPerfect (Coe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' These lens mod- els have been used in previous studies in CLASH (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Coe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Zitrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Now with the addition of JWST NIRCam imaging data, which add on many new strongly-lensed multiple-image candidates (thanks to its high spatial resolution and depth), a new lens model has been established using the dPIEeNFW method (Zitrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2015) with some mod- ifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' A detailed information on this lens model- ing of MACS0647+70 cluster along with the list of the multiple-image systems are given in Meena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This new method has also been implemented to several clusters using JWST NIRCam data (Pascale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Roberts-Borsani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We used this lens model constructed by Meena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2022) for galaxies in the MACS0647+70 field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We correct the M∗ and SFR obtained from SED fitting for the lensing magnification by dividing them with µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We also correct size or radius measurement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' half-mass and half-SFR radii;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3) by dividing them with µt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Integrated Properties Before analyzing the spatially resolved properties of our sample galaxies, we first present their integrated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', global) properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' To bring it into the context of the global demographics of galaxies, we plot our sample on the integrated star-forming main sequence (SFMS) diagram, as shown in the left panel of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The integrated M∗ and SFR of a galaxy are derived by sum- ming up the values in pixels obtained from the spatially resolved SED fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Due to our limited sample, we plot all our galaxies on the SFMS diagram instead of dividing them into a number of redshift bins and ex- amine the SFMS relation in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This can cause the broad distribution as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Different symbols represent the fields where the galaxies are lo- cated (WHL0137−08, blank field, and MACS0647+70), whereas color-coding represent redshift grouping, where we divide the redshift range into five bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The dashed lines show the SFMS relations at the median redshifts of the five redshift bins, calculated using the prescription from Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The lines are colored based on the redshift groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We then classify our sample galaxies into star-forming, green valley, and quiescent groups based on their posi- tions with respect to the SFMS ridge line at the red- shift of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We define star-forming, green- valley, and quiescent galaxies as those having SFR > SFRMS(z, M∗) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 dex, SFRMS(z, M∗) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 ≥ SFR > SFRMS(z, M∗)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 dex, and SFR ≤ SFRMS(z, M∗)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 dex, respectively, where SFRMS(z, M∗) is the SFMS ridge line for exact z and M∗ of the individual galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' With this selection criteria, we have 219, 108, and 117 total numbers of the star-forming, green-valley, and quiescent galaxies from the three fields, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We will use these classified samples throughout the analysis in this paper to investigate the differences in spatially resolved properties of galaxies in various evolutionary stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The right panel of Figure 5 show the distribu- tions of these groups on the SFMS diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' To get a sense of how the global specific SFR (sSFR≡ SFR/M∗) evolves with cosmic time in our sample galax- ies, we plot the sSFR against redshift in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We can see a clear trend of decreasing global sSFR with cos- mic time and an increasing number of quiescent galaxies along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' In our sample, quiescent galaxies start to emerge from z ∼ 3, ∼ 2 Gyr after the Big Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' To compare our global sSFR trend with the similar trend from previous studies, we plot sSFR(z) inferred from the SFMS normalization based on Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2014) prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We calculate sSFR(z) for four M∗ of 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5, 1010, 1011, and 1012 M⊙ and show them in the figure as black dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We see an overall agreement be- tween the evolutionary trend of sSFR in our sample and that expected based on the evolution of the SFMS nor- malization from Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We also show global sSFR measurements of z ∼ 0 spiral galaxies from Abdurro’uf & Akiyama (2017) and Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2022a) (personal communication) who performed spa- tially resolved SED fitting using piXedfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Previous studies have classified passive galaxies using various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' One of the methods is by compar- ing the Hubble time (tH) with the mass doubling time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', inverse of sSFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Basically, this method defines qui- escent galaxies as those having sSFR < 1/tH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The red 13 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Left panel: Integrated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', global) M∗ and SFR of our sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Different symbols represent different fields where the galaxies are located, whereas the color-coding represents redshift grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The dashed lines show the global SFMS relations at the median redshifts of the 5 redshift groups calculated using the prescription from Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The lines are colored based on the redshift groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Right panel: Distribution the star-forming, green-valley, and quiescent galaxies in our sample that are classified as having SFR > SFRMS(z, M∗) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 dex, SFRMS(z, M∗) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 ≥ SFR > SFRMS(z, M∗) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 dex, and SFR ≤ SFRMS(z, M∗) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 dex, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' SFRMS(z, M∗) is the SFMS ridge line that is calculated for exact z and M∗ of the individual galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' dashed line in Figure 6 represents 1/tH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' To compare our quiescent classification with this method, we plot 1/tH in Figure 6, which is shown with red dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We can see that our quiescent galaxies lie below this line, indicating that our classification method is consistent with that based on tH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Radial Profiles of the Stellar Population Properties As we have shown the global properties of the sam- ple galaxies and classified them into star-forming, green- valley, and quiescent groups, now we will analyze their spatially resolved properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We start by presenting the radial profiles of the stellar population properties to get a sense of how the properties vary radially within the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' To derive the radial profiles, first, we per- form 2D single-component S´ersic fitting using GALFIT (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2002) on F444W stamp image of each galaxy to get their ellipticities, position angles, and central co- ordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We then use this information to define ellip- tical annuli in the radial profile calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The radial profiles are derived from the 2D maps of properties ob- tained from the spatially resolved SED fitting by aver- aging values of pixels within the annuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Since galaxies have a wide range of size, we normalize the radius by the half-mass radius (Re), which is the radius that covers half of the integrated M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We use radial increment (δr) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Evolution of the integrated specific SFR (sSFR) with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The black dashed lines represent sSFR evolu- tion of SFMS galaxies with M∗ = 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5, 1010, 1011, and 1012 M⊙ (decreasing normalization) as inferred from the normal- ization of the SFMS, calculated using the prescription from Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The red dashed line represents 1/tH where tH is the Hubble time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Almost all our quiescent galax- ies lie below this line, indicating that their mass doubling timescale is longer than tH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Thanks to the gravitational lensing effect, we 20 0 4 3 log(SFR[Moyr-1]) 2 0 Star-forming 3 Green valley Quiescent 8 9 10 11 12 0 25 log(M*[Mo])z2 0 1 3 4 5 6 7 8 WHL0137-08cluster Star-forming WHL0137-08blankfield 6 Green valley MACS0647+70cluster Quiescent 7 log(sSFR[yr-1]) 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='9 10 11 12 1/Hubbletime 13 SFMSSpeagle etal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=" (2014) z=0Abdurro'uf&Akivama(2o17） log(M*[Mo])=8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content="0 ¥z=0Abdurro'ufetal." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2022a) 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0 log(1 + z)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='00 C WHL0137-08cluster ★ WHL0137-08blankfield 3 MACS0647+70cluster 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='86 2 log(SFR[Moyr-1 ]) 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='72 Redshift 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='58 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='44 3 Lines:SFMSSpeagleetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' (2014) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='30 8 9 10 11 12 log(M*[Mo])14 can resolve many of our galaxies down to sub-kpc scales (109 galaxies in our sample have delensed Re < 1 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Figure 7 shows the radial profiles of the stellar mass surface density (Σ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We divide the sample into 5 bins of redshift and 4 bins of M∗ to see how the radial pro- files vary with global M∗ and cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Moreover, we indicate the star-forming, green valley, and quiescent galaxies with different colors, in a similar way as in Fig- ure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For groups that contain at least 5 galaxies, we show average radial profiles with tick line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Some inter- esting trends from Figure 7 is that at each redshift bin, more massive galaxies tend to have higher Σ∗(r) normal- ization than less massive galaxies, indicating that the excess in mass happens across the entire radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' More- over, we also see that quiescent galaxies tend to have higher Σ∗(r) normalization than the star-forming and green-valley galaxies in all redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This is especially clear in the most massive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' It is also interesting to see that Σ∗(r) profiles have negative gradient (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', de- creasing mass with increasing radius) in all redshift and mass bins, although the profiles seem to be shallower at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' To see how galaxies quench their star formation, specifically where in the galaxies the suppression of star formation first happens and how it progresses over cos- mic time, next we analyze the radial profiles of sSFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The radial profiles of sSFR are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' As we can see from this figure, the sSFR radial profiles of the majority of our sample galaxies at z ≳ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 are broadly flat, while they show more diversity in shape at lower redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8 ≲ z ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5, star-forming galax- ies in our sample tend to have a flat or centrally-peaked sSFR(r), while quiescent galaxies tend to have centrally- suppressed sSFR(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' On the other hand, green-valley galaxies in our sample seem to have broadly flat ra- dial profile up to z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='0, except in the most massive group, where some of them show a sSFR suppression in their central regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' At lower redshifts, the ma- jority of our sample galaxies have centrally-suppressed sSFR(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' It is also interesting to see that the majority of star-forming galaxies at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8 ≲ z ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 (in which the cosmic noon epoch is covered), have a centrally-peaked sSFR(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This central elevation of sSFR is not observed at higher redshifts, instead they have roughly flat radial profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Next, we analyze the radial profiles of the stellar popu- lation age to see how this quantity varies radially within our sample galaxies and investigate the underlying stel- lar population properties causing the diversity in the sSFR radial profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' From our spatially resolved SED fitting, we obtain maps of the mass-weighted ages, which is the average age of stars in a stellar population as weighted by the stellar mass formed over the course of the star formation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The age radial profiles are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' As can be seen from this figure, there is a trend of increasing overall age of the stellar popu- lations in galaxies over cosmic time, as indicated by the increasing normalization of the radial profiles with de- creasing redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The star-forming galaxies that have a centrally-peaked sSFR(r) at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8 ≲ z ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 (possibly around the cosmic noon epoch) as shown in Figure 8 are likely in a phase of rapid star formation in their centers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', a nuclear starburst;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Dekel & Burk- ert 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Zolotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Tacchella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2016b), as indicated by the young stellar populations (age ≲ 100 Myr) in their central regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' At this epoch, green-valley and quiescent galaxies tend to have radially decreasing age profiles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', negative gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='8, low-mass galaxies (log(M∗/M⊙) < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5) in all stages of star formation have radially decreasing age radial pro- files (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', negative gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' A similar trend is still hold for star-forming and green-valley galaxies in higher mass group (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 < log(M∗/M⊙) < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' On the other hand, quiescent galaxies at this epoch tend to have overall flat and old stellar populations across their entire radial ex- tents, with higher normalization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', older) than that of star-forming and quiescent galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Compactness of the Spatial Distributions of Stellar Mass and SFR The centrally-peaked sSFR(r) of star-forming galax- ies at around the cosmic noon epoch indicates that they are likely undergoing a nuclear starburst that builds the bulge component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The centrally-suppressed sSFR(r) profiles which start to emerge in quiescent galaxies at around the same epoch can be caused by the cessation of star formation in the center and/or a matured bulge that has been formed in these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This trend pro- vides a hint on how galaxies quench their star formation, which seems to progress in an inside-to-outside manner (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', quenching starts from the center and then prop- agates outward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' At the same time, this trends may indicate that galaxies build their central regions first, forming a mature bulge, and then subsequently assem- ble their disk through star formation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', inside-out growth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' To further investigate this, next we compare the compactness of the spatial distributions of M∗ and SFR by means of the half-mass and half-SFR radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' We compare the half-mass radius and the half-SFR ra- dius in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The half-SFR radius is a radius (mea- sured along the elliptical semi-major axis) that covers half of the total SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Similar as in Figure 6, the star- forming, green-valley, and quiescent galaxies are shown in blue, green, and red colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' To com- 15 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Radial profiles of the stellar mass surface density (Σ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' The sample galaxies are divided into 4 bins of global M∗ and 5 bins of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' At each group, we further classify the galaxies into star-forming, green-valley, and quiescent groups and indicate them with different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' For sub-groups that contain at least 5 galaxies, we show average radial profiles with tick line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' At each redshift bin, more massive galaxies tend to have higher Σ∗(r) normalization than less massive galaxies, indicating that the excess in mass happens across the galaxy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Quiescent galaxies tend to have higher Σ∗(r) normalization than star-forming in all redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This is especially clear in high M∗ bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' pare the distributions of our star-forming and quies- cent galaxies on this diagram, we plot the density con- tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' As can be seen from this figure, star-forming galaxies broadly follow the one-to-one line, whereas qui- escent galaxies are excess above the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' This means that in quiescent galaxies, the spatial distribution of SFR is more extended than that of stellar mass, indi- cating that star formation is on-going in the disk and less active in the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' It is also possible that a massive bulge might has been formed in the cen- ters, making a more compact stellar mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' On the other hand, star-forming galaxies are equally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Some star-forming galaxies have spatially more compact star formation distribution than the stel- lar mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', below the one-to-one line), which indicates that active star-formation happens at their centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' On the other hand, in the star-forming galaxies that have extended star formation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', above the one-to-one line), the bulge might has been built and active star formation is now progressing outward and building the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' It has been known that galaxy size correlates with global M∗ for galaxies out to at least z ∼ 3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', the size–mass relation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=', Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' van der Wel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Morishita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' However, most of the previous studies rely on galaxy half-light radii as a measure of galaxy size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Since mass- to-light ratios are not constant across a galaxy’s region, but instead has a gradient, the half-light radii are not a direct probe of the underlying stellar mass profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content=' Therefore, it is expected that the half-mass and half- 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfRgCz/content/2301.02209v1.pdf'} +page_content='5 ai+1 [ai < ai+1], where at+1 +denotes a1. We also say that s is oriented if s is cyclic or s is anti-cyclic. See [4, 24, 25]. Given a partial +transformation α ∈ PT n such that Dom(α) = {a1 < · · · < at}, with t ⩾ 0, we say that α is orientation- +preserving [orientation-reversing, oriented] if the sequence of its images (a1α, . . . , atα) is cyclic [anti-cyclic, +oriented]. +It is easy to show that the product of two orientation-preserving or of two orientation-reversing +transformations is orientation-preserving and the product of an orientation-preserving transformation by an +orientation-reversing transformation, or vice-versa, is orientation-reversing. We denote by POPn the submonoid +of PT n of all orientation-preserving transformations and by PORn the submonoid of PT n of all oriented +2 + +transformations. Consider also the inverse submonoids POPIn = POPn ∩ In, of all orientation-preserving +partial permutations, and PORIn = PORn ∩ In, of all oriented partial permutations, of PT n. +Notice that, POIn ⊆ PODIn ⊆ PORIn and POIn ⊆ POPIn ⊆ PORIn, by definition. +Now, let us consider the following permutations of Ωn of order n and 2, respectively: +g = +�1 +2 +· · · +n − 1 +n +2 +3 +· · · +n +1 +� +and +h = +�1 +2 +· · · +n − 1 +n +n +n − 1 +· · · +2 +1 +� +. +It is clear that g, h ∈ DIn. Moreover, for n ⩾ 3, g together with h generate the well-known dihedral group +D2n of order 2n (considered as a subgroup of Sn). In fact, for n ⩾ 3, +D2n = ⟨g, h | gn = 1, h2 = 1, hg = gn−1h⟩ = {1, g, g2, . . . , gn−1, h, hg, hg2, . . . , hgn−1} +and we have +gk = +� +1 +2 +· · · +n − k +n − k + 1 +· · · +n +1 + k +2 + k +· · · +n +1 +· · · +k +� +, +i.e. +igk = +� i + k +if 1 ⩽ i ⩽ n − k +i + k − n +if n − k + 1 ⩽ i ⩽ n, +and +hgk = +�1 +· · · +k +k + 1 +· · · +n +k +· · · +1 +n +· · · +k + 1 +� +, +i.e. +ihgk = +� k − i + 1 +if 1 ⩽ i ⩽ k +n + k − i + 1 +if k + 1 ⩽ i ⩽ n, +for 0 ⩽ k ⩽ n − 1. Denote also by Cn the cyclic group of order n generated by g, i.e. +Cn = ⟨g | gn = 1⟩ = {1, g, g2, . . . , gn−1}. +Until the end of this paper, we will consider n ⩾ 3. +For any two vertices x and y of Cn, we now denote the distance dCn(x, y) simply by d(x, y). Notice that, +we have +d(x, y) = min{|x − y|, n − |x − y|} = +� |x − y| +if |x − y| ⩽ n +2 +n − |x − y| +if |x − y| > n +2 +and so 0 ⩽ d(x, y) ⩽ n +2 , for all x, y ∈ {1, 2, . . . , n}. Observe also that +d(x, y) = n +2 +⇔ +|x − y| = n +2 +⇔ +n − |x − y| = n +2 +⇔ +|x − y| = n − |x − y|, +in which case n is even. +Recall that DIn is the submonoid of the monoid PORIn whose elements are precisely all restrictions of the +dihedral group D2n of order 2n. Moreover, it is also known exactly how many extensions in D2n each element +of DIn has: +Lemma 1.1 ([17, Lemma 1.1]) Let α ∈ PT n. Then α ∈ DIn if and only if there exists σ ∈ D2n such that +α = σ|Dom(α). Furthermore, for α ∈ DIn, one has: +1. If either | Dom(α)| = 1 or | Dom(α)| = 2 and d(min Dom(α), max Dom(α)) = n +2 (in which case n is even), +then there exist exactly two (distinct) permutations σ, σ′ ∈ D2n such that α = σ|Dom(α) = σ′|Dom(α); +2. If either | Dom(α)| = 2 and d(min Dom(α), max Dom(α)) ̸= n +2 or | Dom(α)| ⩾ 3, then there exists exactly +one permutation σ ∈ D2n such that α = σ|Dom(α). +Notice that, for an even n, we have +B2 += +{α ∈ DIn | |Dom(α)| = 2 and d(min Dom(α), max Dom(α)) = n +2 } += +��i +i + n +2 +j +j + n +2 +� +, +� +i +i + n +2 +j + n +2 +j +� +| 1 ⩽ i, j ⩽ n +2 +� +3 + +and so |B2| = 2(n +2 )2 = 1 +2n2. +In this paper, we study three remarkable submonoids of DIn, namely OPDIn = DIn∩POPIn, the monoid +of all orientation-preserving partial isometries of Cn, MDIn = DIn ∩ PODIn, the monoid of all monotone +partial isometries of Cn, and ODIn = DIn ∩POIn, the monoid of all order-preserving partial isometries of Cn. +Observe that DIn, OPDIn, MDIn and ODIn are all inverse submonoids of the symmetric inverse monoid +In, ODIn ⊆ MDIn and ODIn ⊆ OPDIn. Also, notice that OPDI3 = POPI3, MDI3 = PODI3 and +ODI3 = POI3. +In Section 2, we compute the cardinals of ODIn, MDIn and OPDIn and, in Section 3, we describe their +Green’s relations. Finally, Section 4 is dedicated to establish generating sets and to determine the ranks of +these three monoids. +For general background on Semigroup Theory and standard notations, we refer to Howie’s book [23]. +We would like to point out that we made considerable use of computational tools, namely GAP [21]. +2 +Cardinals +In this section, we determine the number of elements of each of the monoids ODIn, MDIn and OPDIn. +Let α ∈ PT n. Recall that the rank of α, denoted by rank(α), is the size of Im(α). +By applying Lemma 1.1 and counting all possible distinct orientation-preserving and order-preserving re- +strictions of permutations from D2n, we have: +Theorem 2.1 One has +|ODIn| = 3 · 2n + (n + 1)n(n − 1) +6 +− 1 + (−1)n +8 +n2 − 2n − 2 +and +|OPDIn| = n2n + n2(n − 1) +2 +− 1 + (−1)n +4 +n2 − n + 1. +Proof. Let A = {α ∈ DIn | | Dom(α)| ⩽ 1}, B = {α ∈ ODIn | | Dom(α)| ⩾ 2} and C = {α ∈ OPDIn | +| Dom(α)| ⩾ 2}. +Clearly, A = {α ∈ OPDIn | | Dom(α)| ⩽ 1} = {α ∈ ODIn | | Dom(α)| ⩽ 1} and so +|ODIn| = |A| + |B| and |OPDIn| = |A| + |C|. +It is also clear that |A| = 1 + n2. +Therefore, in view of +Lemma 1.1, to determine the sizes of ODIn and OPDIn, it suffices to count how many distinct restrictions +of permutations of D2n with rank greater than or equal to 2 are order-preserving and orientation-preserving, +respectively. +First, we determine B. Let k ∈ {0, 1, . . . , n − 1}. +Clearly, the only order-preserving restrictions of hgk, with rank greater than or equal to 2, are of the form +hgk|{i n +2 and jr+1 − jr < n +2 . Moreover, r is the only index in {1, 2, . . . , k − 1} such that ir+1 − ir ⩾ n +2 . +6 + +We begin by admiting that jp+1 − jp < +n +2 for all p = 1, 2, . . . , k − 1. +Since ip+1 − ip < +n +2 for all p ∈ +{1, 2, . . . , k − 1} \ {r}, then +d(ip, ip+1) = d(jp, jp+1), for p = 1, 2, . . . , k − 1 +=⇒ +k−1 +� +p=1 +d(ip, ip+1) = +k−1 +� +p=1 +d(jp, jp+1) +=⇒ +r−1 +� +p=1 +(ip+1 − ip) + (n − ir+1 + ir) + +k−1 +� +p=r+1 +(ip+1 − ip) = +k−1 +� +p=1 +(jp+1 − jp) +=⇒ +(ir − i1) + (n − ir+1 + ir) + (ik − ir+1) = jk − j1 +=⇒ +(ik − i1) + (n − ir+1 + ir) + (ir − ir+1) = jk − j1. +On the other hand, as d(j1, jk) = d(i1, ik) then jk − j1 = ik − i1 or jk − j1 = n − ik + i1. If jk − j1 = ik − i1 then +n−ir+1 +ir = ir+1 −ir > n +2 , which is a contradiction. Thus jk −j1 = n−ik +i1, whence 2(ik −i1 +ir −ir+1) = 0 +and so ik − i1 = ir+1 − ir, which is again a contradiction (since k ⩾ 3). +Therefore, there exists s ∈ {1, 2, . . . , k − 1} such that js+1 − js ⩾ n +2 , which is the only index under these +conditions. Moreover, js+1 − js > n +2 and s > r. Then, we have +d(ip, ip+1) = d(jp, jp+1), for p = 1, 2, . . . , k − 1 +=⇒ +k−1 +� +p=1 +d(ip, ip+1) = +k−1 +� +p=1 +d(jp, jp+1) +=⇒ +r−1 +� +p=1 +(ip+1 − ip) + (n − ir+1 + ir) + +k−1 +� +p=r+1 +(ip+1 − ip) = +s−1 +� +p=1 +(jp+1 − jp) + (n − js+1 + js) + +k−1 +� +p=s+1 +(jp+1 − jp) +=⇒ +(ir − i1) + (n − ir+1 + ir) + (ik − ir+1) = (js − j1) + (n − js+1 + js) + (jk − js+1) +=⇒ +(n + ik − i1) + 2(ir − ir+1) = (n + jk − j1) + 2(js − js+1). +Next, as ik − i1 ⩾ ir+1 − ir > n +2 and jk − j1 ⩾ js+1 − js > n +2, we have +n − ik + i1 = d(i1, ik) = d(j1, jk) = n − jk + j1 +and so ir+1 − ir = js+1 − js. On the other hand, since is+1 − is < n +2 and js+1 − js > n +2, we have +is+1 − is = d(is, is+1) = d(js, js+1) = n − js+1 + js, +whence ir+1 −ir = n−is+1 +is and so n−1 ⩾ is+1 −ir = n+is −ir+1 ⩾ n, which is once again a contradiction. +Thus, we proved that ip+1 − ip = jp+1 − jp, for all p ∈ {1, 2, . . . , k − 1}. +Now, let 1 ⩽ p < q ⩽ k. Then, we have iq − ip = �q−1 +t=p(it+1 − it) = �q−1 +t=p(jt+1 − jt) = jq − jp, from which +follows also that n − iq + ip = n − jq + jp. Hence +d(ip, iq) = +� iq − ip +if iq − ip ⩽ n +2 +n − iq + ip +if iq − ip > n +2 += +� jq − jp +if jq − jp ⩽ n +2 +n − jq + jp +if jq − jp > n +2 += d(jp, jq). +Thus α ∈ DIn, as required. +Let us denote by id the identity transformation on Ωn and, for X ⊆ Ωn, by idX the partial identity with +domain X, i.e. idX = id|X. +Now, for A = {i1 < i2 < · · · < ik} ⊆ Ωn with 2 ⩽ k ⩽ n, define +d(A) = (d1, d2, . . . , dk), +7 + +with dp = d(ip, ip+1), for p = 1, . . . , k − 1, and dk = d(i1, ik). Take also B = {j1 < j2 < · · · < jk} ⊆ Ωn and +define δA,B as the only order-preserving transformation from A onto B, i.e. +δA,B = +� i1 +i2 +· · · +ik +j1 +j2 +· · · +jk +� +. +Then, we have: +Lemma 3.2 Let A = {i1 < i2 < · · · < ik} ⊆ Ωn and B = {j1 < j2 < · · · < jk} ⊆ Ωn with 2 ⩽ k ⩽ n. Then: +1. d(A) = d(B) if and only if there exists an order-preserving partial isometry from A onto B (i.e. if and +only if δA,B ∈ ODIn); +2. d(A) = d(Bh) if and only if there exists an order-reversing partial isometry from A onto B; +3. d(A) = d(Bg−s) for some 0 ⩽ s ⩽ n − 1 if and only if there exists an orientation-preserving partial +isometry from A onto B. +Proof. In order to prove Property 1, first suppose that d(A) = d(B). Then, we have, for 1 ⩽ p ⩽ k − 1, +d(ip, ip+1) = d(jp, jp+1) = d(ipδA,B, ip+1δA,B) and d(i1, ik) = d(j1, jk) = d(i1δA,B, ikδA,B), whence δA,B ∈ DIn, +by Lemma 3.1, and so δA,B ∈ ODIn. +Conversely, suppose that δA,B ∈ ODIn. Then, in particular, d(ip, ip+1) = d(ipδA,B, ip+1δA,B) = d(jp, jp+1), +for 1 ⩽ p ⩽ k − 1, and d(i1, ik) = d(i1δA,B, ikδA,B) = d(j1, jk), whence d(A) = d(B). +Next, we prove Property 2. If d(A) = d(Bh) then, by Property 1, δA,Bh ∈ ODIn and so, as k ⩾ 2 and h|Bh +is an order-reversing partial isometry from Bh onto B, it follows that δA,Bhh|Bh is an order-reversing partial +isometry from A onto B. +Conversely, suppose there exists an order-reversing partial isometry ξ from A onto B. Then +ξ = +� i1 +i2 +· · · +ik +jk +jk−1 +· · · +j1 +� +and Bh = {n − jk + 1 < n − jk−1 + 1 < · · · < n − j1 + 1}, whence +δA,Bh = +� +i1 +i2 +· · · +ik +n − jk + 1 +n − jk−1 + 1 +· · · +n − j1 + 1 +� += ξh|B ∈ ODIn +and so, by Property 1, d(A) = d(Bh). +Finally, we prove Property 3. First, suppose that d(A) = d(Bg−s) for some 0 ⩽ s ⩽ n − 1. Then, we have +δA,Bg−s ∈ ODIn, by Property 1. Since gs|Bg−s is an orientation-preserving partial isometry from Bg−s onto B, +then δA,Bg−sgs|Bg−s is an orientation-preserving partial isometry from A onto B. +Conversely, suppose there exists an orientation-preserving partial isometry ξ from A onto B. If k = 2 then +ξ = +� i1 +i2 +j1 +j2 +� += δA,B +or +ξ = +� i1 +i2 +j2 +j1 +� +and so, in both cases, we get δA,B ∈ ODIn, whence d(A) = d(B)(= d(Bg−s), with s = 0), by Property 1. Thus, +suppose that k > 2. Then, since an orientation-preserving restriction of an orientation-reversing permutation +must have rank less than or equal to two (cf. proof of Theorem 2.1), there exists 0 ⩽ s ⩽ n − 1 such that +ξ = gs|A. Therefore, A = Bg−s and so δA,Bg−s = δA,A = idA ∈ ODIn, since any partial identity is an order- +preserving partial isometry. Hence, by Property 1, it follows that d(A) = d(Bg−s), as required. +8 + +Next, recall that, given an inverse submonoid M of In, it is well known that Green’s relations L, R and H +of M can be described as following: for α, β ∈ M, +• αLβ if and only if Im(α) = Im(β); +• αRβ if and only if Dom(α) = Dom(β); +• αHβ if and only if Im(α) = Im(β) and Dom(α) = Dom(β). +In In we also have +• αJβ if and only if | Dom(α)| = | Dom(β)| (if and only if | Im(α)| = | Im(β)|). +Observe that, for a finite monoid, we always have J = D(= L ◦ R = R ◦ L). +Since the monoids ODIn, MDIn, and OPDIn are inverse submonoids of In, it remains to give a description +of Green’s relation J: +Theorem 3.3 Let M ∈ {ODIn, MDIn, OPDIn} and let α, β ∈ M. Then, αJβ if and only if one of the +following properties is satisfied: +1. | Dom(α)| = | Dom(β)| ⩽ 1; +2. | Dom(α)| = | Dom(β)| ⩾ 2 and +d(Dom(α)) = + + + +d(Dom(β)) +if M = ODIn +d(Dom(β)) or d(Dom(hβ)) +if M = MDIn +d(Dom(gsβ)) for some 0 ⩽ s ⩽ n − 1 +if M = OPDIn. +Proof. First, suppose that αJβ (in M). +Then αJβ in In and so | Dom(α)| = | Dom(β)|. +If | Dom(α)| = +| Dom(β)| ⩽ 1 there is nothing more to prove. +Thus, suppose that | Dom(α)| = | Dom(β)| ⩾ 2 and let γ, λ ∈ M be such that α = γβλ. We can assume, +without loss of generality (by considering γ|Dom(α) instead of γ, if necessary), that Dom(γ) = Dom(α). Hence +Im(γ) = Dom(β). Then, γ is an order-preserving partial isometry from Dom(α) onto Dom(β), if M = ODIn, γ +is an order-preserving or order-reversing partial isometry from Dom(α) onto Dom(β), if M = MDIn, and γ is +an orientation-preserving partial isometry from Dom(α) onto Dom(β), if M = OPDIn. Therefore, by Lemma +3.2, we have +d(Dom(α)) = d(Dom(β)), if M = ODIn, +d(Dom(α)) = d(Dom(β)) or d(Dom(α)) = d(Dom(β)h) = d(Dom(β)h−1) = d(Dom(hβ)), if M = MDIn, +and +d(Dom(α)) = d(Dom(β)g−s) = d(Dom(gsβ)), for some 0 ⩽ s ⩽ n − 1, if M = OPDIn. +Conversely, suppose that Property 1 or 2 is satisfied. If | Dom(α)| = | Dom(β)| ⩽ 1 then, as M contains all +partial permutations of rank less than or equal to one, it is clear that αJβ. So, suppose that Property 2 holds. +Since Dom(hβ) = Dom(β)h and Dom(gsβ) = Dom(β)g−s for all 0 ⩽ s ⩽ n − 1, by Lemma 3.2, we can conclude +that M possesses a partial transformation γ from Dom(α) onto Dom(β). Take also λ = β−1γ−1α ∈ M. Hence, +since γββ−1γ−1 and γ−1αα−1γ are idempotents, we have +γβλ = γββ−1γ−1α = idDom(α)α = α +and +γ−1αλ−1 = γ−1αα−1γβ = idDom(β)β = β +and so αJβ, as required. +9 + +4 +Generators and ranks +Let +ei = idΩn\{i} = +�1 +· · · +i − 1 +i + 1 +· · · +n +1 +· · · +i − 1 +i + 1 +· · · +n +� +∈ DIn, +for 1 ⩽ i ⩽ n. Clearly, for 1 ⩽ i, j ⩽ n, we have e2 +i = ei and eiej = idΩn\{i,j} = ejei. More generally, for any +X ⊆ Ωn, we get Πi∈Xei = idΩn\X. +Now, take α ∈ DIn. +Then, since the elements of DIn are precisely the restrictions of D2n, we have +α = hjgi|Dom(α), for some j ∈ {0, 1} and i ∈ {0, 1, . . . , n − 1}. Hence α = hjgiidDom(α) = hjgiΠk∈Ωn\Dom(α)ek. +Therefore {g, h, e1, e2, . . . , en} is a generating set of DIn. Moreover, since ei = gn−iengi for all i ∈ {1, 2, . . . , n}, +it follows that {g, h, en} is also a generating set of DIn (in fact, as gn = 1, we also have en = gieign−i and so +each set {g, h, ei}, with 1 ⩽ i ⩽ n, generates DIn). See [17]. +Notice that e1, e2, . . . , en are elements of ODIn, MDIn and OPDIn. +Consider the elements +x = +�1 +2 +· · · +n − 1 +2 +3 +· · · +n +� +and +y = x−1 = +�2 +3 +· · · +n +1 +2 +· · · +n − 1 +� +of ODIn with rank n − 1 and the elements +xi = +�1 +1 + i +1 +n − i + 1 +� +and +yi = x−1 +i += +�1 +n − i + 1 +1 +1 + i +� +, +for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, of ODIn with rank 2. Observe that d(1, 1 + i) = i, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, and ⌊n−1 +2 ⌋ < n +2 . +Proposition 4.1 The monoids ODIn, MDIn and OPDIn are generated by +{x, y, e2, . . . , en−1, x1, x2, . . . , x⌊ n−1 +2 +⌋, y1, y2, . . . , y⌊ n−1 +2 +⌋}, +{h, x, e2, . . . , e⌊ n+1 +2 ⌋, x1, x2, . . . , x⌊ n−1 +2 ⌋, y1, y2, . . . , y⌊ n−1 +2 ⌋} +and +{g, ei, x1, x2, . . . , x⌊ n−1 +2 +⌋}, +with 1 ⩽ i ⩽ n, +respectively. +Proof. First, we show that {x, y, e2, . . . , en−1, x1, x2, . . . , x⌊ n−1 +2 +⌋, y1, y2, . . . , y⌊ n−1 +2 +⌋} generates ODIn. +Let M be the monoid generated by {x, y, e2, . . . , en−1, x1, x2, . . . , x⌊ n−1 +2 +⌋, y1, y2, . . . , y⌊ n−1 +2 +⌋} ⊆ ODIn. Then +M is contained in ODIn. In order to show the converse inclusion, notice first that e1 = yx and en = xy, whence +e1, e2, . . . , en ∈ M and so M contains all restrictions of each of its elements. +Next, since the elements of DIn are the restrictions of D2n, then the elements of ODIn are the order- +preserving restrictions of gk and hgk for 0 ⩽ k ⩽ n − 1, which are, in turn, the restrictions of +gk|{1,2,...,n−k}, +gk|{n−k+1,...,n} +and +hgk|{i,j}, +with 1 ⩽ i ⩽ k and k + 1 ⩽ j ⩽ n. Therefore, it suffices to show that these elements belong to M. +Notice that, if k = 0 then gk|{1,2,...,n−k} and gk|{n−k+1,...,n} are the identity transformation and the empty +transformation, respectively, and so both belong to M. So, let 1 ⩽ k ⩽ n − 1. Then, we have gk|{1,2,...,n−k} = +xk ∈ M and gk|{n−k+1,...,n} = yn−k ∈ M. On the other hand, for 1 ⩽ i ⩽ k and k + 1 ⩽ j ⩽ n, we get +hgk|{i,j} = + + + + + +� +ℓ∈Ωn\{i,j} eℓ +if i = k+1 +2 +� +ℓ∈Ωn\{i,j} eℓxk−2i+1 +if i < k+1 +2 +� +ℓ∈Ωn\{i,j} eℓy2i−k−1 +if i > k+1 +2 , +10 + +if j − i = n +2 , and +hgk|{i,j} = +� yi−1xj−ixk−i +if j − i ⩽ ⌊n−1 +2 ⌋ +yi−1yn−j+ixk−i +if j − i > ⌊n−1 +2 ⌋, +if j − i ̸= n +2 (as usual, putting x0 = y0 = 1), and so hgk|{i,j} ∈ M. +Thus, we proved that M = ODIn. +Next, regarding the monoid MDIn, we have α = (αh)h and αh ∈ ODIn for all α ∈ MDIn \ ODIn, which +allows us to deduce that MDIn is generated by ODIn ∪ {h}. On the other hand, we have y = hxh and heih = +en−i+1 for all 1 ⩽ i ⩽ n. +Thus, we conclude that {h, x, e2, . . . , e⌊ n+1 +2 ⌋, x1, x2, . . . , x⌊ n−1 +2 +⌋, y1, y2, . . . , y⌊ n−1 +2 +⌋} +generates MDIn. +Finally, we turn our attention to the monoid OPDIn. +Let α ∈ OPDIn. Then α ∈ POPIn and so, by [10, Proposition 3.1], there exist 0 ⩽ k ⩽ n−1 and β ∈ POIn +such that α = gkβ. Since β = gn−kα ∈ DIn, we get β ∈ ODIn. So α = gkβ, with β ∈ ODIn. Therefore, +OPDIn is generated by ODIn ∪ {g}. On the other hand, we have ej = gn−jengj for all 1 ⩽ j ⩽ n, gℓxℓgℓ = yℓ +for all 1 ⩽ ℓ ⩽ ⌊n−1 +2 ⌋, x = eng and y = gn−1en. Hence, OPDIn is generated by {g, en, x1, x2, . . . , x⌊ n−1 +2 ⌋}. +Let 1 ⩽ i ⩽ n. Since en = gieign−i, then {g, ei, x1, x2, . . . , x⌊ n−1 +2 +⌋} also generates OPDIn, as required. +In order to determine the ranks of these monoids, we first prove the following lemma: +Lemma 4.2 Let 1 ⩽ i ⩽ ⌊n−1 +2 ⌋ and let γ1, γ2, . . . , γk, λ1, λ2, . . . , λℓ be k + ℓ (k, ℓ ⩾ 1) elements of DIn such +that xi = γ1γ2 · · · γk and yi = λ1λ2 · · · λℓ. +1. If γ1, γ2, . . . , γk, λ1, λ2, . . . , λℓ ∈ MDIn then there exist 1 ⩽ p ⩽ k, 1 ⩽ q ⩽ ℓ, 1 ⩽ a < b ⩽ n and +1 ⩽ c < d ⩽ n such that Dom(γp) = {a, b}, Dom(λq) = {c, d}, b − a = i and d − c = n − i. +2. If γ1, γ2, . . . , γk ∈ OPDIn then there exist 1 ⩽ p ⩽ k and 1 ⩽ a < b ⩽ n such that Dom(γp) = {a, b} and +b − a ∈ {i, n − i}. +Consequently, any generating set of ODIn, MDIn and OPDIn has at least 2⌊n−1 +2 ⌋, 2⌊n−1 +2 ⌋ and ⌊n−1 +2 ⌋ trans- +formations of rank two, respectively. +Proof. First, observe that the last statement of this lemma follows immediately from Properties 1 (notice that +ODIn ⊆ MDIn) and 2 and from the fact that {1, 2, . . . , ⌊n−1 +2 ⌋} ∩ {n − i | 1 ⩽ i ⩽ ⌊n−1 +2 ⌋} = ∅ . +We begin by making some considerations about the elements of MDIn. +Let ξ be an element of MDIn with rank greater than or equal to 2 and take 0 ⩽ t ⩽ n − 1 such that +ξ = gt|Dom(ξ) or ξ = hgt|Dom(ξ). +If either ξ is order-reversing and ξ = gt|Dom(ξ) or ξ is order-preserving and ξ = hgt|Dom(ξ) then ξ must have +rank 2: Dom(ξ) = {a < b}, with 1 ⩽ a ⩽ n − t < b ⩽ n, in the first case, and 1 ⩽ a ⩽ t < b ⩽ n, in the last +one. We say that such an element ξ of MDIn is inverted. +On the other hand, if either ξ is order-preserving and ξ = gt|Dom(ξ) or ξ is order-reversing and ξ = hgt|Dom(ξ) +then, for all a, b ∈ Dom(ξ), we have +|aξ − bξ| = |a − b|. +(1) +Notice that if a, b ∈ Dom(ξ) are such that a < b then, in the first case, 1 ⩽ a < b ⩽ n−t or n+t+1 ⩽ a < b ⩽ n +and, in the second case, 1 ⩽ a < b ⩽ t or t + 1 ⩽ a < b ⩽ n. We say that such an element ξ of MDIn is +non-inverted. +Next, let ξ1, ξ2, . . . , ξr be r ( r ⩾ 1) non-inverted elements of MDIn such that rank(ξ1ξ2 · · · ξr) ⩾ 2. Then, +for all a, b ∈ Dom(ξ1ξ2 · · · ξr), by applying consecutively (1) to ξr, ξr−1, . . . , ξ1, clearly, we obtain +|aξ1ξ2 · · · ξr − bξ1ξ2 · · · ξr| = |a − b|. +(2) +11 + +Now, in order to prove Property 1, suppose that γ1, γ2, . . . , γk, λ1, λ2, . . . , λℓ ∈ MDIn (keep in mind that +γ1γ2 · · · γk = xi and λ1λ2 · · · λℓ = yi). +If γ1, γ2, . . . , γk are all non-inverted elements of MDIn then, by (2), we have +n − i = |1 − (n − 1 + i)| = |1xi − (1 + i)xi| = |1γ1γ2 · · · γk − (1 + i)γ1γ2 · · · γk| = |1 − (1 + i)| = i, +which is a contradiction. Thus, at least one of the elements γ1, γ2, . . . , γk is inverted. Let 1 ⩽ p ⩽ k be the +smallest index such that γp is inverted. Then, γp has rank 2 and, since 1γ1 · · · γp−1, (1+i)γ1 · · · γp−1 ∈ Dom(γp), +we have Dom(γp) = {1γ1 · · · γp−1, (1 + i)γ1 · · · γp−1} and, by (2), +|1γ1 · · · γp−1 − (1 + i)γ1 · · · γp−1| = |1 − (1 + i)| = i. +Similarly, if λ1, λ2, . . . , λℓ are all non-inverted elements of MDIn then, by (2), we have +i = |1 − (1 + i)| = |1yi − (n − 1 + i)yi| = |1λ1λ2 · · · λℓ − (n − 1 + i)λ1λ2 · · · λℓ| = |1 − (n − i + 1)| = n − i, +which is also a contradiction. Thus, at least one of the elements λ1, λ2, . . . , λℓ is inverted and we may take the +smallest index 1 ⩽ q ⩽ ℓ such that λq is inverted. Since 1λ1 · · · λq−1, (n + i − 1)λ1 · · · λq−1 ∈ Dom(λq) and λq +has rank 2, we have Dom(λq) = {1λ1 · · · λq−1, (n − i + 1)λ1 · · · λq−1} and, by (2), +|1λ1 · · · λq−1 − (n − i + 1)λ1 · · · λq−1| = |1 − (n − i + 1)| = n − i. +Therefore, we proved Property 1. +Next, with the purpose of proving Property 2, suppose that γ1, γ2, . . . , γk ∈ OPDIn (remember we have +γ1γ2 · · · γk = xi). +We begin by observing that xi = hg|{1,1+i}. +Since d(1, 1 + i) = i < +n +2 , then hg is the only extension +in D2n of xi, by Lemma 1.1. +If for all 1 ⩽ j ⩽ k there exists 0 ⩽ tj ⩽ n − 1 such that γj = gtj|Dom γj, +then xi = g +�k +j=1 tj|{1,1+i}, which contradicts the previous conclusion. +Hence, there exists 1 ⩽ p ⩽ k such +that γp = hgt|Dom(γp), for some 0 ⩽ t ⩽ n − 1. Let us assume that the index p is the smallest under these +conditions. +Since γp preserves the orientation, then Dom(γp) = {a, b}, for some 1 ⩽ a ⩽ t < b ⩽ n. As +1γ1 · · · γp−1, (1 + i)γ1 · · · γp−1 ∈ Dom(γp), it follows that Dom(γp) = {1γ1 · · · γp−1, (1 + i)γ1 · · · γp−1}. +On the other hand, by the minimality of p, we have γ1 · · · γp−1 = gs|Dom(γ1···γp−1), for some 0 ⩽ s ⩽ n − 1. +Hence +|1γ1 · · · γp−1 − (1 + i)γ1 · · · γp−1| = |1gs − (1 + i)gs| ∈ {i, n − i}, +as required. +Recall that ODI3 = POI3, MDI3 = PODI3 and OPDI3 = POPI3. Then, the monoids ODI3, MDI3 +and OPDI3 have ranks 3, 3 and 2 (see [10, 11, 13]), respectively. For n greater than 3, we have: +Theorem 4.3 For n ⩾ 4, the monoids ODIn, MDIn and OPDIn have ranks n + 2⌊n−1 +2 ⌋, 2 + 3⌊n−1 +2 ⌋ and +2 + ⌊n−1 +2 ⌋, respectively. +Proof. Let M ∈ {ODIn, MDIn, OPDIn} and let G be a generating set of the monoid M. Notice that the +partial identities e1, . . . , en belong to M. +Suppose that M = ODIn. Then, the only permutation of M is the identity and so, for 1 ⩽ i ⩽ n, we +have ei = γ1γ2 · · · γk, for some γ1, γ2, . . . , γk ∈ G \ {1} (k ⩾ 1), and so Im(γk) = Im(ei) = Ωn \ {i}. Hence, G +possesses at least n elements with rank n − 1. Thus, taking into account Lemma 4.2, we get |G| ⩾ n + 2⌊n−1 +2 ⌋. +Next, suppose that M = MDIn. Recall that, in addition to the identity, M has only h as a permutation +and so, in particular, we must have h ∈ G. Let 1 ⩽ i ⩽ n. Then, there exist γ1, γ2, . . . , γk ∈ G \ {1} (k ⩾ 1) +such that ei = γ1γ2 · · · γk and: γk ̸= h; or k ⩾ 2, γk = h and γk−1 ̸= h. Hence, Im(γk) = Im(ei) = Ωn \ {i} or +12 + +Im(γk−1) = Im(ei)h = Ωn \ {n − i + 1}. Therefore, we can conclude that G possesses at least ⌊n+1 +2 ⌋ elements +with rank n − 1. Thus, in view of Lemma 4.2, we obtain |G| ⩾ 1 + ⌊n+1 +2 ⌋ + 2⌊n−1 +2 ⌋ = 2 + 3⌊n−1 +2 ⌋. +Finally, suppose that M = OPDIn. Since OPDIn contains the permutation g and a partial identity of +rank n−1, we can conclude that G has at least one permutation and one transformation with rank n−1. 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Thesis, School of Science and +Technology of NOVA University Lisbon, 2022. +Ilinka Dimitrova, Department of Mathematics, Faculty of Mathematics and Natural Science, South-West University ”Neofit +Rilski”, 2700 Blagoevgrad, Bulgaria; e-mail: ilinka dimitrova@swu.bg. +V´ıtor H. Fernandes, Center for Mathematics and Applications (NovaMath) and Department of Mathematics, FCT NOVA, +Faculdade de Ciˆencias e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516 Caparica, Portugal; e-mail: +vhf@fct.unl.pt. +J¨org Koppitz, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria; e-mail: +koppitz@math.bas.bg. +Teresa M. Quinteiro, Instituto Superior de Engenharia de Lisboa, 1950-062 Lisboa, Portugal. Also: Center for Mathematics +and Applications (NovaMath), Faculdade de Ciˆencias e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516 +Caparica, Portugal; e-mail: tmelo@adm.isel.pt. +14 + diff --git a/IdAzT4oBgHgl3EQfjf31/content/tmp_files/load_file.txt b/IdAzT4oBgHgl3EQfjf31/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9c612d5a3e76eb886751c16d67902741b10075c --- /dev/null +++ b/IdAzT4oBgHgl3EQfjf31/content/tmp_files/load_file.txt @@ -0,0 +1,815 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf,len=814 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='01519v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='RA] 4 Jan 2023 On three remarkable submonoids of the dihedral inverse monoid on a finite set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Dimitrova, V´ıtor H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Fernandes∗ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Koppitz and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Quinteiro† January 5, 2023 Abstract In this paper we consider three notable submonoids of the dihedral inverse monoid DIn, namely its submonoids OPDIn, MDIn and ODIn of all orientation-preserving, monotone and order-preserving trans- formations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' For each of these three monoids, we compute the cardinal, give descriptions of Green’s relations and determine the rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' 2020 Mathematics subject classification: 20M10, 20M20, 05C12, 05C25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Keywords: dihedral inverse monoid, transformations, orientation, monotonicity, partial isometries, cycle graphs, rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' 1 Introduction and preliminaries Let Ω be a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Denote by PT (Ω) the monoid (under composition) of all partial transformations on Ω, by T (Ω) the submonoid of PT (Ω) of all full transformations on Ω, by I(Ω) the symmetric inverse monoid on Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' the inverse submonoid of PT (Ω) of all partial permutations on Ω, and by S(Ω) the symmetric group on Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' the subgroup of PT (Ω) of all permutations on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' If Ω is a finite set with n elements (n ∈ N), say Ω = Ωn = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , n}, as usual, we denote PT (Ω), T (Ω), I(Ω) and S(Ω) simply by PT n, Tn, In and Sn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Recall that the rank of a monoid M is the minimum size of a generating set of M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' the minimum of the set {|X| | X ⊆ M and X generates M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' For n ⩾ 3, it is well-known that Sn has rank 2 (as a semigroup, a monoid or a group) and Tn, In and PT n have ranks 3, 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' The survey [12] presents these results and similar ones for other classes of transformation monoids, in particular, for monoids of order-preserving transformations and for some of their extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' For example, the rank of the extensively studied monoid of all order-preserving transformations of a chain with n elements is n, a result proved by Gomes and Howie [22] in 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' More recently, for instance, the papers [3, 6, 7, 8, 9, 14, 15, 18, 20] are dedicated to the computation of the ranks of certain classes of transformation semigroups or monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Now, let G = (V, E) be a finite simple connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' The (geodesic) distance between two vertices x and y of G, denoted by dG(x, y), is the length of a shortest path between x and y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' the number of edges in a shortest path between x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Let α ∈ PT (V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' We say that α is a partial isometry or distance preserving partial transformation of G if dG(xα, yα) = dG(x, y), ∗This work is funded by national funds through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=', under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (NovaMath - Center for Mathematics and Applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' †This work is funded by national funds through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=', under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (NovaMath - Center for Mathematics and Applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' 1 for all x, y ∈ Dom(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Denote by DP(G) the subset of PT (V ) of all partial isometries of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Clearly, DP(G) is a submonoid of PT (V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' As a consequence of the property dG(x, y) = 0 if and only if x = y, for all x, y ∈ V , it immediately follows that DP(G) ⊆ I(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Moreover, DP(G) is an inverse submonoid of I(V ) (see [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Observe that, if G = (V, E) is a complete graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' E = {{x, y} | x, y ∈ V, x ̸= y}, then DP(G) = I(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' For n ∈ N, consider the undirected path Pn with n vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Pn = ({1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , n}, {{i, i + 1} | i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , n − 1}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Then, obviously, DP(Pn) coincides with the monoid DPn = {α ∈ In | |iα − jα| = |i − j|, for all i, j ∈ Dom(α)} of all partial isometries on Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' The study of partial isometries on Ωn was initiated by Al-Kharousi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' The first of these two papers is dedicated to investigating some combinatorial properties of the monoid DPn and of its submonoid ODPn of all order-preserving (considering the usual order of N) partial isometries, in particular, their cardinalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' The second paper presents the study of some of their algebraic properties, namely Green’s structure and ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Presentations for both the monoids DPn and ODPn were given by Fernandes and Quinteiro in [19] and the maximal subsemigroups of ODPn were characterized by Dimitrova in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' The monoid DP(Sn) of all partial isometries of a star graph Sn with n vertices (n ⩾ 1) was considered by Fernandes and Paulista in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' They determined the rank and size of DP(Sn) as well as described its Green’s relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' A presentation for DP(Sn) was also exhibited in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Next, for n ⩾ 3, consider the cycle graph Cn = ({1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , n}, {{i, i + 1} | i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , n − 1} ∪ {{1, n}}) with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Notice that, cycle graphs and cycle subgraphs play a fundamental role in Graph Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' The monoid DP(Cn) of all partial isometries of the cycle graph Cn was studied by Fernandes and Paulista in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' They showed that DP(Cn) is an inverse submonoid of the monoid of all oriented partial permutations on a chain with n elements and, moreover, that it coincides with the inverse submonoid of In formed by all restrictions of a dihedral subgroup of Sn of order 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Therefore, in [17], DP(Cn) was called the dihedral inverse monoid on Ωn and, in this paper, from now on, we denote DP(Cn) by the most appropriate notation DIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Recall also that in [17] it was determined the cardinal and rank of DIn as well as descriptions of its Green’s relations and, furthermore, presentations for DIn were also given in that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Next, suppose that Ωn is a chain, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Ωn = {1 < 2 < · · · < n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' A partial transformation α ∈ PT n is called order-preserving [order-reversing] if, x ⩽ y implies xα ⩽ yα [xα ⩾ yα], for all x, y ∈ Dom(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' A partial transformation is said to be monotone if it is order-preserving or order-reversing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' It is clear that the product of two order-preserving or of two order-reversing transformations is order-preserving and the product of an order-preserving transformation by an order-reversing transformation, or vice-versa, is order-reversing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' We denote by POn the submonoid of PT n of all order-preserving transformations and by PODn the submonoid of PT n of all monotone transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Let also POIn = POn∩In, the monoid of all order-preserving partial permutations of Ωn, and PODIn = PODn ∩ In, the monoid of all monotone partial permutations of Ωn, which are inverse submonoids of PT n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Let s = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , at) be a sequence of t (t ⩾ 0) elements from the chain Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' We say that s is cyclic [anti-cyclic] if there exists no more than one index i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , t} such that ai > ai+1 [ai < ai+1], where at+1 denotes a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' We also say that s is oriented if s is cyclic or s is anti-cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' See [4, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Given a partial transformation α ∈ PT n such that Dom(α) = {a1 < · · · < at}, with t ⩾ 0, we say that α is orientation- preserving [orientation-reversing, oriented] if the sequence of its images (a1α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , atα) is cyclic [anti-cyclic, oriented].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' It is easy to show that the product of two orientation-preserving or of two orientation-reversing transformations is orientation-preserving and the product of an orientation-preserving transformation by an orientation-reversing transformation, or vice-versa, is orientation-reversing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' We denote by POPn the submonoid of PT n of all orientation-preserving transformations and by PORn the submonoid of PT n of all oriented 2 transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Consider also the inverse submonoids POPIn = POPn ∩ In, of all orientation-preserving partial permutations, and PORIn = PORn ∩ In, of all oriented partial permutations, of PT n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Notice that, POIn ⊆ PODIn ⊆ PORIn and POIn ⊆ POPIn ⊆ PORIn, by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Now, let us consider the following permutations of Ωn of order n and 2, respectively: g = �1 2 · · n − 1 n 2 3 · · n 1 � and h = �1 2 · · n − 1 n n n − 1 · · 2 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' It is clear that g, h ∈ DIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Moreover, for n ⩾ 3, g together with h generate the well-known dihedral group D2n of order 2n (considered as a subgroup of Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' In fact, for n ⩾ 3, D2n = ⟨g, h | gn = 1, h2 = 1, hg = gn−1h⟩ = {1, g, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , gn−1, h, hg, hg2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , hgn−1} and we have gk = � 1 2 · · n − k n − k + 1 · · n 1 + k 2 + k · · n 1 · · k � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' igk = � i + k if 1 ⩽ i ⩽ n − k i + k − n if n − k + 1 ⩽ i ⩽ n, and hgk = �1 · · k k + 1 · · n k · · 1 n · · k + 1 � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' ihgk = � k − i + 1 if 1 ⩽ i ⩽ k n + k − i + 1 if k + 1 ⩽ i ⩽ n, for 0 ⩽ k ⩽ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Denote also by Cn the cyclic group of order n generated by g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Cn = ⟨g | gn = 1⟩ = {1, g, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , gn−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Until the end of this paper, we will consider n ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' For any two vertices x and y of Cn, we now denote the distance dCn(x, y) simply by d(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Notice that, we have d(x, y) = min{|x − y|, n − |x − y|} = � |x − y| if |x − y| ⩽ n 2 n − |x − y| if |x − y| > n 2 and so 0 ⩽ d(x, y) ⩽ n 2 , for all x, y ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Observe also that d(x, y) = n 2 ⇔ |x − y| = n 2 ⇔ n − |x − y| = n 2 ⇔ |x − y| = n − |x − y|, in which case n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Recall that DIn is the submonoid of the monoid PORIn whose elements are precisely all restrictions of the dihedral group D2n of order 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Moreover, it is also known exactly how many extensions in D2n each element of DIn has: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='1 ([17, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='1]) Let α ∈ PT n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Then α ∈ DIn if and only if there exists σ ∈ D2n such that α = σ|Dom(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Furthermore, for α ∈ DIn, one has: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' If either | Dom(α)| = 1 or | Dom(α)| = 2 and d(min Dom(α), max Dom(α)) = n 2 (in which case n is even), then there exist exactly two (distinct) permutations σ, σ′ ∈ D2n such that α = σ|Dom(α) = σ′|Dom(α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' If either | Dom(α)| = 2 and d(min Dom(α), max Dom(α)) ̸= n 2 or | Dom(α)| ⩾ 3, then there exists exactly one permutation σ ∈ D2n such that α = σ|Dom(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Notice that, for an even n, we have B2 = {α ∈ DIn | |Dom(α)| = 2 and d(min Dom(α), max Dom(α)) = n 2 } = ��i i + n 2 j j + n 2 � , � i i + n 2 j + n 2 j � | 1 ⩽ i, j ⩽ n 2 � 3 and so |B2| = 2(n 2 )2 = 1 2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' In this paper, we study three remarkable submonoids of DIn, namely OPDIn = DIn∩POPIn, the monoid of all orientation-preserving partial isometries of Cn, MDIn = DIn ∩ PODIn, the monoid of all monotone partial isometries of Cn, and ODIn = DIn ∩POIn, the monoid of all order-preserving partial isometries of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Observe that DIn, OPDIn, MDIn and ODIn are all inverse submonoids of the symmetric inverse monoid In, ODIn ⊆ MDIn and ODIn ⊆ OPDIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Also, notice that OPDI3 = POPI3, MDI3 = PODI3 and ODI3 = POI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' In Section 2, we compute the cardinals of ODIn, MDIn and OPDIn and, in Section 3, we describe their Green’s relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Finally, Section 4 is dedicated to establish generating sets and to determine the ranks of these three monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' For general background on Semigroup Theory and standard notations, we refer to Howie’s book [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' We would like to point out that we made considerable use of computational tools, namely GAP [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' 2 Cardinals In this section, we determine the number of elements of each of the monoids ODIn, MDIn and OPDIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Let α ∈ PT n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Recall that the rank of α, denoted by rank(α), is the size of Im(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' By applying Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='1 and counting all possible distinct orientation-preserving and order-preserving re- strictions of permutations from D2n, we have: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='1 One has |ODIn| = 3 · 2n + (n + 1)n(n − 1) 6 − 1 + (−1)n 8 n2 − 2n − 2 and |OPDIn| = n2n + n2(n − 1) 2 − 1 + (−1)n 4 n2 − n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Let A = {α ∈ DIn | | Dom(α)| ⩽ 1}, B = {α ∈ ODIn | | Dom(α)| ⩾ 2} and C = {α ∈ OPDIn | | Dom(α)| ⩾ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Clearly, A = {α ∈ OPDIn | | Dom(α)| ⩽ 1} = {α ∈ ODIn | | Dom(α)| ⩽ 1} and so |ODIn| = |A| + |B| and |OPDIn| = |A| + |C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' It is also clear that |A| = 1 + n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Therefore, in view of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content='1, to determine the sizes of ODIn and OPDIn, it suffices to count how many distinct restrictions of permutations of D2n with rank greater than or equal to 2 are order-preserving and orientation-preserving, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' First, we determine B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Let k ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfjf31/content/2301.01519v1.pdf'} +page_content=' Clearly, the only order-preserving restrictions of hgk, with rank greater than or equal to 2, are of the form hgk|{i 1. If this condition +is never met, the model implies that the star +forms a black hole. +2.1.2. Explosion phase +During the first episode after shock revival +(Phase I ), outflow and inflow of materials co- +exist in the post-shock region. +This phase is +treated similarly as the pre-explosion phase ex- +cept that the explosion energy Eexp is grad- +ually increasing due to the recombination of +ejected neutrino-heated material. The relevant +mass outflow rate is computed from the neu- +trino heating rate and the binding energy at the +gain radius based on the heating model from +the pre-explosion phase. As the post-shock ve- +locity (which is computed from the explosion +energy, ejecta mass and pre-shock density) ex- +ceeds the escape velocity, accretion is assumed +to cease, and Eexp changes mainly due to explo- +sive nuclear burning and the addition of binding +energy of the outer shells (Phase II ). We deter- +mine MNS,by at the end of Phase I, and compute + +Type IIP SNe +5 +Eexp by integration throughout the envelope up +to the stellar surface. +The explosive yields of iron-group (IG) el- +ements are computed in a crude way by +“flashing” shocked material into IG elements +when the post-shock temperature exceeds 4.5 × +109 K3, but is less than the temperature for +50% dissociation into α-particles. The original +model of M¨uller et al. (2016) did not account for +the contribution of the neutrino-heated ejecta +to the IG yields. To improve upon the origi- +nal prescription, we take half of these IG ele- +ments to be 56Ni and add another contribution +from neutrino-driven outflows, which we assume +to be proportional to Eexp (as Eexp is by con- +struction determined by the amount of ejected +neutrino-heated material Mν), i.e., +MNi = 1 +2MIG + 1 +2αEexp ≈ 1 +2MIG + 1 +2Mν , (1) +where the proportional constant α is set to +mB/5 MeV. The second term represents a rough +upper limit for the production of nickel by +neutrino-driven outflows, corresponding to the +optimistic assumption that about half of the +neutrino-heated ejecta recombine to 56Ni. We +emphasize that an accurate MNi can only be +obtained by multi-D neutrino-transport simula- +tions and that Eq. (1) only represents a rough +estimate. +Our semi-analytic model includes several pa- +rameters that can be used for calibration +against more sophisticated multi-D simulations +or observational constraints (M¨uller 2015), i.e., +the shock compression factor, the conversion ef- +ficiency of accretion to neutrino luminosity, the +PNS cooling timescale (Table 1 of M¨uller et al. +2016). These parameters can be used to tune +the CCSN explosion landscape, including the +explodability and magnitude of Eexp consider- +3 We use 4.5 × 109 K instead of 5 × 109 K in the original +prescription. +ably. As a first step, we use the default param- +eter set and keep the tunability in mind. +Finally, we treat fallback as an all-or-nothing +process as in the original prescription (M¨uller +et al. 2016). We remark that fallback can signif- +icantly influence the properties of explosions for +near-critically exploding models. Also, for some +failed CCSNe, mass ejection is still possible due +to the decrease of the PNS gravitational mass by +neutrino emission (Piro 2013; Fern´andez et al. +2018; Schneider & O’Connor 2022). However, +whereas fallback is now recognized as important +for understanding the black-hole mass distribu- +tion (Mandel & M¨uller 2020; Mandel et al. 2021; +Antoniadis et al. 2022), these extreme events +may not contribute to the SNe IIP population. +2.2. RSG models and the explosion landscape +We apply the semi-analytic approach to two +sets of single-star solar-metallicity RSG models +as CCSN progenitors, which we refer to as M16 +(M¨uller et al. 2016) and S16 (Sukhbold et al. +2016). +Both sets were evolved with the stel- +lar evolution code KEPLER (Weaver et al. 1978; +Heger & Woosley 2010) but with two major +known differences in the physical inputs. One +is that the erroneous pair-neutrino loss rate was +updated to a corrected version in M16 but not +in S16 (see §2 of Sukhbold et al. 2018). This +can affect the late burning stages after core he- +lium depletion. The other difference is that a +fixed, large boundary pressure was used at the +stellar surface in M16 to keep the models stable. +This affected the RSG structure, making them +more compact and affecting the mass loss dur- +ing the reg giant phase. Other differences may +exist, such as the helium burning rates that im- +pact the size of the carbon oxygen core after +core helium depletion(Imbriani et al. 2001; Tur +et al. 2007; West et al. 2013). +The differences between the two sets at the +onset of collapse are shown by the comparison + +6 +Zha et al. +0 +4 +8 +12 +16 +Enclosed mass [M⊙] +10−11 +10−7 +10−3 +101 +105 +109 +Density [g cm−3] +9.5 M⊙ +14.9 M⊙ +19.9 M⊙ +M16 +S16 +108 +1010 +1012 +1014 +Radius [cm] +Figure 1. Pre-SN density profiles as a function of enclosed mass (left panel) and radius (right panel) for +selected progenitor models with MZAMS = 9.5, 14.9 and 19.9 M⊙ from M¨uller et al. (2016, M16, solid lines) +and Sukhbold et al. (2016, S16, dotted lines). All the models successfully explode. In particular, 9.5 M⊙ +is the minimum mass common to both sets, and 14.9M⊙ and 19.9 M⊙ are the closest progenitor masses to +15 M⊙ and 20 M⊙ with explosions in both sets. +Table 1. Presupernova, explosion and light-curve properties for the progenitor models shown in Fig. 1. +MZAMS +Source +Mprog +Rprog +MFe +Menv +ξ2.5 +Eexp +MNi +MNS,by +Lpl +tpl +(M⊙) +(M⊙) +(1013 cm) +(M⊙) +(M⊙) +(1051 erg) +(10−2 M⊙) +(M⊙) +(108 L⊙) +(days) +9.5 +M16 +9.11 +10.19 +1.29 +6.77 +1.6 × 10−5 +0.25 +2.3 +1.35 +3.37 +144 +S16 +9.16 +2.87 +1.30 +7.13 +6.1 × 10−5 +0.32 +2.8 +1.34 +1.48 +114 +14.9 +M16 +11.4 +10.5 +1.56 +7.27 +0.15 +0.99 +5.2 +2.18 +10.9 +96 +S16 +12.8 +5.70 +1.50 +8.62 +0.16 +1.06 +5.5 +2.18 +6.19 +95 +19.9 +M16 +14.3 +10.3 +1.56 +8.18 +0.20 +1.20 +11.0 +1.74 +11.3 +95 +S16 +15.8 +7.41 +1.53 +9.63 +0.22 +1.32 +11.4 +1.76 +8.42 +96 +Note—Here, MZAMS is the ZAMS mass for the pre-SN model. M16 and S16 stand for progenitor from the sets of +M¨uller et al. (2016) and Sukhbold et al. (2016), respectively. Mprog and Rprog are the stellar mass and radius, MFe +and Menv are the masses of the iron core and hydrogen envelope, and ξ2.5 is the compactness (Eq. 2), all defined at +the onset of collapse. Eexp, MNi and MNS,by are the resulting explosion energy, nickel mass and remnant neutron-star +mass obtained by the semi-analytic model of M¨uller et al. (2016). Lpl and tpl are the plateau luminosity and duration +of the resultant SN IIP light curve obtained by SNEC simulations. + +Type IIP SNe +7 +of the pre-SN density profiles for three selected +values of MZAMS (Fig. 1), and the global param- +eters for the pre-SN stellar structure (Fig. 2). +Figure 1 clearly shows that the larger pressure +cut results in a more dilute hydrogen envelope +for M16 models whereas the core structures are +nearly the same. +This can also be inferred +from the larger pre-SN stellar radiiRprog for M16 +models with MZAMS ≲ 24 M⊙ (panel (b) of +Fig. 2), although the different pre-SN stellar +masses (Mprog, panel (a) of Fig. 2) also indicate +subtle differences in the mass loss rates as a re- +sult of feedback processes that requires further +study but is beyond the scope of this work. A +striking difference is the opposite trends of Rprog +versus MZAMS. +The progenitor radius, Rprog +is positively correlated with progenitor mass in +the S16 models, but decreases slightly with mass +in the M16 models. +Figure 2 also illustrates differences in the core +structure between the two sets. The S16 models +have a smaller mass of the carbon-oxygen core +than M16 models for the same MZAMS, which +carries through to later evolutionary phases. +This is reflected by the final iron-core mass MFe +(panel (c) of Fig. 2), and can also be inferred +from the progenitor compactness ξ2.5 (panel (d) +of Fig. 2). Here ξ2.5 is defined as (O’Connor & +Ott 2011) +ξM = +M/M⊙ +R(Mbaryon = M)/1,000 km +���� +t=t0 +, +(2) +where M is set to be 2.5 M⊙, and t0 is the time +at the onset of collapse. Structures in the land- +scape of ξ2.5 are systematically shifted to higher +MZAMS in the S16 models. Except for this shift, +M16 and S16 models have quite similar core +structures, with stochastic variations in ξ2.5 for +MZAMS ≃ 15–20 M⊙ due to the chaotic merging +of oxygen and carbon- and neon-burning shells +(Sukhbold et al. 2018; Collins et al. 2018; Yadav +et al. 2020). The impact of the erroneous neu- +trino loss rate is most significant for stars with +MZAMS ≳ 20 M⊙, which constitute only ∼ 18% +8 +10 +12 +14 +16 +Mprog [M⊙] +(a) +M16 +S16 +0.0 +2.5 +5.0 +7.5 +10.0 +Rprog [1013 cm] +(b) +1.2 +1.4 +1.6 +1.8 +MFe [M⊙] +(c) +10 +15 +20 +25 +30 +MZAMS [M⊙] +0.0 +0.1 +0.2 +0.3 +0.4 +ξ2.5 +(d) +Figure 2. +Comparison of progenitor properties +as a function of ZAMS mass between M16 (blue +squares) and S16 (red dots) models. From top to +bottom, the panels show the pre-SN stellar mass +(Mprog), pre-SN stellar radius (Rprog), iron-core +mass (MFe) and compactness parameter (ξ2.5) at +the onset of collapse. A choice of ξ2.5,crit = 0.263 +(0.243) best discriminates the explodability for the +M16 (S16) models. + +8 +Zha et al. +0.0 +0.5 +1.0 +1.5 +2.0 +Eexp [1051erg] +M16 +S16 +0.00 +0.05 +0.10 +0.15 +MNi [M⊙] +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +ξ2.5 +1.5 +2.0 +MNS,by [M⊙] +Figure 3. Similar to Fig. 2, but for the comparison +of explosion properties as a function of progenitor +compactness ξ2.5 between the M16 (blue squares) +and S16 (red dots) models. From top to bottom, +the panels show the explosion energy Eexp, 56Ni +mass MNi and baryonic neutron star mass MNS,by. +of all the progenitors and even less for explod- +ing models. +Therefore, the overall impact on +the ensemble of SNe IIP LCs is small. +We +only +consider +pre-SN +models +with +MZAMS ≤ 30 M⊙, because models with a larger +MZAMS would exceed the Humphreys-Davidson +limit and experience significant mass loss and +result in SNe other than type IIP, aside from +the fact that few explosions are predicted in +this region in the first place. For M16 we have +1891 models with a mass resolution of 0.01 M⊙, +for which 991 successfully explode. +For S16 +we have 187 models with a mass resolution of +0.1 (0.25) M⊙ at MZAMS above (below) 13 M⊙, +for which 115 models successfully explode. In +Fig. 3 we show the explosion properties pre- +dicted by the semi-analytic supernova model as +a function of the ξ2.5. We find good agreement +between the two sets of progenitors and deter- +mine a critical ξ2.5 = 0.263 (0.243) that best +discriminates the explodability for M16 (S16) +models. +3. COMPARISON OF ALTERNATIVE +PHENOMENOLOGICAL EXPLOSION +MODELS (S16 SET) +It is currently not feasible to perform 3D sim- +ulations with neutrino transport to determine +the properties of CCSN explosions for a suffi- +ciently large number of progenitors required for +population studies. +Our semi-analytic model +is among several efficient phenomenological ap- +proaches to predict the outcome of collapse (ex- +plosion or non-explosion) as well as explosion +and remnant properties (O’Connor & Ott 2011; +Ugliano et al. 2012; Pejcha & Thompson 2015; +Perego et al. 2015; Sukhbold et al. 2016; Couch +et al. 2020; Ertl et al. 2020; Barker et al. 2022a; +Ghosh et al. 2022). +Most other studies rely +on 1D simulations that mimic the supportive +role of multi-dimensional flow instabilities in +enabling shock revival either by increasing the +neutrino emission, the neutrino energy depo- +sition, or by means of 1D turbulence models +(but see M¨uller 2019b for a critical discussion +of this approach). Qualitative and quantitative +differences and similarities between the various +phenomenological models have been discussed +in the literature, and Pejcha (2020) also pro- +vides a side-by-side comparison of important +outcomes such as the relation between explo- +sion energy and nickel mass or the predicted +neutron star mass distribution. Such compar- +isons can be somewhat skewed by differences in + +Type IIP SNe +9 +0.0 +0.5 +1.0 +1.5 +2.0 +Eexp [1051erg] +Sukhbold et al. +Barker et al. +this work +0.00 +0.05 +0.10 +0.15 +MNi [M⊙] +10 +15 +20 +25 +30 +MZAMS [M⊙] +1.5 +2.0 +MNS,by [M⊙] +Figure 4. Comparison of the explosion properties +of S16 progenitors as obtained in Sukhbold et al. +(black crosses), Barker et al. (green open squares) +and this work (red dots). Note that Barker et al. +did not calculate MNi but used the nickel masses of +S16 instead. +the size, mass range, and input physics of un- +derlying stellar evolution model sets. +For this reason, it is useful to compare our re- +sults to those obtained by different 1D simula- +tion studies for the S16 progenitor set, namely +from the study of Sukhbold et al. (Sukhbold +et al. 2016) and Barker et al. +(Barker et al. +2022a). Sukhbold et al. used the P-HOTB code +(Ugliano et al. 2012; Ertl et al. 2016) with a +gray neutrino-transport scheme and a proto- +neutron star core model, and is calibrated by +two well-observed CCSNe. +Their models are +calibrated to inferred explosion properties for +SN 1054 and SN 1987A at the respective pro- +genitor masses. +The SN 1987A calibration is +used for all progenitors with MZAMS > 12 M⊙, +and for MZAMS < 12 M⊙ interpolation between +the relevant model parameters for the two cal- +ibration cases is applied. +Barker et al. +used +the FLASH code with a multi-group two-moment +neutrino-transport scheme (O’Connor & Couch +2018) plus the STIR method for simulating tur- +bulence in 1D (Couch et al. 2020). Their STIR +method is calibrated to fit full 3D simulations +run in the same code (O’Connor & Couch 2018). +The comparison is shown in Fig. 4 for Eexp, +MNi and MNS,by. +Although with quite differ- +ent implementations and degrees of approxima- +tions, we find considerable agreements among +the results from Sukhbold et al., Barker et al. +and this work. The agreement is especially re- +markable for the baryonic neutron star mass +MNS,by, which once again confirms the impor- +tant role of the Si-O shell interface as a natural +point for the onset of the explosion and a strong +predictor for the final neutron star mass. +Discrepancies are noteworthy mainly in the +mass ranges with near-critical explodability +(gray shaded bands in Fig. 4), with MZAMS ≃ +12–15 M⊙ and 22–25 M⊙. +For MZAMS ≃ 12– +15 M⊙, Barker et al. predicts no explosion while +both Sukhbold et al. +and the semi-analytic +model obtain explosions. +Eexp and MNi in +Sukhbold et al. are, however, larger by about a +factor of 2.5 than those in this work, which may +be related to the change in calibration case of +P-HOTB from SN 1054 to SN 1987A at 12 M⊙. +On the other hand, for MZAMS ≃ 22–25 M⊙, +Sukhbold et al. +and our semi-analytic model +predict no explosion, whereas Barker et al. +yields relatively large explosion energies Eexp +(≥ 2 × 1051 erg). +The explodability of these +critical models is still under debate with state- +of-the-art 3D simulations (e.g., Ott et al. 2018; + +10 +Zha et al. +Melson et al. 2020; Burrows et al. 2020). The +mass distribution of observed SN IIP progeni- +tors (Smartt 2015) and first observational ev- +idence for the quiet disappearance of a RSG +(Adams et al. 2017), presumably by stellar col- +lapse favor a lower probability of explosion in +this mass range. +The overall trends and patterns in explosion +energy are qualitatively compatible between +the three phenomenological models outside the +gray-shaded areas. They all predict low explo- +sion energies at the low-mass end, a general +trend towards higher explosion energies in the +range of 15–22 M⊙ with considerable scatter at +higher masses. Above 25 M⊙, the agreement is +less convincing. It is noteworthy, however, that +even in the region of 22–25 M⊙, where Barker +et al. disagrees qualitatively with the other two +models for, the high explosion energies reflect a +similar pattern in M¨uller et al. (2016) with pa- +rameter choices that increase explodability (i.e., +higher turbulent pressure in the gain region or +a higher accretion efficiency for neutrino emis- +sion). +The situation for the nickel masses, MNi, +which are only available for Sukhbold et al. and +our semi-analytic model, is similar to the ex- +plosion energies. +There is rather good agree- +ment between Sukhbold et al. and our work be- +low 22 M⊙, which is rather striking considering +the relatively simple model for nickel production +used in our approach. +These results demonstrate that predictions +of explosion and remnant properties from the +three phenomenological models are quite robust +to differences in the methodology, once some +form of calibration (e.g., for one or two specific +supernovae or for the typical energy range of +observed explosions) is applied. +4. THEORETICAL LIGHT CURVES OF +TYPE IIP SNE +With the explosion properties (Eexp, MNi and +MNS,by) obtained in §2, we utilize SNEC (Mo- +rozova et al. 2015b) to generate LCs of SNe +IIP from M16 and S16 progenitors. +SNEC is +an open-source spherically-symmetrical radia- +tion hydrodynamics code with the capability to +follow the shock propagation through the stel- +lar envelope. It solves the Lagrangian hydro- +dynamics equations supplemented with a radi- +ation diffusion term. Note that SNEC assumes +local thermal equilibrium between matter and +radiation, which fails during the shock breakout +and nebular phase, but is reasonably reliable for +LCs during the plateau phase (Blinnikov & Bar- +tunov 1993) that is of interest here. We refer to +the code paper (Morozova et al. 2015b) and doc- +umentation (Morozova et al. 2015a) for details +on the numerical implementation. +We employ the default settings of SNEC, such +as the equation of state, ionization treatment +and opacities. +The newborn NS with MNS,by +is excised from the numerical grid and a ther- +mal bomb is used to initialize the shock. The +sum of Eexp and binding energy of the mass con- +tent above the excised NS is spread into the +0.1 M⊙ above the excised boundary so that the +final explosion energy equals the desired value +Eexp (Morozova et al. 2015a). For the mixing +of nickel, we simply spread MNi homogeneously +up to 3 M⊙ as our semi-analytic approach can- +not treat the mixing. The mixing of nickel is +beyond the scope of this paper but its impact +on SNe IIP LCs may be worth further investiga- +tion (see, e.g., Utrobin et al. 2017). We evolve +all the models to ∼ 200 days, by which time all +models have reached the radioactively-powered +tail phase. For comparison to observations, we +are particularly interested in two LC parame- +ters: the plateau luminosity Lpl and the plateau +duration tpl. We take the bolometric luminos- +ity at 50 days after the shock break out as Lpl. +The determination of tpl is more tricky; we ten- +tatively pick the time of the steepest gradient +of the B-band magnitude as the end of plateau +phase. The key LC and explosion parameters + +Type IIP SNe +11 +0 +50 +100 +150 +200 +t − t0 [day] +107 +108 +109 +1010 +Lbol [L⊙] +9.5 M⊙ +14.9 M⊙ +19.9 M⊙ +M16 +S16 +Figure 5. Bolometric light curves of SNe IIP from +the M16 (solid lines) and S16 (dotted lines) pre-SN +models shown in Fig. 1. t0 denotes the time upon +which the explosion shock breaks out of the stellar +surface. +for all models are publicly available at Zenodo: +doi:10.5281/zenodo.7354733 in the same form +as listed in Table 1 . +As representative examples, we plot in Fig. 5 +the bolometric LCs of SNe IIP from the pre-SN +models shown in Fig. 1, with their respective +Lpl and tpl given in Table 1. +It is clear at a +first glance that the M16 models are brighter +than S16 models during the plateau phase for +the same MZAMS, despite the similar explosion +properties (also listed in Table 1). This feature +is further exemplified in Fig. 6, which compares +Lpl and tpl as a function of MZAMS between all +M16 and S16 models that successfully explode. +Whereas tpl is quite similar for the two sets of +models, Lpl of M16 models is in general larger +by a factor of ∼2 than that of S16 models. This +difference cannot be accounted for even by ap- +pealing to large uncertainties in the explosion +energy. Similar values of Lpl as in S16 can only +be realized for M16 models by artificially di- +viding Eexp by three, which is unrealistic and +would affect tpl considerably. Indeed, the differ- +ence in Lpl reflects the systematically different +envelope structure between M16 and S16 pro- +8.0 +8.5 +9.0 +log10(Lpl/L⊙) +M16 +S16 +10 +15 +20 +25 +30 +MZAMS [M⊙] +50 +100 +150 +200 +tpl [day] +Figure 6. Comparison of light curve parameters +as a function of ZAMS mass between M16 (blue +squares) and S16 (red dots) models. +The upper +and lower panels show the plateau luminosity and +length, respectively. +genitors (see the density profiles in Fig. 1 and +the pre-SN masses and radii in Fig. 2). As we +shall see in §5, the comparison with observations +suggests a preference for the S16 models as re- +alistic progenitors as they match the observed +plateau luminosities better. +Lastly, we compare our results to analytic +scaling relations often used by observers to infer +the properties of progenitor and explosion from +LC parameters, both to guide the interpretation +of our results and to check the validity of the +analytic relations. For Lpl, we use the relation +derived in Popov (1993) +Lpl = L0E5/6 +51 M −1/2 +10 +R2/3 +0,500, +(3) +where E51 is the explosion energy in units of +1051 erg, M10 is the mass of the hydrogen en- +velope (the progenitor mass minus the helium +core mass) in units of 10 M⊙, and R0,500 is the + +12 +Zha et al. +0.0 +0.5 +1.0 +1.5 +2.0 +Predicted Lpl [109L⊙] +0.0 +0.5 +1.0 +1.5 +2.0 +SNEC Lpl [109L⊙] +M16 +S16 +50 +100 +150 +200 +Predicted tpl [day] +50 +100 +150 +200 +SNEC tpl [day] +M16 +S16 +Figure 7. Comparison of light curve parameters, i.e., plateau luminosity (Lpl, left panel) and duration (tpl, +right panel), between SNEC simulations (ordinate) and the analytic scaling relations in Eqs. (3) and (5, +abscissa) for M16 and S16 progenitors. Open symbols indicate the models with tpl ≤ 80 days, which leads +to discrepancy of Lpl between SNEC results and the scaling relation for M16 models. The black lines in both +panels mark the diagonal. +pre-SN stellar radius Rprog in units of 500 R⊙. +Our preferred values of L0 are 1.69×1042 erg s−1 +and 1.51×1042 erg s−1 for M16 and S16 models, +respectively. The left panel of Fig. 7 shows that +Eq. (3) predicts Lpl well overall, with a relative +error ≲ 10% for most models. The discrepancy +for models with a large Lpl with a relative error +up to 40% is due to their short plateau for which +Lbol at 50 days may not well represent Lpl. +The scaling relation for the plateau duration +from Popov (1993) assumes no energy input +from radioactive decay of nickel and cobalt and +reads +tpl,0 = t0 E−1/6 +51 +M 1/2 +10 R1/6 +0,500. +(4) +Following Sukhbold et al. (2016), we use a mod- +ified relation for tpl that takes into account that +energy input from radioactive decay can prolong +the plateau, +tpl = tpl,0 × f 1/6 +rad , +frad = 1 + CfMNiE−1/2 +51 +M −1/2 +10 +R−1 +0,500, +(5) +where we set the constant Cf = 21 as suggested +in Sukhbold et al. (2016). Comparing the LCs +from SNEC to Eq. (5) is more appropriate, as +SNEC includes the energy release from radioac- +tive decay. The fitted t0 are 93.0 d and 89.7 d +for M16 and S16 models, respectively. The right +panel of Fig. 7 shows that Eq. (5) predicts tpl +well at tpl ≳ 100 days, with a relative error +≲ 15%. For tpl ≲ 100 days, the relative error +can be up to ∼ 25%. +5. COMPARISON TO OBSERVATIONS +5.1. Global statistics +Our large ensemble of stellar models allows for +a statistical comparison to observational data. +As a first step towards such a quantitative com- +parison, we choose the volume-limited set of +well-observed nearby SNe IIP from PP15, who +provide Lpl, tpl, and MNi, using their own LC fit- +ting method consistently across the photomet- +ric data of the entire sample instead of just col- +lecting LC parameters from the literature. Fol- +lowing Pejcha & Prieto (2015b), we use a sub- + +Type IIP SNe +13 +set from the PP15 sample including 17 SNe IIP +with well-determined photometry4. +Here, we compare global statistical param- +eters in theoretical models to observations. +For theoretical model sets, we calculate the +weighted means of the LC parameters, defined +as +⟨a⟩ = +� +i aiw(Mi)∆Mi +� +i w(Mi)∆Mi +. +(6) +Here, +a +stands +for +any +of +the +variables +log10(Lpl/L⊙), tpl, or log10(MNi/M⊙). +The +Salpeter initial mass function (IMF, Salpeter +1955) is used as the weighting function, i.e., +w(Mi) ∝ M −2.35 +i +, and ∆Mi is the resolution +of the ZAMS mass grid around Mi. +We set +the minimum and maximum Mi to 9 M⊙ and +30 M⊙, respectively. For the observational data, +we give each SN the same weight as appropri- +ate for a volume-limited sample. The standard +deviation σ of the LC parameters is evaluated +as +σ = +�� +i(ai − ⟨a⟩)2w(Mi)∆Mi +� +i w(Mi)∆Mi +. +(7) +The M16 set has a deficit of models with MZAMS +from 9 M⊙ to 12 M⊙ (see gaps in Fig. 2). The +pre-SN evolutionary simulation of stars near the +low-mass end is difficult and beset with uncer- +tainties due to the increasing influence of degen- +eracy in the core (Woosley & Heger 2015b), and +awaits for further improvement. To accommo- +date the deficit of low-mass models, we assign +the weight in a 1 M⊙ bin to the existing models +4 Pejcha & Prieto (2015b) include SN2013am in their +analysis, but no quantitative results were given for this +particular SN. Also, we exclude SN1980K, which is a +Type IIL. +w(M) = w0(M) +� M0+1M⊙ +M0 +w0(M ′)dM ′ +� +Mi∈[M0,M0+1M⊙] +w0(Mi)∆Mi +, +(8) +where w0(M) is the original weight from the +IMF and M0 = 9, 10 , 11 M⊙. +Table 2 summarizes the global statistical pa- +rameters for the LCs from the two theoretical +model sets and the PP15 sample. This is supple- +mented by the cumulative distribution functions +(CDF) of the LC parameters as shown in Fig. 8. +Due to generally smaller progenitor radii and +slightly higher envelope masses, the S16 mod- +els generally have a lower Lpl that better agrees +with the PP15 sample. However, the CDF of +theoretical Lpl shows a deficit of models with +low luminosity Lpl ≤ 108 L⊙. M16 models give +a longer mean plateau duration of ∼ 123 days +because low-mass models (MZAMS ≤ 12 M⊙) +have tpl ≥ 120 days (Fig. 6). The comparison +of the CDF of tpl shows both theoretical models +struggle to reproduce all the observational con- +straints. However, this discrepancy may partly +be due to the different definition of tpl between +this work and PP15. +For MNi, M16 and S16 +models give very similar mean values and CDFs. +This is expected as MNi mainly depends on the +core structure and the explosion model, which +are similar in both model sets (Fig. 3). Com- +paring to the PP15 data, our theoretical models +have a slightly larger mean MNi, and, based on +the CDF, this is likely due to a lack of models +with very small nickel masses MNi < 0.01 M⊙. +The scarcity of models with low nickel mass and +(to a lesser extent for the S16 models) low lumi- +nosity, might be due to the absence of electron- +capture supernovae in the model sets (Kozyreva +et al. 2021; Zha et al. 2022)5; the mass range for +5 Note, however, that adding electron-capture supernovae +may only help to add explosions with low nickel mass, +but not with low luminosity (Moriya et al. 2014). + +14 +Zha et al. +Table 2. +Global statistical parameters of light curves from observations and +theoretical models. +Data set +log10(Lpl/L⊙) +tpl (day) +log10(MNi/M⊙) +Mean +σ +p-value +Mean +σ +p-value +Mean +σ +p-value +PP15 +8.39 +0.39 +— +119 +13 +— +-1.52 +0.48 +— +M16 +8.76 +0.17 +4 × 10−8 +123 +16 +0.04 +-1.37 +0.19 +0.005 +S16 +8.49 +0.23 +0.21 +113 +13 +2 × 10−4 +-1.35 +0.22 +0.03 +7.5 +8.0 +8.5 +9.0 +9.5 +log10(Lpl/L⊙) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Distribution Function +PP15 +M16 +S16 +60 +80 +100 +120 +140 +160 +tpl [day] +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +log10(MNi/M⊙) +Figure 8. Comparison of the cumulative distribution function of the light curve parameters between the +theoretical model sets (M16 and S16) and the observational data set (PP15). +electron-capture supernovae remains quite un- +certain (Doherty et al. 2017; Poelarends et al. +2008). +Another cause could be uncertainties +for models with near-critical explodability (the +gray shaded regions in Fig. 4). It is possible that +some of these models might result in low-energy +explosions that produce little nickel and may ex- +perience fallback (which could remove nickel as +well). +To further assess discrepancies between the +observed and predicted distribution of explosion +properties, we perform individual Kolmogorov- +Smirnov (K-S) tests for each LC parameter to +estimate the goodness of fit of our theoretical +models to the PP15 sample. For each theoreti- +cal model set, we generate a large random sam- +ple of SN IIP models following the IMF. For the +M16 set, we assign the weight for MZAMS be- +low 12 M⊙ to the existing models according to +Eq. (8). We choose a sample size of 105 so that +the random sample well reproduces the theoret- +ical CDFs. The large sample size ensures that +the random generation process does not affect +the resultant p-values of K-S tests, which are +listed in Table 2. +The K-S test for Lpl sug- +gests an obvious preference of S16 models over +M16 models, agreeing with our assessment of +the mean Lpl. The K-S test for tpl favors the +M16 models, but the fit is far from perfect with +indications of possibly significant differences to +the observed distribution (p-value of 0.04). Note +that the test statistic is subject to uncertainties +in obtaining tpl for both models and observa- +tions. As expected from the lack of models with + +Type IIP SNe +15 +low 56Ni yields and smaller mean MNi in our +models, the K-S tests for MNi show both model +sets struggle to fit the PP15 sample. +5.2. Correlations between explosion properties +Correlations have been found between LC pa- +rameters in observations and inferred explosion +properties (e.g., Lpl and MNi, see Hamuy 2003; +Poznanski et al. 2012; Chugai & Utrobin 2014; +Pejcha & Prieto 2015b; M¨uller et al. 2017b). +Correlations can also allow to put constraints on +the theoretical progenitor and explosion mod- +els. Figure 9 shows three pairs of LC parame- +ters from the PP15 sample and the two model +sets in this work. +Visually, one can see that +both M16 and S16 model sets possess corre- +lations between all three pairs of parameters, +whereas PP15 only exhibits a clear correlation +between Lpl and MNi. Comparison of the theo- +retically predicted two-dimensional distribution +of Lpl and MNi to that in the PP15 sample also +suggests a preference for S16 models due to their +smaller Lpl, agreeing with the conclusion drawn +from the global statistical parameter. +To quantify the strength of the predicted and +observed correlations, we calculate the weighted +correlation matrix elements as +ρ(a, b) = +� +i(ai − ⟨a⟩)(bi − ⟨b⟩)w(Mi)∆Mi +σaσb +� w(Mi)∆Mi +, +for any pair of parameters a and b, and i runs +over all data/bins. Here we take log10(Lpl/L⊙), +tpl, and log10(MNi/M⊙) as the LC parameters. +Similar to our analysis in the previous section, +we use the Salpeter IMF as the weight w for the- +oretical models and assign the same weight for +each SN in the PP15 sample. The three non- +trivial correlation matrix elements for PP15, +M16, and S16 are given in Table 3. The corre- +lation between Lpl and MNi are similar between +either of our model sets and the PP15 sample, +while the pronounced correlation between Lpl +and tpl found in both sets is clearly absent in the +PP15 sample. +This discrepancy indicates the +need for further investigation with a larger SN +IIP sample. Although the presence or absence +of a correlation may be somewhat altered by a +more consistent determination of tpl in models +and observations, the discrepancy may indicate +missing physics in the explosion models or the +progenitor structure. Specifically, the effect of +adding Type IIP progenitors that have under- +gone binary interactions (Podsiadlowski et al. +1992; Zapartas et al. 2021) needs to be inves- +tigated. +Although it is plausible that binary +interactions could destroy the predicted corre- +lation between Lpl and tpl (which may be spu- +rious), it is not clear how binary effects could +reduce the overly large spread in tpl; in fact they +might even exacerbate this problem. +6. CONCLUSIONS +In this paper, we presented ∼ 1100 light +curves of SNe Type IIP generated by SNEC from +two sets of single-star solar-metallicity progeni- +tor models in M16 (M¨uller et al. 2016) and S16 +(Sukhbold et al. 2016), with very high resolu- +tion in ZAMS mass grid as fine as 0.01 M⊙ in +the former set. We assume that SNe IIP are +driven by neutrinos and calculate the key explo- +sion parameters Eexp, MNS,by and MNi using a +semi-analytical approach derived in M¨uller et al. +(2016). +The explosion parameters agree well globally +between the M16 and S16 model sets and be- +tween the semi-analytic model and alternative +phenomenological explosion models from previ- +ous studies of exploding S16 models (Sukhbold +et al. 2016; Barker et al. 2022a). +In particu- +lar, the agreement between the prediction of the +semi-analytic model and the 1D simulations of +Sukhbold et al. (2016) for the same progenitor +set is striking. The plateaus of SNe Type IIP are +systematically fainter by a factor of ∼2 in bolo- +metric luminosity for the S16 set due to denser +hydrogen envelopes of S16 progenitors. +The + +16 +Zha et al. +Table 3. Correlation matrix elements of the LC parameters in the observational sample and +our model sets. +Data set +ρ(log10(Lpl/L⊙), log10(MNi/M⊙)) +ρ(log10(Lpl/L⊙), tpl) +ρ(log10(MNi/M⊙), tpl) +PP15 +0.92 +-0.18 +-0.12 +M16 +0.85 +-0.91 +-0.71 +S16 +0.83 +-0.61 +-0.41 +more extended envelope structure of the M16 +models lead to brighter plateaus and is likely +artificial because of simplification of the surface +boundary condition in the stellar evolution cal- +culations. This reinforces previous findings on +the sensitivity of Type IIP explosions to the en- +velope structure (Dessart et al. 2013) and im- +plies that difference in theoretical light curves +may rather reflect assumptions about stellar +structure and evolution, in particular those that +affect the structure of the convective RSG en- +velope, than the modeling of the explosion en- +gine. +As already pointed out by Dessart & +Hillier (2019), this may cause problems in in- +ferring progenitor properties from observables, +e.g., inferring the ZAMS mass from the plateau +luminosity (Barker et al. 2022b). It is important +to highlight that even among available stellar +evolution models computed with the same code, +there may be subtle different in the treatment +of the convective envelope and outer boundary +due to code improvements and model parameter +choices that may have significant repercussions +for supernova light curve modeling. To fully ex- +ploit the diagnostic potential of SNe Type IIP +light curves, more theoretical and observational +work on RSG envelopes and environments is +critical. +We compare the parameters of the predicted +light curves to the volume-limited PP15 sam- +ple of well-observed SNe IIP (Pejcha & Prieto +2015a). We construct a mock supernova popu- +lation from the two progenitor sets by weight- +ing the models with the Salpeter IMF. Based +on the mean value of the plateau luminosity +Lpl and a K-S test, the S16 models fit the ob- +served brightness distribution in the PP15 SNe +IIP sample better. We find a similar correlation +between Lpl and MNi in both model sets and +in the PP15 sample. However, we find tensions +with the observational data for both model sets, +which may either indicate an incomplete under- +standing of the progenitor-explosion connection +or of the pre-supernova progenitor structure. +Both progenitor sets lack models with explo- +sions that produce the very small nickel masses +MNi < 0.01 M⊙ that are observed in some IIP +explosions. This discrepancy may be related to +the uncertainties in models at the low-mass end +(Woosley & Heger 2015b) or to models close +to black-hole formation. The comparison of the +plateau duration tpl remains beset with ambigu- +ities in the definition of the plateau length, but +we tentatively find a significantly larger spread +in plateau duration in the models compared to +the observed sample. +Furthermore, the mod- +els predict an anti-correlation between Lpl and +tpl, which is not found in the PP15 sample. In- +dications of an anti-correlation have, however, +been found in other samples (Faran et al. 2014), +and future studies need to assess whether big- +ger volume-limited samples confirm the tension +between theory and observations. +These results provide an interesting lead for +further comparisons of the theoretical models +and observational data for SNe IIP LCs. +In +particular, the predicted correlation between +plateau luminosity and plateau duration would + +Type IIP SNe +17 +present a challenge to current stellar evolu- +tion models of massive stars, if the tension to +observations can be corroborated using larger +volume-limited transient samples. Future stud- +ies should explore variations of single-star and +binary evolution models and pit them against +bigger volume-limited transient samples from +recent and upcoming surveys. By obviating the +need for time-critical 1D (let alone 3D) super- +nova simulations, our semi-analytic model may +be useful for conducting such large-scale com- +parisons more efficiently with little loss of accu- +racy, given the remarkable agreement with the +explosion properties obtained by Sukhbold et al. +(2016). +We thank Evan O’Connor for useful discus- +sion. SZ is supported by the China Postdoc- +toral Science Foundation (2022M712082). The +simulations were run on the Siyuan-1 clus- +ter supported by the Center for High Perfor- +mance Computing at Shanghai Jiao Tong Uni- +versity. +BM was supported by ARC Future +Fellowship FT160100035. +BM and AH are +supported by the Australian Research Council +(ARC) Centre of Excellence (CoE) for Grav- +itational Wave Discovery (OzGrave) project +number CE170100004. +AH is supported by +the ARC CoE for All Sky Astrophysics in +3 Dimensions (ASTRO 3D) project number +CE170100013. 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M., Leung, +S.-C., & Nomoto, K. 2022, MNRAS, 513, 1317, +doi: 10.1093/mnras/stac1035 + diff --git a/MdAyT4oBgHgl3EQfgfhQ/content/tmp_files/load_file.txt b/MdAyT4oBgHgl3EQfgfhQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..531d39890ee9bff6a32c7c002bfc6ec1a67130d6 --- /dev/null +++ b/MdAyT4oBgHgl3EQfgfhQ/content/tmp_files/load_file.txt @@ -0,0 +1,1412 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf,len=1411 +page_content='Draft version January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2023 Typeset using LATEX preprint2 style in AASTeX631 Light Curves of Type IIP Supernovae from Neutrino-driven Explosions of Red Supergiants Obtained by a Semi-analytic Approach Shuai Zha ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1 Bernhard M¨uller ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 3 Amy Weir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='2 and Alexander Heger 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 5 1Tsung-Dao Lee Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Shanghai Jiao Tong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Shanghai 200240,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' China 2School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Monash University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Clayton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Victoria 3800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Australia 3Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Clayton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' VIC 3800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Australia 4Center of Excellence for Astrophysics in Three Dimensions (ASTRO-3D),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Canberra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' ACT 2611,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Australia 5The Joint Institute for Nuclear Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Michigan State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' East Lansing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' MI 48824,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' USA ABSTRACT Type IIP supernovae (SNe IIP) mark the explosive death of red supergiants (RSGs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' evolved massive stars with an extended hydrogen envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' They are the most common supernova type and allow for benchmarking of supernova explosion models by statistical comparison to observed population properties rather than comparing individual models and events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We construct a large synthetic set of SNe IIP light curves (LCs) using the radiation hydrodynamics code SNEC and explosion energies and nickel masses obtained from an efficient semi-analytic model for two different sets of stellar progenitor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' By direct comparison we demonstrate that the semi-analytic model yields very similar predictions as alternative phenomenological explosion models based on one-dimensional simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We find systematic differences of a factor of ∼2 in plateau luminosities between the two progenitor sets due to different stellar radii, which highlights the importance of the RSG envelope structure as a major uncertainty in interpreting LCs of SNe IIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' A comparison to a volume-limited sample of observed SNe IIP shows decent agreement in plateau luminosity, plateau duration and nickel mass for at least one of the synthetic LC sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The models, however, do not produce sufficient events with very small nickel mass MNi < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='01 M⊙ and predict an anticorrelation between plateau luminosity and plateau duration that is not present in the observed sample, a result that warrants further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Our results suggest that a better understanding of RSG stellar structure is no less important for reliably explaining the light curves of SNe IIP than the explosion physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' INTRODUCTION Core-collapse supernovae (CCSNe) are the spectacular explosions that mark the death Corresponding author: Shuai Zha szha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='astrop@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='com of massive stars with zero-age main-sequence masses (MZAMS) greater than ∼8–10 M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Ibeling & Heger 2013) in the case of single- star progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Understanding the explosion mechanism of CCSNe has become the equiv- alent of a millennium problem of modern as- trophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' CCSNe have great importance as a arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='00359v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='HE] 1 Jan 2023 ID2 Zha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' source of multi-messenger events1 and of com- pact remnants that they leave behind, and are the origin of most heavy elements in the uni- verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Many of the latest three-dimensional (3D) simulations with sophisticated physics inputs, such as accurate modeling for the neutrino transport, have yielded successful CCSN explo- sions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Lentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Ott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Burrows et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Bollig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2021), which supports the view that most CCSNe are powered by the neutrino- driven mechanism aided by hydrodynamical in- stabilities (see the reviews of Bethe 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Janka 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Burrows & Vartanyan 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' M¨uller 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' There is, however, still an ongoing discussion on several key issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Using phenomenological models, considerable progress has been made in determining how the pre-collapse stellar struc- ture impacts which stars successfully explode and which ones fail and make black holes (O’Connor & Ott 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Ugliano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Ertl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Pejcha & Thompson 2015), but the best structural correlates for “explodabil- ity” and the parameter space for neutron star and black hole formation are still debated in supernova theory (Couch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Tsang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Beyond the question of explod- ability, both detailed multi-dimensional simula- tions and phenomenological models have shed some light on the relation between progenitors and their explosion properties, such as the rem- nant mass, explosion energy, and nucleosynthe- sis yields as critical input for chemogalactic evo- lution (Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2013), but the key chal- lenge is now to more rigorously validate the emerging theoretical picture using observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1 SN 1987A was the first-ever extragalactic astronomical multi-messenger event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Direct multi-messenger probes of the CCSN mechanism include gravitational waves (GWs) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Fryer & New 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Evans & Zanolin 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Kalogera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Abdikamalov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022) and neutrinos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Janka 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Horiuchi & Kneller 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' M¨uller 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Current neutrino and GW detectors, however, are only sensitive to events within ∼100 kpc (Scholberg 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Ab- bott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Electromagnetic signals are more readily available, especially in today’s era of large-scale surveys (Bellm 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Tonry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Masci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Ivezi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Type IIP supernovae (SNe IIP) are of par- ticular interest for comparing theoretical CCSN model predictions to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' They are the most common observed supernova type and originate from hydrogen-rich red super- giants (RSGs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Smartt 2015) that are predom- inantly, though not exclusively, unaffected by binary mass transfer (Podsiadlowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Zapartas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' They thus represent the supernova sub-population that most closely matches the progenitor models underlying pop- ulation studies based on phenomenological ex- plosion models2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' A Type IIP supernova exhibits a ∼100-day phase with nearly constant lumi- nosity (“plateau” – P) in its light curve (LC) during the inward propagation of a recombina- tion wave through the shock-heated hydrogen envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The plateau luminosity (Lpl) and du- ration (tpl) are related to the CCSN explosion energy, the progenitor radius, and the mass of the hydrogen envelope (Popov 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Kasen & Woosley 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The plateau phase is followed by an exponential luminosity tail that is at first powered by the radioactive decays of 56Ni and 56Co and by other radioactive species later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2 Note that phenomenological explosions models for stripped stars in binary systems have also been pre- sented recently by Ertl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2020) and Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Type IIP SNe 3 Historically, supernova explosion and progen- itor properties have most often been inferred by fitting individual SN LCs with semi-analytic so- lutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Arnett 1980, 1982) or radiation hy- drodynamic simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Blinnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Kasen & Woosley 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Bersten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Dessart & Hillier 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This approach, however, can suffer from a degeneracy of the explosion energy and progenitor mass as key parameters that determine the LC (Dessart & Hillier 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The problem of degeneracies can be reduced by considering larger samples of ob- served transients from surveys or compilations (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Faran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Pejcha & Pri- eto 2015a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Guti´errez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Most work on inferring progenitor and explosion parameters for larger supernova sam- ples to date has relied on LC fitting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', on re- verse modeling (Morozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The complementary approach is to use forward modeling of entire supernova pop- ulations for validating or constraining CCSN explosion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Several recent studies pro- duced a considerable number of LCs derived from phenomenological CCSN explosion models (Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Cur- tis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' What is still missing, however, is a global comparison between such a suite of the- oretical models and a representative, volume- limited supernova sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Such a comparison also needs to explore the sensitivity and robustness of explosion param- eter and LC predictions to variations in model assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This is particularly important to ascertain the potential for determining physical parameters of individual supernovae or entire populations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', the recent idea to exploit pro- posed correlations between iron core mass and plateau luminosity (Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022a,b) for use in parameter inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' In this work, we use the radiation hydrody- namics code SNEC (Morozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2015b) to calculate LCs of SNe IIP based on two sets of progenitor and explosion models from M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016, hereafter M16) and Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016, hereafter S16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We obtain explo- sion properties using the efficient semi-analytic model for neutrino-driven explosions from M16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Though both evolved with the stellar evolu- tion code KEPLER (Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Heger & Woosley 2010), M16 and S16 progenitors have been evolved with slightly different physics as- sumptions, and illustrate that LC predictions are especially sensitive to model variations that affect the hydrogen envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We then quantita- tively compare the models to the volume-limited SNe IIP sample of Pejcha & Prieto (2015a, here- after PP15) to highlight salient points of agree- ment and disagreement between the predictions and the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' In particular, we high- light that even though the models reproduce the well-known correlation between plateau lu- minosity Lpl and nickel mass MNi, there are still tensions between models and observations in distribution of plateau luminosity, plateau du- ration and nickel mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Similar to Dessart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2013), our results underscore the sensitivity of the LCs to the envelope structure of the progen- itor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' In §2 we review the semi-analytic approach for obtain- ing neutrino-driven CCSN explosions and the resulting explosion landscape for supernova pro- genitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We compare our approach to two other phenomenological explosion models for the S16 progenitor set in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' In §4 we present the theoretical SN IIP LCs from radiation hy- drodynamic simulations, and they are compared to the observational sample in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Our conclu- sions are given in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' CCSN EXPLOSION MODEL In this section, we first review the semi- analytic approach of M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016) for neutrino-driven CCSN explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Then we ap- ply this semi-analytic model to obtain the prop- 4 Zha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' erties of CCSN explosions for two sets of progen- itor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The semi-analytic approach We use the semi-analytic approach of M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016) to obtain the properties of suc- cessful neutrino-driven CCSN explosion such as the explosion energy Eexp, the baryonic mass of the remnant neutron star MNS,by, and the ejected 56Ni mass MNi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The semi-analytic ap- proach uses physically-motivated scaling laws and solves simple differential equations instead of performing detailed hydrodynamic simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' A current laptop computer can process nearly 2,000 models in just a few minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This allows us to explore a large parameter space such as detailed studies in stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Here, we provide an overview of the treatment of the CCSN dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The full description can be found in M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The iron core of a massive star starts to col- lapse when it reaches a critical mass that de- pends on its temperature and neutron excess (Clayton 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' As the central density reaches nuclear densities, the equation of state stiffens due to nuclear repulsive force, abruptly halt- ing the collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' An outgoing bounce shock is launched, but it quickly stalls because of en- ergy losses due to neutrinos and nuclear photo- disintegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The bounce shock turns into a quasi-stationary accretion shock within a few milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Material that passes through the shock gets accreted by the proto-neutron star (PNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Eventually, the shock may be revived to a runaway expansion – a successful explo- sion – or the PNS collapses to a black hole – a failed explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' During this accretion phase, copious amounts of neutrinos emanate from the PNS and heat up the matter inside the accre- tion shock (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', the reviews in Janka 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Burrows & Vartanyan 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The semi-analytic model of M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016) treats both the pre-explosion neutrino heating phase and the subsequent explosion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Pre-explosion phase The region roughly above the PNS and be- low the accretion shock, dubbed gain region, re- ceives net heating by neutrinos emanating from the PNS due to accretion and PNS cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The gain region is treated as an adiabatically stratified and radiation-dominated layer follow- ing Janka (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The mass accretion rate ˙M is computed following Woosley & Heger (2015a) assuming that the stellar interior nearly col- lapses in free fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The time evolution of the PNS radius and shock radius and thus the mass in the gain region can be determined from ˙M and the mass behind the shock, from which one can, in turn, compute the advection timescale τadv and the heating timescale τheat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The time of shock revival is determined from the assump- tion that material must have spent enough time in the gain region for neutrino heating to over- come the binding energy, leading to the criti- cal condition τadv/τheat > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' If this condition is never met, the model implies that the star forms a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Explosion phase During the first episode after shock revival (Phase I ), outflow and inflow of materials co- exist in the post-shock region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This phase is treated similarly as the pre-explosion phase ex- cept that the explosion energy Eexp is grad- ually increasing due to the recombination of ejected neutrino-heated material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The relevant mass outflow rate is computed from the neu- trino heating rate and the binding energy at the gain radius based on the heating model from the pre-explosion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' As the post-shock ve- locity (which is computed from the explosion energy, ejecta mass and pre-shock density) ex- ceeds the escape velocity, accretion is assumed to cease, and Eexp changes mainly due to explo- sive nuclear burning and the addition of binding energy of the outer shells (Phase II ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We deter- mine MNS,by at the end of Phase I, and compute Type IIP SNe 5 Eexp by integration throughout the envelope up to the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The explosive yields of iron-group (IG) el- ements are computed in a crude way by “flashing” shocked material into IG elements when the post-shock temperature exceeds 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 × 109 K3, but is less than the temperature for 50% dissociation into α-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The original model of M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016) did not account for the contribution of the neutrino-heated ejecta to the IG yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' To improve upon the origi- nal prescription, we take half of these IG ele- ments to be 56Ni and add another contribution from neutrino-driven outflows, which we assume to be proportional to Eexp (as Eexp is by con- struction determined by the amount of ejected neutrino-heated material Mν), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', MNi = 1 2MIG + 1 2αEexp ≈ 1 2MIG + 1 2Mν , (1) where the proportional constant α is set to mB/5 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The second term represents a rough upper limit for the production of nickel by neutrino-driven outflows, corresponding to the optimistic assumption that about half of the neutrino-heated ejecta recombine to 56Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We emphasize that an accurate MNi can only be obtained by multi-D neutrino-transport simula- tions and that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (1) only represents a rough estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Our semi-analytic model includes several pa- rameters that can be used for calibration against more sophisticated multi-D simulations or observational constraints (M¨uller 2015), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', the shock compression factor, the conversion ef- ficiency of accretion to neutrino luminosity, the PNS cooling timescale (Table 1 of M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' These parameters can be used to tune the CCSN explosion landscape, including the explodability and magnitude of Eexp consider- 3 We use 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 × 109 K instead of 5 × 109 K in the original prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' ably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' As a first step, we use the default param- eter set and keep the tunability in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Finally, we treat fallback as an all-or-nothing process as in the original prescription (M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We remark that fallback can signif- icantly influence the properties of explosions for near-critically exploding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Also, for some failed CCSNe, mass ejection is still possible due to the decrease of the PNS gravitational mass by neutrino emission (Piro 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Fern´andez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Schneider & O’Connor 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' However, whereas fallback is now recognized as important for understanding the black-hole mass distribu- tion (Mandel & M¨uller 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Mandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Antoniadis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022), these extreme events may not contribute to the SNe IIP population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' RSG models and the explosion landscape We apply the semi-analytic approach to two sets of single-star solar-metallicity RSG models as CCSN progenitors, which we refer to as M16 (M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016) and S16 (Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Both sets were evolved with the stel- lar evolution code KEPLER (Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Heger & Woosley 2010) but with two major known differences in the physical inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' One is that the erroneous pair-neutrino loss rate was updated to a corrected version in M16 but not in S16 (see §2 of Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This can affect the late burning stages after core he- lium depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The other difference is that a fixed, large boundary pressure was used at the stellar surface in M16 to keep the models stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This affected the RSG structure, making them more compact and affecting the mass loss dur- ing the reg giant phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Other differences may exist, such as the helium burning rates that im- pact the size of the carbon oxygen core after core helium depletion(Imbriani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Tur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The differences between the two sets at the onset of collapse are shown by the comparison 6 Zha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 0 4 8 12 16 Enclosed mass [M⊙] 10−11 10−7 10−3 101 105 109 Density [g cm−3] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 M⊙ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='9 M⊙ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='9 M⊙ M16 S16 108 1010 1012 1014 Radius [cm] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Pre-SN density profiles as a function of enclosed mass (left panel) and radius (right panel) for selected progenitor models with MZAMS = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='9 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='9 M⊙ from M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016, M16, solid lines) and Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016, S16, dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' All the models successfully explode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' In particular, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 M⊙ is the minimum mass common to both sets, and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='9M⊙ and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='9 M⊙ are the closest progenitor masses to 15 M⊙ and 20 M⊙ with explosions in both sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Presupernova, explosion and light-curve properties for the progenitor models shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' MZAMS Source Mprog Rprog MFe Menv ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 Eexp MNi MNS,by Lpl tpl (M⊙) (M⊙) (1013 cm) (M⊙) (M⊙) (1051 erg) (10−2 M⊙) (M⊙) (108 L⊙) (days) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 M16 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='11 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='6 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='37 144 S16 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='48 114 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='9 M16 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='56 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='99 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='9 96 S16 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='50 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='19 95 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='9 M16 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='56 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='20 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='74 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='3 95 S16 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='53 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='32 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='76 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='42 96 Note—Here, MZAMS is the ZAMS mass for the pre-SN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' M16 and S16 stand for progenitor from the sets of M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016) and Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Mprog and Rprog are the stellar mass and radius, MFe and Menv are the masses of the iron core and hydrogen envelope, and ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 is the compactness (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2), all defined at the onset of collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Eexp, MNi and MNS,by are the resulting explosion energy, nickel mass and remnant neutron-star mass obtained by the semi-analytic model of M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Lpl and tpl are the plateau luminosity and duration of the resultant SN IIP light curve obtained by SNEC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Type IIP SNe 7 of the pre-SN density profiles for three selected values of MZAMS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1), and the global param- eters for the pre-SN stellar structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Figure 1 clearly shows that the larger pressure cut results in a more dilute hydrogen envelope for M16 models whereas the core structures are nearly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This can also be inferred from the larger pre-SN stellar radiiRprog for M16 models with MZAMS ≲ 24 M⊙ (panel (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2), although the different pre-SN stellar masses (Mprog, panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2) also indicate subtle differences in the mass loss rates as a re- sult of feedback processes that requires further study but is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' A striking difference is the opposite trends of Rprog versus MZAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The progenitor radius, Rprog is positively correlated with progenitor mass in the S16 models, but decreases slightly with mass in the M16 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Figure 2 also illustrates differences in the core structure between the two sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The S16 models have a smaller mass of the carbon-oxygen core than M16 models for the same MZAMS, which carries through to later evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This is reflected by the final iron-core mass MFe (panel (c) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2), and can also be inferred from the progenitor compactness ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 (panel (d) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Here ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 is defined as (O’Connor & Ott 2011) ξM = M/M⊙ R(Mbaryon = M)/1,000 km ���� t=t0 , (2) where M is set to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 M⊙, and t0 is the time at the onset of collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Structures in the land- scape of ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 are systematically shifted to higher MZAMS in the S16 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Except for this shift, M16 and S16 models have quite similar core structures, with stochastic variations in ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 for MZAMS ≃ 15–20 M⊙ due to the chaotic merging of oxygen and carbon- and neon-burning shells (Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Yadav et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The impact of the erroneous neu- trino loss rate is most significant for stars with MZAMS ≳ 20 M⊙, which constitute only ∼ 18% 8 10 12 14 16 Mprog [M⊙] (a) M16 S16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 Rprog [1013 cm] (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='8 MFe [M⊙] (c) 10 15 20 25 30 MZAMS [M⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='4 ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 (d) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Comparison of progenitor properties as a function of ZAMS mass between M16 (blue squares) and S16 (red dots) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' From top to bottom, the panels show the pre-SN stellar mass (Mprog), pre-SN stellar radius (Rprog), iron-core mass (MFe) and compactness parameter (ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5) at the onset of collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' A choice of ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5,crit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='263 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='243) best discriminates the explodability for the M16 (S16) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 8 Zha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 Eexp [1051erg] M16 S16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='15 MNi [M⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='30 ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 MNS,by [M⊙] Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2, but for the comparison of explosion properties as a function of progenitor compactness ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 between the M16 (blue squares) and S16 (red dots) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' From top to bottom, the panels show the explosion energy Eexp, 56Ni mass MNi and baryonic neutron star mass MNS,by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' of all the progenitors and even less for explod- ing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Therefore, the overall impact on the ensemble of SNe IIP LCs is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We only consider pre-SN models with MZAMS ≤ 30 M⊙, because models with a larger MZAMS would exceed the Humphreys-Davidson limit and experience significant mass loss and result in SNe other than type IIP, aside from the fact that few explosions are predicted in this region in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For M16 we have 1891 models with a mass resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='01 M⊙, for which 991 successfully explode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For S16 we have 187 models with a mass resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='25) M⊙ at MZAMS above (below) 13 M⊙, for which 115 models successfully explode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 3 we show the explosion properties pre- dicted by the semi-analytic supernova model as a function of the ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We find good agreement between the two sets of progenitors and deter- mine a critical ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='263 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='243) that best discriminates the explodability for M16 (S16) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' COMPARISON OF ALTERNATIVE PHENOMENOLOGICAL EXPLOSION MODELS (S16 SET) It is currently not feasible to perform 3D sim- ulations with neutrino transport to determine the properties of CCSN explosions for a suffi- ciently large number of progenitors required for population studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Our semi-analytic model is among several efficient phenomenological ap- proaches to predict the outcome of collapse (ex- plosion or non-explosion) as well as explosion and remnant properties (O’Connor & Ott 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Ugliano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Pejcha & Thompson 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Perego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Couch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Ertl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Most other studies rely on 1D simulations that mimic the supportive role of multi-dimensional flow instabilities in enabling shock revival either by increasing the neutrino emission, the neutrino energy depo- sition, or by means of 1D turbulence models (but see M¨uller 2019b for a critical discussion of this approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Qualitative and quantitative differences and similarities between the various phenomenological models have been discussed in the literature, and Pejcha (2020) also pro- vides a side-by-side comparison of important outcomes such as the relation between explo- sion energy and nickel mass or the predicted neutron star mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Such compar- isons can be somewhat skewed by differences in Type IIP SNe 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 Eexp [1051erg] Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' this work 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='15 MNi [M⊙] 10 15 20 25 30 MZAMS [M⊙] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 MNS,by [M⊙] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Comparison of the explosion properties of S16 progenitors as obtained in Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (black crosses), Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (green open squares) and this work (red dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Note that Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' did not calculate MNi but used the nickel masses of S16 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' the size, mass range, and input physics of un- derlying stellar evolution model sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For this reason, it is useful to compare our re- sults to those obtained by different 1D simula- tion studies for the S16 progenitor set, namely from the study of Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016) and Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' used the P-HOTB code (Ugliano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Ertl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016) with a gray neutrino-transport scheme and a proto- neutron star core model, and is calibrated by two well-observed CCSNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Their models are calibrated to inferred explosion properties for SN 1054 and SN 1987A at the respective pro- genitor masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The SN 1987A calibration is used for all progenitors with MZAMS > 12 M⊙, and for MZAMS < 12 M⊙ interpolation between the relevant model parameters for the two cal- ibration cases is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' used the FLASH code with a multi-group two-moment neutrino-transport scheme (O’Connor & Couch 2018) plus the STIR method for simulating tur- bulence in 1D (Couch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Their STIR method is calibrated to fit full 3D simulations run in the same code (O’Connor & Couch 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The comparison is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 4 for Eexp, MNi and MNS,by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Although with quite differ- ent implementations and degrees of approxima- tions, we find considerable agreements among the results from Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' and this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The agreement is especially re- markable for the baryonic neutron star mass MNS,by, which once again confirms the impor- tant role of the Si-O shell interface as a natural point for the onset of the explosion and a strong predictor for the final neutron star mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Discrepancies are noteworthy mainly in the mass ranges with near-critical explodability (gray shaded bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 4), with MZAMS ≃ 12–15 M⊙ and 22–25 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For MZAMS ≃ 12– 15 M⊙, Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' predicts no explosion while both Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' and the semi-analytic model obtain explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Eexp and MNi in Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' are, however, larger by about a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 than those in this work, which may be related to the change in calibration case of P-HOTB from SN 1054 to SN 1987A at 12 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' On the other hand, for MZAMS ≃ 22–25 M⊙, Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' and our semi-analytic model predict no explosion, whereas Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' yields relatively large explosion energies Eexp (≥ 2 × 1051 erg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The explodability of these critical models is still under debate with state- of-the-art 3D simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Ott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 10 Zha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Melson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Burrows et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The mass distribution of observed SN IIP progeni- tors (Smartt 2015) and first observational ev- idence for the quiet disappearance of a RSG (Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2017), presumably by stellar col- lapse favor a lower probability of explosion in this mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The overall trends and patterns in explosion energy are qualitatively compatible between the three phenomenological models outside the gray-shaded areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' They all predict low explo- sion energies at the low-mass end, a general trend towards higher explosion energies in the range of 15–22 M⊙ with considerable scatter at higher masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Above 25 M⊙, the agreement is less convincing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' It is noteworthy, however, that even in the region of 22–25 M⊙, where Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' disagrees qualitatively with the other two models for, the high explosion energies reflect a similar pattern in M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016) with pa- rameter choices that increase explodability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', higher turbulent pressure in the gain region or a higher accretion efficiency for neutrino emis- sion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The situation for the nickel masses, MNi, which are only available for Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' and our semi-analytic model, is similar to the ex- plosion energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' There is rather good agree- ment between Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' and our work be- low 22 M⊙, which is rather striking considering the relatively simple model for nickel production used in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' These results demonstrate that predictions of explosion and remnant properties from the three phenomenological models are quite robust to differences in the methodology, once some form of calibration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', for one or two specific supernovae or for the typical energy range of observed explosions) is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' THEORETICAL LIGHT CURVES OF TYPE IIP SNE With the explosion properties (Eexp, MNi and MNS,by) obtained in §2, we utilize SNEC (Mo- rozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2015b) to generate LCs of SNe IIP from M16 and S16 progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' SNEC is an open-source spherically-symmetrical radia- tion hydrodynamics code with the capability to follow the shock propagation through the stel- lar envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' It solves the Lagrangian hydro- dynamics equations supplemented with a radi- ation diffusion term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Note that SNEC assumes local thermal equilibrium between matter and radiation, which fails during the shock breakout and nebular phase, but is reasonably reliable for LCs during the plateau phase (Blinnikov & Bar- tunov 1993) that is of interest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We refer to the code paper (Morozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2015b) and doc- umentation (Morozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2015a) for details on the numerical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We employ the default settings of SNEC, such as the equation of state, ionization treatment and opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The newborn NS with MNS,by is excised from the numerical grid and a ther- mal bomb is used to initialize the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The sum of Eexp and binding energy of the mass con- tent above the excised NS is spread into the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1 M⊙ above the excised boundary so that the final explosion energy equals the desired value Eexp (Morozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For the mixing of nickel, we simply spread MNi homogeneously up to 3 M⊙ as our semi-analytic approach can- not treat the mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The mixing of nickel is beyond the scope of this paper but its impact on SNe IIP LCs may be worth further investiga- tion (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Utrobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We evolve all the models to ∼ 200 days, by which time all models have reached the radioactively-powered tail phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For comparison to observations, we are particularly interested in two LC parame- ters: the plateau luminosity Lpl and the plateau duration tpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We take the bolometric luminos- ity at 50 days after the shock break out as Lpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The determination of tpl is more tricky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' we ten- tatively pick the time of the steepest gradient of the B-band magnitude as the end of plateau phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The key LC and explosion parameters Type IIP SNe 11 0 50 100 150 200 t − t0 [day] 107 108 109 1010 Lbol [L⊙] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 M⊙ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='9 M⊙ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='9 M⊙ M16 S16 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Bolometric light curves of SNe IIP from the M16 (solid lines) and S16 (dotted lines) pre-SN models shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' t0 denotes the time upon which the explosion shock breaks out of the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' for all models are publicly available at Zenodo: doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='7354733 in the same form as listed in Table 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' As representative examples, we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 5 the bolometric LCs of SNe IIP from the pre-SN models shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1, with their respective Lpl and tpl given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' It is clear at a first glance that the M16 models are brighter than S16 models during the plateau phase for the same MZAMS, despite the similar explosion properties (also listed in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This feature is further exemplified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 6, which compares Lpl and tpl as a function of MZAMS between all M16 and S16 models that successfully explode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Whereas tpl is quite similar for the two sets of models, Lpl of M16 models is in general larger by a factor of ∼2 than that of S16 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This difference cannot be accounted for even by ap- pealing to large uncertainties in the explosion energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Similar values of Lpl as in S16 can only be realized for M16 models by artificially di- viding Eexp by three, which is unrealistic and would affect tpl considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Indeed, the differ- ence in Lpl reflects the systematically different envelope structure between M16 and S16 pro- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 log10(Lpl/L⊙) M16 S16 10 15 20 25 30 MZAMS [M⊙] 50 100 150 200 tpl [day] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Comparison of light curve parameters as a function of ZAMS mass between M16 (blue squares) and S16 (red dots) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The upper and lower panels show the plateau luminosity and length, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' genitors (see the density profiles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1 and the pre-SN masses and radii in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' As we shall see in §5, the comparison with observations suggests a preference for the S16 models as re- alistic progenitors as they match the observed plateau luminosities better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Lastly, we compare our results to analytic scaling relations often used by observers to infer the properties of progenitor and explosion from LC parameters, both to guide the interpretation of our results and to check the validity of the analytic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For Lpl, we use the relation derived in Popov (1993) Lpl = L0E5/6 51 M −1/2 10 R2/3 0,500, (3) where E51 is the explosion energy in units of 1051 erg, M10 is the mass of the hydrogen en- velope (the progenitor mass minus the helium core mass) in units of 10 M⊙, and R0,500 is the 12 Zha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 Predicted Lpl [109L⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 SNEC Lpl [109L⊙] M16 S16 50 100 150 200 Predicted tpl [day] 50 100 150 200 SNEC tpl [day] M16 S16 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Comparison of light curve parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', plateau luminosity (Lpl, left panel) and duration (tpl, right panel), between SNEC simulations (ordinate) and the analytic scaling relations in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (3) and (5, abscissa) for M16 and S16 progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Open symbols indicate the models with tpl ≤ 80 days, which leads to discrepancy of Lpl between SNEC results and the scaling relation for M16 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The black lines in both panels mark the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' pre-SN stellar radius Rprog in units of 500 R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Our preferred values of L0 are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='69×1042 erg s−1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='51×1042 erg s−1 for M16 and S16 models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 7 shows that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (3) predicts Lpl well overall, with a relative error ≲ 10% for most models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The discrepancy for models with a large Lpl with a relative error up to 40% is due to their short plateau for which Lbol at 50 days may not well represent Lpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The scaling relation for the plateau duration from Popov (1993) assumes no energy input from radioactive decay of nickel and cobalt and reads tpl,0 = t0 E−1/6 51 M 1/2 10 R1/6 0,500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (4) Following Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016), we use a mod- ified relation for tpl that takes into account that energy input from radioactive decay can prolong the plateau, tpl = tpl,0 × f 1/6 rad , frad = 1 + CfMNiE−1/2 51 M −1/2 10 R−1 0,500, (5) where we set the constant Cf = 21 as suggested in Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Comparing the LCs from SNEC to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (5) is more appropriate, as SNEC includes the energy release from radioac- tive decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The fitted t0 are 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 d and 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='7 d for M16 and S16 models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 7 shows that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (5) predicts tpl well at tpl ≳ 100 days, with a relative error ≲ 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For tpl ≲ 100 days, the relative error can be up to ∼ 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' COMPARISON TO OBSERVATIONS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Global statistics Our large ensemble of stellar models allows for a statistical comparison to observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' As a first step towards such a quantitative com- parison, we choose the volume-limited set of well-observed nearby SNe IIP from PP15, who provide Lpl, tpl, and MNi, using their own LC fit- ting method consistently across the photomet- ric data of the entire sample instead of just col- lecting LC parameters from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Fol- lowing Pejcha & Prieto (2015b), we use a sub- Type IIP SNe 13 set from the PP15 sample including 17 SNe IIP with well-determined photometry4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Here, we compare global statistical param- eters in theoretical models to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For theoretical model sets, we calculate the weighted means of the LC parameters, defined as ⟨a⟩ = � i aiw(Mi)∆Mi � i w(Mi)∆Mi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (6) Here, a stands for any of the variables log10(Lpl/L⊙), tpl, or log10(MNi/M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The Salpeter initial mass function (IMF, Salpeter 1955) is used as the weighting function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', w(Mi) ∝ M −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='35 i , and ∆Mi is the resolution of the ZAMS mass grid around Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We set the minimum and maximum Mi to 9 M⊙ and 30 M⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For the observational data, we give each SN the same weight as appropri- ate for a volume-limited sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The standard deviation σ of the LC parameters is evaluated as σ = �� i(ai − ⟨a⟩)2w(Mi)∆Mi � i w(Mi)∆Mi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (7) The M16 set has a deficit of models with MZAMS from 9 M⊙ to 12 M⊙ (see gaps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The pre-SN evolutionary simulation of stars near the low-mass end is difficult and beset with uncer- tainties due to the increasing influence of degen- eracy in the core (Woosley & Heger 2015b), and awaits for further improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' To accommo- date the deficit of low-mass models, we assign the weight in a 1 M⊙ bin to the existing models 4 Pejcha & Prieto (2015b) include SN2013am in their analysis, but no quantitative results were given for this particular SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Also, we exclude SN1980K, which is a Type IIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' w(M) = w0(M) � M0+1M⊙ M0 w0(M ′)dM ′ � Mi∈[M0,M0+1M⊙] w0(Mi)∆Mi , (8) where w0(M) is the original weight from the IMF and M0 = 9, 10 , 11 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Table 2 summarizes the global statistical pa- rameters for the LCs from the two theoretical model sets and the PP15 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This is supple- mented by the cumulative distribution functions (CDF) of the LC parameters as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Due to generally smaller progenitor radii and slightly higher envelope masses, the S16 mod- els generally have a lower Lpl that better agrees with the PP15 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' However, the CDF of theoretical Lpl shows a deficit of models with low luminosity Lpl ≤ 108 L⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' M16 models give a longer mean plateau duration of ∼ 123 days because low-mass models (MZAMS ≤ 12 M⊙) have tpl ≥ 120 days (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The comparison of the CDF of tpl shows both theoretical models struggle to reproduce all the observational con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' However, this discrepancy may partly be due to the different definition of tpl between this work and PP15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For MNi, M16 and S16 models give very similar mean values and CDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This is expected as MNi mainly depends on the core structure and the explosion model, which are similar in both model sets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Com- paring to the PP15 data, our theoretical models have a slightly larger mean MNi, and, based on the CDF, this is likely due to a lack of models with very small nickel masses MNi < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='01 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The scarcity of models with low nickel mass and (to a lesser extent for the S16 models) low lumi- nosity, might be due to the absence of electron- capture supernovae in the model sets (Kozyreva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Zha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022)5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' the mass range for 5 Note, however, that adding electron-capture supernovae may only help to add explosions with low nickel mass, but not with low luminosity (Moriya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 14 Zha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Global statistical parameters of light curves from observations and theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Data set log10(Lpl/L⊙) tpl (day) log10(MNi/M⊙) Mean σ p-value Mean σ p-value Mean σ p-value PP15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='39 — 119 13 — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='48 — M16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='17 4 × 10−8 123 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='005 S16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='21 113 13 2 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 log10(Lpl/L⊙) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 Cumulative Distribution Function PP15 M16 S16 60 80 100 120 140 160 tpl [day] −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 log10(MNi/M⊙) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Comparison of the cumulative distribution function of the light curve parameters between the theoretical model sets (M16 and S16) and the observational data set (PP15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' electron-capture supernovae remains quite un- certain (Doherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Poelarends et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Another cause could be uncertainties for models with near-critical explodability (the gray shaded regions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' It is possible that some of these models might result in low-energy explosions that produce little nickel and may ex- perience fallback (which could remove nickel as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' To further assess discrepancies between the observed and predicted distribution of explosion properties, we perform individual Kolmogorov- Smirnov (K-S) tests for each LC parameter to estimate the goodness of fit of our theoretical models to the PP15 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For each theoreti- cal model set, we generate a large random sam- ple of SN IIP models following the IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' For the M16 set, we assign the weight for MZAMS be- low 12 M⊙ to the existing models according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We choose a sample size of 105 so that the random sample well reproduces the theoret- ical CDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The large sample size ensures that the random generation process does not affect the resultant p-values of K-S tests, which are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The K-S test for Lpl sug- gests an obvious preference of S16 models over M16 models, agreeing with our assessment of the mean Lpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The K-S test for tpl favors the M16 models, but the fit is far from perfect with indications of possibly significant differences to the observed distribution (p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Note that the test statistic is subject to uncertainties in obtaining tpl for both models and observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' As expected from the lack of models with Type IIP SNe 15 low 56Ni yields and smaller mean MNi in our models, the K-S tests for MNi show both model sets struggle to fit the PP15 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Correlations between explosion properties Correlations have been found between LC pa- rameters in observations and inferred explosion properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Lpl and MNi, see Hamuy 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Poznanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Chugai & Utrobin 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Pejcha & Prieto 2015b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Correlations can also allow to put constraints on the theoretical progenitor and explosion mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Figure 9 shows three pairs of LC parame- ters from the PP15 sample and the two model sets in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Visually, one can see that both M16 and S16 model sets possess corre- lations between all three pairs of parameters, whereas PP15 only exhibits a clear correlation between Lpl and MNi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Comparison of the theo- retically predicted two-dimensional distribution of Lpl and MNi to that in the PP15 sample also suggests a preference for S16 models due to their smaller Lpl, agreeing with the conclusion drawn from the global statistical parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' To quantify the strength of the predicted and observed correlations, we calculate the weighted correlation matrix elements as ρ(a, b) = � i(ai − ⟨a⟩)(bi − ⟨b⟩)w(Mi)∆Mi σaσb � w(Mi)∆Mi , for any pair of parameters a and b, and i runs over all data/bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Here we take log10(Lpl/L⊙), tpl, and log10(MNi/M⊙) as the LC parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Similar to our analysis in the previous section, we use the Salpeter IMF as the weight w for the- oretical models and assign the same weight for each SN in the PP15 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The three non- trivial correlation matrix elements for PP15, M16, and S16 are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The corre- lation between Lpl and MNi are similar between either of our model sets and the PP15 sample, while the pronounced correlation between Lpl and tpl found in both sets is clearly absent in the PP15 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This discrepancy indicates the need for further investigation with a larger SN IIP sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Although the presence or absence of a correlation may be somewhat altered by a more consistent determination of tpl in models and observations, the discrepancy may indicate missing physics in the explosion models or the progenitor structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Specifically, the effect of adding Type IIP progenitors that have under- gone binary interactions (Podsiadlowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Zapartas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2021) needs to be inves- tigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Although it is plausible that binary interactions could destroy the predicted corre- lation between Lpl and tpl (which may be spu- rious), it is not clear how binary effects could reduce the overly large spread in tpl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' in fact they might even exacerbate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' CONCLUSIONS In this paper, we presented ∼ 1100 light curves of SNe Type IIP generated by SNEC from two sets of single-star solar-metallicity progeni- tor models in M16 (M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016) and S16 (Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016), with very high resolu- tion in ZAMS mass grid as fine as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='01 M⊙ in the former set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We assume that SNe IIP are driven by neutrinos and calculate the key explo- sion parameters Eexp, MNS,by and MNi using a semi-analytical approach derived in M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The explosion parameters agree well globally between the M16 and S16 model sets and be- tween the semi-analytic model and alternative phenomenological explosion models from previ- ous studies of exploding S16 models (Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' In particu- lar, the agreement between the prediction of the semi-analytic model and the 1D simulations of Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016) for the same progenitor set is striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The plateaus of SNe Type IIP are systematically fainter by a factor of ∼2 in bolo- metric luminosity for the S16 set due to denser hydrogen envelopes of S16 progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The 16 Zha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Correlation matrix elements of the LC parameters in the observational sample and our model sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Data set ρ(log10(Lpl/L⊙), log10(MNi/M⊙)) ρ(log10(Lpl/L⊙), tpl) ρ(log10(MNi/M⊙), tpl) PP15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='12 M16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='71 S16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='41 more extended envelope structure of the M16 models lead to brighter plateaus and is likely artificial because of simplification of the surface boundary condition in the stellar evolution cal- culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This reinforces previous findings on the sensitivity of Type IIP explosions to the en- velope structure (Dessart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2013) and im- plies that difference in theoretical light curves may rather reflect assumptions about stellar structure and evolution, in particular those that affect the structure of the convective RSG en- velope, than the modeling of the explosion en- gine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' As already pointed out by Dessart & Hillier (2019), this may cause problems in in- ferring progenitor properties from observables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', inferring the ZAMS mass from the plateau luminosity (Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' It is important to highlight that even among available stellar evolution models computed with the same code, there may be subtle different in the treatment of the convective envelope and outer boundary due to code improvements and model parameter choices that may have significant repercussions for supernova light curve modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' To fully ex- ploit the diagnostic potential of SNe Type IIP light curves, more theoretical and observational work on RSG envelopes and environments is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We compare the parameters of the predicted light curves to the volume-limited PP15 sam- ple of well-observed SNe IIP (Pejcha & Prieto 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We construct a mock supernova popu- lation from the two progenitor sets by weight- ing the models with the Salpeter IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Based on the mean value of the plateau luminosity Lpl and a K-S test, the S16 models fit the ob- served brightness distribution in the PP15 SNe IIP sample better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We find a similar correlation between Lpl and MNi in both model sets and in the PP15 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' However, we find tensions with the observational data for both model sets, which may either indicate an incomplete under- standing of the progenitor-explosion connection or of the pre-supernova progenitor structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Both progenitor sets lack models with explo- sions that produce the very small nickel masses MNi < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='01 M⊙ that are observed in some IIP explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' This discrepancy may be related to the uncertainties in models at the low-mass end (Woosley & Heger 2015b) or to models close to black-hole formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The comparison of the plateau duration tpl remains beset with ambigu- ities in the definition of the plateau length, but we tentatively find a significantly larger spread in plateau duration in the models compared to the observed sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Furthermore, the mod- els predict an anti-correlation between Lpl and tpl, which is not found in the PP15 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' In- dications of an anti-correlation have, however, been found in other samples (Faran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2014), and future studies need to assess whether big- ger volume-limited samples confirm the tension between theory and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' These results provide an interesting lead for further comparisons of the theoretical models and observational data for SNe IIP LCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' In particular, the predicted correlation between plateau luminosity and plateau duration would Type IIP SNe 17 present a challenge to current stellar evolu- tion models of massive stars, if the tension to observations can be corroborated using larger volume-limited transient samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Future stud- ies should explore variations of single-star and binary evolution models and pit them against bigger volume-limited transient samples from recent and upcoming surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' By obviating the need for time-critical 1D (let alone 3D) super- nova simulations, our semi-analytic model may be useful for conducting such large-scale com- parisons more efficiently with little loss of accu- racy, given the remarkable agreement with the explosion properties obtained by Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We thank Evan O’Connor for useful discus- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' SZ is supported by the China Postdoc- toral Science Foundation (2022M712082).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' The simulations were run on the Siyuan-1 clus- ter supported by the Center for High Perfor- mance Computing at Shanghai Jiao Tong Uni- versity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' BM was supported by ARC Future Fellowship FT160100035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' BM and AH are supported by the Australian Research Council (ARC) Centre of Excellence (CoE) for Grav- itational Wave Discovery (OzGrave) project number CE170100004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' AH is supported by the ARC CoE for All Sky Astrophysics in 3 Dimensions (ASTRO 3D) project number CE170100013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' We acknowledge computer time allocations from Astronomy Australia Limited’s ASTAC scheme and the National Computa- tional Merit Allocation Scheme (NCMAS), and from and Australasian Leadership Computing Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Software: SNEC (Morozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2015b), NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2020), Matplotlib (Hunter 2007) , SciPy (Virtanen et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 log10(MNi/M⊙) PP15 M16 S16 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 log10(Lpl/L⊙) 60 80 100 120 140 160 tpl [day] −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='5 log10(MNi/M⊙) 60 80 100 120 140 160 tpl [day] Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' Correlations between light curve param- eters for theoretical predictions of the M16 (blue 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2020, Living Reviews in Computational Astrophysics, 6, 3, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1007/s41115-020-0008-5 M¨uller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Heger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', Liptai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=', & Cameron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' B.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content=' 2022, MNRAS, 513, 1317, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} +page_content='1093/mnras/stac1035' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfgfhQ/content/2301.00359v1.pdf'} diff --git a/O9FJT4oBgHgl3EQfISxO/content/tmp_files/2301.11455v1.pdf.txt b/O9FJT4oBgHgl3EQfISxO/content/tmp_files/2301.11455v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e448b4b86f56bd528f604dfde9a93a8d69470527 --- /dev/null +++ b/O9FJT4oBgHgl3EQfISxO/content/tmp_files/2301.11455v1.pdf.txt @@ -0,0 +1,2771 @@ +DRAFT VERSION 30TH JANUARY, 2023 +Typeset using LATEX twocolumn style in AASTeX63 +First Observations of the Brown Dwarf HD 19467 B with JWST +ALEXANDRA Z. GREENBAUM +,1 JORGE LLOP-SAYSON +,2 BEN LEW +,3 GEOFFREY BRYDEN +,4 THOMAS ROELLIG,3 +MARIE YGOUF +,4 B.J. FULTON +,5 DANIEL R. HEY +,6 DANIEL HUBER +,6 SAGNICK MUKHERJEE +,7 MICHAEL MEYER +,8 +JARRON LEISENRING +,9 MARCIA RIEKE +,9 MARTHA BOYER,10 JOSEPH J. GREEN,4 DOUG KELLY,9 KARL MISSELT,9 +EUGENE SERABYN,4 JOHN STANSBERRY,10 LAURIE E. U. CHU +,11 MATTHEW DE FURIO +,8 DOUG JOHNSTONE +,12, 13 +JOSHUA E. SCHLIEDER,14 AND CHARLES BEICHMAN +4, 5 +1IPAC, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA +2California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA +3NASA Ames Research Center, Mountain View, CA, 94035, USA +4Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA +5NASA Exoplanet Science Institute, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA +6Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA +7Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA +8Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA +9Steward Observatory, University of Arizona, Tucson, AZ 85721, USA +10Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +11NASA Postdoctoral Program Fellow, NASA Ames Research Center, M/S 245-1, Moffett Field, CA 94035, USA +12NRC Herzberg Astronomy and Astrophysics, 5071 West Saanich Rd, Victoria, BC, V9E 2E7, Canada +13Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada +14Exoplanets and Stellar Astrophysics Laboratory, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD, USA +(Accepted January 25, 2023) +ABSTRACT +We observed HD 19467 B with JWST’s NIRCam in six filters spanning 2.5-4.6 µm with the Long Wavelength +Bar coronagraph. The brown dwarf HD 19467 B was initially identified through a long-period trend in the +radial velocity of G3V star HD 19467. HD 19467 B was subsequently detected via coronagraphic imaging +and spectroscopy, and characterized as a late-T type brown dwarf with approximate temperature ∼ 1000 K. We +observed HD 19467 B as a part of the NIRCam GTO science program, demonstrating the first use of the NIRCam +Long Wavelength Bar coronagraphic mask. The object was detected in all 6 filters (contrast levels of 2×10−4 to +2×10−5) at a separation of 1.6′′ using Angular Differential Imaging (ADI) and Synthetic Reference Differential +Imaging (SynRDI). Due to a guidestar failure during acquisition of a pre-selected reference star, no reference +star data was available for post-processing. However, RDI was successfully applied using synthetic Point Spread +Functions (PSFs) developed from contemporaneous maps of the telescope’s optical configuration. Additional +radial velocity data (from Keck/HIRES) are used to constrain the orbit of the HD 19467 B. Photometric data +from TESS are used to constrain the properties of the host star, particularly its age. NIRCam photometry, spectra +and photometry from literature, and improved stellar parameters are used in conjunction with recent spectral +and evolutionary substellar models to derive physical properties for HD 19467 B. Using an age of 9.4±0.9 Gyr +inferred from spectroscopy, Gaia astrometry, and TESS asteroseismology, we obtain a model-derived mass of +62±1 MJ, which is consistent within 2-σ with the dynamically derived mass of 81+14 +−12 MJ. +1. INTRODUCTION +Brown dwarfs provide a unique testbed for confronting +evolutionary and atmospheric models of sub-stellar objects +Corresponding author: Alexandra Z. Greenbaum +azg@ipac.caltech.edu +with well-defined observations. Those brown dwarfs which +are companions to main sequence stars, as opposed to free- +floating, are particularly valuable since they are presumed to +inherit observable stellar properties such as metallicity and +share similar ages. This knowledge constrains many of the +free parameters in the comparison of models with observa- +tion. +arXiv:2301.11455v1 [astro-ph.SR] 26 Jan 2023 + +ID2 +GREENBAUM ET AL. +Low-mass brown dwarf companions to main-sequence +stars were initially found through blind imaging searches, +e.g. GL229 B (Nakajima et al. 1995), and subsequently as a +by-product of planet searches using the radial velocity (RV) +technique. In the case of HD 19467, Crepp et al. (2014) iden- +tified it as a star with a significant RV trend suggestive of +a massive brown dwarf companion. Coronagraphic imag- +ing with Keck NIRC2 first confirmed the presence of the +companion (Crepp et al. 2014). This was followed by spec- +troscopy with Palomar’s P1640 instrument that characterized +HD 19467 B as a brown dwarf with effective temperature of +∼1000 K corresponding to a T5.5 spectral type (Crepp et al. +2015). More recently, proper motion measurements from the +Hipparcos and Gaia catalogs have been used to identify sys- +tems with companions or help characterize them, including +HD 19467 (Brandt et al. 2021a). +Multiple JWST programs will provide imaging and spec- +troscopy of HD 19467 B across the near- and mid-IR where +brown dwarfs emit most of their energy. The program pre- +sented here (PID #1189) uses NIRCam (Rieke et al. in press) +to provide medium and narrow band imaging and photometry +of HD 19467 B in 6 bands, spanning 2.5 to 4.5 µm. At a later +date, another JWST program (PID #1414) will use NIRSpec +(Jakobsen et al. 2022) to obtain high-resolution (R ≃ 2700) +3–5 µm spectra of HD 19467 B. +JWST observations of the G3V star HD 19467 with its T5+ +brown dwarf companion, HD 19467 B (Crepp et al. 2014), +represent one of the earliest exercises of the NIRCam Coro- +nagraphic LW Bar (Krist et al. 2007; Beichman et al. 2010; +Girard et al. 2022), providing an opportunity for an early sci- +entific result and a demonstration of the capabilities of the +instrument. +The NIRCam observations presented in this study are de- +signed to accomplish three main goals: +1. Provide an early dataset that exercises the bar mask on +NIRCam, especially without a reference star (§2 & 4); +2. Refine the orbital parameters of HD 19467 B with a +new imaging data point along with new RV data from +Keck/HIRES (§5); and +3. Add additional photometric measurements to better +constrain the physical properties of the brown dwarf +(§6). +We also include new analysis of TESS observations to con- +strain properties of the host star (§3). +2. OBSERVATIONS +2.1. NIRCam Observations +NIRCam observed HD 19467 on 2022-Aug-12 with the +long-wavelength bar (LWB) coronagraphic mask in subar- +ray mode with six filters: F250M, F300M, F360M, F410M, +F430M, and F460M. The target star was observed at two tele- +scope roll angles separated by 7.72 degrees. Table 1 shows a +summary of the observations and settings per filter. Observa- +tions of HD 19467 were taken with the long-wavelength bar +(MASKLWB) coronagraph, providing a test of NIRCam’s +capabilities at smaller inner working angles than are possi- +ble with the round masks (4λ/D for MASKLWB vs 6λ/D +for MASK210R, MASK335R, and MASK430R; Krist et al. +(2007)). At the time of these observations, the MASKLWB +positions were not well-defined, with a y-offset ∼70 mas. +Future use of this mode with updated position definition +will improve the ability to center the star on the mask and +therefore contrast performance, especially close in to the +mask. +These observations represent one of the first post- +commissioning uses of the bar coronagraph. +The observation plan was initially scheduled to include se- +quential observations of the reference star HD 19096 in or- +der to perform PSF subtraction using Reference Differential +Imaging (RDI). However, the reference observations were +unsuccessful because the telescope failed to acquire a guide +star. Instead, we performed post-processing using only an- +gular diversity along with models of the telescope and in- +strument’s optical performance enabled by regular measure- +ments of the telescope wavefront error, simulating Reference +Star Differential Imaging but without the actual observation +of a reference star. A very similar approach has been applied +to enable high contrast imaging with the Hubble Space Tele- +scope by modeling the instrument PSF (e.g. Krist et al. 1997); +JWST’s stability and regular measurements of the wavefront +further enable this technique. +High contrast observations +with only angular diversity can significantly reduce the ob- +servation time and overhead. We demonstrate that this can +be an appropriate strategy for bright and widely separated +companions. +2.2. Radial Velocity Observations +New radial velocity measurements of HD 19467 were ob- +tained in July through August 2022 using the High Resolu- +tion spectrometer (HIRES) on the Keck I Telescope. The new +RV measurements are processed using standard data reduc- +tion techniques described in Butler et al. (1996) and Butler +et al. (2017). The majority of the RVs come from Rosenthal +et al. (2021), where the reduction techniques are described +in more detail. In brief, the HIRES RV values are measured +using an iodine cell-based design in order to wavelength cal- +ibrate the stellar spectrum. The spectral region from 5000- +6200 ˚A is used for measuring the radial velocities. We com- +bine the new observations with previous measurements for a +total of 53 RV measurements spanning 25 years for the data +analysis. The data including the new measurements are listed +in Table 8 in Appendix B. +2.3. TESS Observations + +JWST OBSERVATIONS OF HD 19467 B +3 +Table 1. NIRCam Observing Parameters (PID:#1189) +Target +Filter +Readout +Groups/Int +Ints/Exp +Dithers +Exp Time (s) +Subarray SUB320; Roll 1 +HD 19467 +F250M +MEDIUM2 +10 +10 +1 +983.517 +HD 19467 +F300M +MEDIUM2 +10 +5 +1 +491.758 +HD 19467 +F360M +MEDIUM2 +10 +5 +1 +491.758 +HD 19467 +F410M +MEDIUM2 +10 +5 +1 +491.758 +HD 19467 +F430M +MEDIUM2 +10 +5 +1 +491.758 +HD 19467 +F460M +MEDIUM2 +10 +5 +1 +491.758 +Subarray SUB320; Roll 2 +HD 19467 +F250M +MEDIUM2 +10 +10 +1 +983.517 +HD 19467 +F300M +MEDIUM2 +10 +5 +1 +491.758 +HD 19467 +F360M +MEDIUM2 +10 +5 +1 +491.758 +HD 19467 +F410M +MEDIUM2 +10 +5 +1 +491.758 +HD 19467 +F430M +MEDIUM2 +10 +5 +1 +491.758 +HD 19467 +F460M +MEDIUM2 +10 +5 +1 +491.758 +NOTE—Observations of reference star HD19096 were not executed. +NOTE—Total Time refers to the effective exposure time reported in the data headers, keyword +XPOSURE. +HD 19467 was observed by the TESS spacecraft (Ricker +et al. 2015) in Sectors 4 and 31, resulting in ≈ 60 days of +high-precision optical photometry. Sector 31 includes data +obtained with 20-second cadence, a new observing mode in- +troduced in the TESS extended mission. TESS 20-second +data shows improved photometric precision for bright stars +such as HD 19467 (Huber et al. 2022), and we therefore fo- +cus on 20-second data here. We used the PDC-MAP light +curves provided by the Science Processing Operations Cen- +ter (SPOC, Jenkins et al. 2016), which have been optimized +to remove instrumental variability (Smith et al. 2012; Stumpe +et al. 2012), and remove all data with quality flags not equal +to zero which yields the best precision for 20-second data +(Huber et al. 2022). +3. HOST STAR PROPERTIES +HD 19467 is a slightly metal poor G3 main sequence star +(Gomes da Silva et al. 2021) as summarized in Table 2. +In some cases multiple values are given for key parame- +ters to give an idea of their spread. +The biggest discrep- +ancy concerns the age estimates which range from 5.41+1.8 +−1.34 +to 11.88±2.56 Gyr (Brandt et al. 2021a; Wood et al. 2019; +Maire et al. 2020; Gomes da Silva et al. 2021). Maire et al. +(2020) apply different approaches to age determination and +provide a thorough discussion of their merits and drawbacks. +Their work generally suggests older ages than the initial +Crepp et al. (2014) estimation but they note that the chem- +ical abundance and kinematics likely place HD 19467 in the +thin disk population, suggesting an age younger than 10 Gyr. +We discuss our choice of age in more detail below, including +new asteroseismology data from TESS, which favor an older +age. +3.1. Analysis of TESS photometry +The top panel of Figure 1 shows the TESS 20-second ca- +dence light curve for HD 19467. +We observe no signifi- +cant long-term variability, with an RMS of 25.5 ppm over 6 +hour timescales. To search for high-frequency variability, we +used the established asteroseismic tools pySYD (Huber et al. +2009; Chontos et al. 2021) and FAMED (Corsaro et al. 2020), +which analyze the data in the frequency domain. Both meth- +ods detected a significant power excess near ≈ 2200 µHz, +consistent with the expected ≈7 minute timescale of solar- +like oscillations (Bedding 2014; Garc´ıa & Ballot 2019) based +on the spectroscopic temperature and surface gravity (Table +2). We also analyzed the data in the time-domain using a +Gaussian Process (GP) model with a stochastically driven +damped harmonic oscillator (Foreman-Mackey et al. 2017), +which has been demonstrated to outperform traditional fre- +quency analysis tools in recovering low S/N oscillations (Hey +et al., in prep). The GP analysis strongly favored a model +with an oscillating component with a ∆BIC=7.1. +The bottom panel of Figure 1 shows the power spectrum +of the 20-second light curve centered on the power excess. +Solar-like oscillations are described by a frequency of max- +imum power (νmax) and a large frequency separation (∆ν), +which approximately scale with log g and the mean stellar +density, respectively (Ulrich 1986; Brown et al. 1991). We +derive νmax = 2180 ± 100 µHz, with the central value taken +from the median of three solutions (pySYD, FAMED, GP), +and uncertainties calculated from the scatter over individual +methods (e.g. Huber et al. 2013). The low S/N of the de- +tection precludes an unambiguous detection of ∆ν. Visual +inspection of an echelle diagram indicates ∆ν ≈ 101µHz, +consistent with the derived νmaxvalue. +3.2. Physical Properties of HD 19467 +We adopted the effective temperature (Teff) and metallic- +ity ([M/H]) from Brewer et al. (2016), derived from a line- +by-line analysis of a Keck/HIRES spectrum. Literature val- +ues from spectroscopy and Gaia color-temperature relations +(Casagrande et al. 2021) are highly consistent, with a range +of 40 K in Teff and 0.04 dex in iron abundance (Table 2). We +used these ranges as an estimate for uncertainties, resulting +in Teff = 5747 ± 40 K and [M/H] = −0.09 ± 0.04 dex. +These uncertainties are smaller than those recommended by +Tayar et al. (2022), which is justified by the fact that star has +properties similar to the Sun and thus suffers from smaller +systematic errors. +We then combined the asteroseismic νmax measurement, +Gaia DR3 parallax, 2MASS K-band magnitude, Teff and +[M/H] with isoclassify (Huber et al. 2017) and BASTA +(Aguirre Børsen-Koch et al. 2022), which perform Bayesian +inference of stellar parameters given input observables us- +ing the stellar evolution models MIST (Choi et al. 2016) and + +4 +GREENBAUM ET AL. +2145 +2150 +2155 +2160 +2165 +Time [BTJD] +−1500 +−1000 +−500 +0 +500 +1000 +1500 +2000 +Flux [ppm] +TESS 20s +Binned +1500 +1750 +2000 +2250 +2500 +2750 +3000 +Frequency [µHz] +0 +20 +40 +60 +80 +100 +120 +140 +160 +PSD [ppm2/µHz] +Smoothed +Model +Figure 1. Top: TESS Sector 31 light curve of HD 19467. Black +points show the original 20-second cadence data, red points show +the data binned to a timescale of 6 hours. Bottom: Power spec- +trum of the data centered on the detected power excess near ≈ 2200 +µHz. The blue line and filled area shows the median and standard +deviation of the GP model posterior +BASTI (Pietrinferni et al. 2004), respectively. Importantly, +νmax tightly constrains the surface gravity to log g = 4.28, +which combined with the radius constraint from the Gaia par- +allax provides a tight constraint on stellar mass, which in turn +constrains stellar age. Both tools consistently imply a mass +of ≈ 0.95 M⊙, which, given that the star has slightly evolved +off the main-sequence (1.2 R⊙), implies an old age. Figure 2 +shows the age posteriors from both evolutionary models and +methods. +The final stellar parameters adopted in our study are listed +in Table 3. +We adopt the self-consistent solution derived +from isoclassify, but add in quadrature the difference to the +BASTA results to account for systematic errors due to differ- +ent model grids (Tayar et al. 2022). +Table 2. Observations of the Host Star HD 19467 +Property +Value +Units +Comments +Spectral Type +G3V +Gomes da Silva et al. (2021) +Teff +5720±10 +K +Gomes da Silva et al. (2021) +Teff +5747±25 +K +Brewer et al. (2016) +Teff +5770±80 +K +Maire et al. (2020) +Teff +5742±10 +K +Nissen et al. (2020) +Mass +0.953±0.022 +M⊙ +Maire et al. (2020) +Mass +0.960±0.02 +M⊙ +This work (§3.2) +Age +5.41+1.8 +−1.34 +Gyr +Brandt et al. (2021a) +Age +8.0+2.0 +−1.0 +Gyr +Maire et al. (2020, Table 1) +Age +10.06+1.16 +−0.82 +Gyr +Wood et al. (2019) +Age +11.882±2.564 +Gyr +Gomes da Silva et al. (2021) +Age +9.4±1.0 +Gyr +This work (§3.3) +[Fe/H] +−0.11±0.01 +dex +Maire et al. (2020) +[Fe/H] +−0.09±0.04 +dex +This work (§3.2) +log(g) +4.32±0.06 +cgs +Maire et al. (2020) +log(g) +4.28±0.04 +cgs +This work (§3.2) +R.A. (Eq 2000; Ep 2000) +03h07m18.570s +Gaia DR3 +Dec. (Eq 2000; Ep 2000) +−13o45′42.419′′ +Gaia DR3 +Distance +32.03±0.03 +pc +Gaia DR3 +Proper Motion (µα, µδ) +(−8.694, −240.64) +mas/yr +Gaia DR3 +RUWE +1.0566 +Gaia DR3 +G +6.814±0.003 +mag +Gaia DR3 +H +5.447±0.033 +mag +2MASS +W1 [3.4 µm] +5.36±0.16 +mag +WISE +W2 [4.6 µm] +5.18±0.06 +mag +WISE +6 +8 +10 +12 +14 +Age [Gyr] +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Density +Isoclassify +BASTA +Figure 2. Posterior distributions for the age of HD 19467 based +on isochrone modeling with isoclassify (blue) and BASTA (orange) +using constraints from asteroseismology, spectroscopy and Gaia. +3.3. The Age of HD 19467 +The age of HD 19467 is important for interpreting the mass +and atmospheric composition of the brown dwarf companion. +As already mentioned, literature estimates have a significant +spread, ranging from 5–12 Gyr (Table 3). Younger, Sun-like + +JWST OBSERVATIONS OF HD 19467 B +5 +Table 3. Adopted Stellar Parameters for HD 19467A +Effective temperature, Teff (K) 5747 ± 40 +Metallicity, [M/H] (dex) +−0.09 ± 0.04 +Luminosity, L ( L⊙) +1.42 ± 0.06 +Stellar radius, R⋆ ( R⊙) +1.20 ± 0.03 +Stellar mass, M⋆ ( M⊙) +0.96 ± 0.02 +Stellar density, ρ⋆ (cgs) +0.56 ± 0.03 +Surface gravity, log g (cgs) +4.28 ± 0.04 +Age, t (Gyr) +9.4 ± 1.0 +NOTE—Teff and [M/H] are adopted from Brewer et al. (2016), with uncertainties +accounting for the spread in literature results. All other properties are derived from the +combination of constraints from asteroseismology, spectroscopy and Gaia (see §3). +ages come from stellar rotation (Maire et al. 2020) and ac- +tivity (Brandt et al. 2021a), while older ages are preferred +by isochrone fitting (Wood et al. 2019; Maire et al. 2020). +The rotation-age is based on the detection of photometric pe- +riod of ≈ 29 days with an amplitude of ≈ 0.5% from ground- +based ASAS data (Maire et al. 2020). The high-precision +of TESS light curve in Figure 1 rules out rotational modu- +lation at the level of 0.5% over 25 day timescale suggested +by the ASAS data, which implies that the rotation period for +HD 19467 is undetermined. This is consistent with results +from the Kepler Mission, which demonstrated that typical ro- +tational amplitudes in mature Sun-like stars are on the order +of a few hundred ppm (McQuillan et al. 2014; Santos et al. +2021) and thus are generally not detectable using ground- +based photometry. Chromospheric activity-based ages also +become more challenging for stars with Sun-like and older +ages due the flattening of the age-activity relation, making +age constraints sensitive to small changes in R′ +HK measure- +ments. While some literature values for R′ +HK favor near so- +lar values (and thus ages) for HD 19467, others are consis- +tent with older, isochrone-based ages (e.g. R′ +HK = 5.1 and +8.8 ± 0.3 Gyr, Lorenzo-Oliveira et al. 2018). +The asteroseismic detection from TESS supports an older +age for HD 19467. While the low S/N precludes a direct +age from a measurement of individual oscillation frequencies +(e.g. Mathur et al. 2012; Metcalfe et al. 2014; Silva Aguirre +et al. 2017), the νmax measurement precisely constrains log g +and thus stellar mass independent of stellar evolutionary +models. +With a mass similar to solar (0.96 ± 0.02 M⊙), +HD 19467 must have an age significantly older than the Sun +to reach a radius of 1.2 R⊙1. As discussed by Maire et al. +(2020), an age of 9.4 ± 1.0 Gyr is compatible with the slight +enhancements in alpha elements and sub-solar metallicity, +1 This analogy is only slightly affected by the sub-stellar metallicity of +HD 19467; a solar-mass star with −0.09 ± 0.04 has an age of ≈ 3.6 Gyr +at solar radius. +placing the star in the transition region between chemical +“thin-disk” and “thick-disk” stars. Overall, we conclude that +HD 19467 is an ≈4-5 Gyr older analog to our Sun and adopt +an age of 9.4±1.0 Gyr (Table 3). +4. NIRCAM DATA REDUCTION AND POST +PROCESSING +We use the processed images retrieved from the Mikulski +Archive for Space Telescopes (MAST)2 that have been cor- +rected for bad pixels, flat-fielding, and background subtrac- +tion with the jwst pipeline. The product we use are the +calints files which result from Stage 2 of the pipeline and +have been through a photometric calibration. The data were +processed with calibrations software version 1.5.3 and cali- +bration reference data context jwst 0943.pmap. In addi- +tion to these data, we take advantage of wavefront informa- +tion provided by Optical Path Difference (OPD) maps taken +by the NIRCam wavefront sensing team3 to generate a NIR- +Cam PSF model close in time to our science observations. +For synthetic PSFs, we utilize the OPD from 2022-08-11, +R2022081102-NRCA3 FP1-1.fits, the closest in time preced- +ing our observations. +4.1. PSF Subtraction +We apply principal component analysis (PCA) (e.g. +Lafreni`ere et al. 2007; Amara & Quanz 2012) via Karhunen +Lo´eve Image Projection (KLIP; Soummer et al. 2012), to +subtract the residual stellar intensity from the science frames +using the images taken in two roll angles for angular di- +versity. We perform PSF subtraction using the open source +Python package pyKLIP (Wang et al. 2015) using Angular +Differential Imaging (ADI) and Reference Differential Imag- +ing (RDI) using a synthetic reference PSF, as described be- +low. The results of the PCA reduction for all filters is dis- +played in Figure 3. +For all filters except filter F250M, the data contained in the +two rolls suffices to obtain an unambiguous detection of the +companion. For the F250M case, although the companion’s +signal is visible using only ADI, we resorted to using RDI +with a set of synthetic PSFs in order to confirm the signal is +indeed from the companion and not due to residual speckles. +In Figure 4 we show the comparison, for the F250M filter, +between only using the roll frames for the PCA reduction +(ADI), and assisting the PCA reduction with a set of synthetic +PSFs (ADI+SynRDI). +The grid of synthetic stellar PSFs is generated us- +ing +WebbPSF +(Perrin +et +al. +2014) +and +tools +from +webbpsf ext4 at offset locations with respect to the coro- +2 https://mast.stsci.edu/ +3 https://webbpsf.readthedocs.io/en/latest/available opds.html +4 https://github.com/JarronL/webbpsf ext + +6 +GREENBAUM ET AL. +Figure 3. Post-processed images of HD 19467 in the six NIRCam filters observed in this program, rotated so that North is up. Images were +reduced using the pyKLIP algorithm as described in the text. Arrows indicate the detected companion. +Figure 4. Top: Comparison between using only ADI for the PCA reduction (left), and using RDI with synthetic PSFs generated with WebbPSF +(right) for the the F250M data. The addition of RDI reduces the speckle noise in the PSF-subtracted images. The data are oriented so that North +is up. Bottom: The forward model compared with the PSF-subtracted data for F250M, using the sythetic PSFs as reference. + +F250M +F300M +F360M +2 +2 +[arcsec] +0 +0 +0 +ec ++ +0 +-2 +-2 +2 +i +0 +-1 +-2 +2 +1 +0 +-1 +-2 +2 +1 +0 +-1 +-2 +F410M +F430M +F460M +12.5 +2 +20 +2 +2 +10.0 +10 +15 +1- +7.5 +((mly) +14 +5.0 +0- +2.5 +0 +Flux +0.0 +-1- +-1 +2.5 +1 +5.0 +2 +2 +-1D +7.5 +2 +1 +0 +-1 +2 +1 +0 +-2 +2 +i +0 +-1 +RA [arcsec]ADI Only +ADI + SynRDI +7.5 +7.5 +5.0 +5.0 +2.5 +2.5 +(Aw) +0.0 +0.0 +Flux +2.5 +2.5 +7.5 +7.5Data +Best-fit Model +Residuals +15 +15 +15 +Counts (DN) +S +10 +0 +10 +2 +(pixel: +0 +> +5 +5 +2 +0 +0 +0 +0 +10 +0 +10 +0 +10 +X (pixels) +X (pixels) +X (pixels)JWST OBSERVATIONS OF HD 19467 B +7 +nagraph focal plane mask. We generate simulated PSFs in +different sets of 9-point grid pattern at even spacings. We +simulate spacings of 2.5, 7, 15, 25, 40 mas, in addition to +a set of rotations of the coronagraphic-PSF with respect to +the detector of 0.1, 0.3, 0.5 degrees. This aims to emulate +the speckles present in the data frames, and assists the PCA +reduction with the diversity in speckle structure needed to +perform a more optimal reference subtraction. +As mentioned above, for filters F300M, F360M, F410M, +F430M, and F460M, ADI suffices for a clear detection. As +a second step we use RDI with the synthetic PSFs to further +subtract the unwanted starlight. This is motivated by the fact +that the companion PSF’s northern lobe falls near the diffrac- +tion speckles caused by the bar coronagraph. +The num- +ber of Karhunen Lo´eve (KL) modes determines how much +of the synthetic PSFs are used for the subtraction. +Since +these have been generated with arbitrary offsets, there is a +risk of subtracting the light from the secondary. We use 15 +KL modes, which minimizes over-subtraction and clears out +slightly more of the residual starlight around the northern +lobe. This was done by visual inspection; a more in-depth +analysis on how to optimally use synthetic PSFs will be ex- +plored in the future. +4.2. Photometry +To accurately extract the flux and position of HD 19467 B, +we account for over-subtraction effects on the PSF that arise +during the reduction process (described in Section 4.1) with +a forward model based on the method described by Pueyo +(2016). +We make use of its implementation on pyKLIP +(Wang et al. 2015). The companion PSF is modeled using +WebbPSF (Perrin et al. 2014) for each filter and accounting +for its position with respect to the bar focal plane mask. An +accurate position of the simulated PSF is particularly impor- +tant in the case of the shorter wavelength filters: poorer spa- +tial sampling of the pixels compared to the diffraction limit +at smaller wavelengths (2.5 µm is sub-Nyquist) results in an +acute sensitivity of the PSF structure as seen in the detector. +An accurate positioning for the case of F250M was done by +trial and error simulating a grid of PSF offsets and selecting +the best fit by least square difference between the simulation +and the coadded science frames. The model PSF are sim- +ulated using the OPD map closest in time and prior to the +observations. +The flux and position of the companion are extracted with +pyKLIP. We fit a model of its photometry and astrometry +to the reduced data using an MCMC approach (emcee; +Foreman-Mackey et al. (2013)). +Figure 5 shows the PSF +model fit to the reduced data for filter F360M, the filter in +which we obtain the highest SNR. Appendix A contains the +full gallery of forward model comparisons with the PSF- +subtracted images in each band. +Figure 5. +The best fit model to the PSF-subtracted signal used +to measure the photometry and astrometry of HD 19467 B in filter +F360M, where the highest SNR was achieved. The residuals show +the companion is fit well by the forward model. +Table 4. Adopted Photometry For HD 19467 A +Filter +Flux (Jy) +F250M flux (Jy) +3.51±0.07 +F300M flux (Jy) +2.63±0.05 +F335M flux (Jy) +2.10±0.04 +F360M flux (Jy) +1.82±0.04 +F410M flux (Jy) +1.49±0.03 +F430M flux (Jy) +1.36±0.03 +F460M flux (Jy) +1.12±0.02 +NOTE—Predicted fluxes in JWST wavebands are based on BOSZ stellar models +(Bohlin et al. 2017). +The flux calibration of the signal is determined based on +the jwst stage 2 pipeline photometric calibration. We apply +a flux correction to the photometry based on measured atten- +uation factor of 0.92 of the Bar mask Lyot stop at ∼ 1.6′′. +We fit a G3V stellar photosphere model to 1-5 µm photom- +etry from 2MASS (Cutri et al. 2003) and WISE (Cutri & et +al. 2012). We also find a ∼ 2% error in fitting the stellar +model to the IR measurements, and apply this error to con- +trast reported. Table 4 shows the estimated stellar flux. Com- +parison of the calibrated flux measured from the acquisition +and astrometric confirmation images, both taken through the +neutral density square, produced from the stage 2 pipeline is +consistent with the estimated stellar spectrum within ∼ 10%. +We therefore apply a 10% uncertainty to reported absolute +photometry of HD 19467 B in this section. +Figure 6 shows the measured photometry in each NIR- +Cam band alongside recent measurements and limits from +the ground (Mesa et al. 2020; Maire et al. 2020). The F460M +flux is consistent with the M band upper limit obtained with +VLT NaCo Maire et al. (2020), however there appears to be +some tension with the NaCo L’ flux compared with F360M +and F410M photometry measurements. Carter et al. (2022) +also noted a discrepancy in measurements from NaCo L′ and +JWST NIRCam photometry. The difference in passband on + +Data +Best-fit Model +Residuals +20 +20 +20 +Y (pixels) +5 +Counts (DN) +10 +10 +1o +0 +上0 +0 +0 +0 +10 +20 +0 +10 +20 +0 +10 +20 +X (pixels) +X (pixels) +X (pixels)8 +GREENBAUM ET AL. +Figure 6. New NIRCam photometry (blue stars) compared with +recent ground-based measurements from VLT-SPHERE and VLT- +NACO. +Figure 7. Left: An example of the raw MASKLWB coronagraph +data for an image in the F360M filter. Right: The best fit model +PSF simulated using the most recent preceding OPD map, used to +measure the centroid of the star. +a steeply rising part of the spectrum, possible water vapor +effects, as well as calibration uncertainties may account for +this discrepancy. Continued refinement of JWST photomet- +ric calibrations will help identify any biases in photometry. +For this study, we do not incorporate NaCo photometry into +the analysis, but rather present the measurement comparison +for future investigation. +Table 5 shows our measured photometry and relative as- +trometry for HD 19467 B (see next section). +4.3. Relative Astrometry +A major challenge of obtaining relative astrometry of a +companion in coronagraphic imaging is that the primary star +is occulted by the focal plane mask. Knowledge of the wave- +front from published OPD maps, and a highly structured PSF +enable a forward model based cross-correlation with the data +to fit for the centroid of the star behind the mask. We per- +form a cross-correlation of model PSFs with the data using +the chi2 shift in the image-registration Python +package5 to measure the best fit position of the star behind the +5 https://image-registration.readthedocs.io/ +Figure 8. Astrometric position of HD 19467 B relative to its parent +star, compared with previous measurements. +mask (Figure 7). We obtain a centroiding error ∼7 mas, con- +sistent with the measured sensitivity in Carter et al. (2022). +The companion position is recovered with the joint astrom- +etry and photometry model fit to the reduced data as de- +scribed in Section 4.2. The model fit errors provide the un- +certainty in the relative position to the measured star position +on the detector. We add the star position uncertainty to the +reported errors (Table 5). Figure 8 shows the new astrometric +measurement compared to previous relative astrometry mea- +surements of HD 19467 B (Crepp et al. 2014, 2015; Bowler +et al. 2020; Maire et al. 2020). +4.4. Performance and Sensitivity +In Figure 9 we show the contrast curves for the reduced +images after PSF subraction. The contrast is measured with +PyKLIP by computing the noise in an azimuthal annulus at +each separation, using a Gaussian cross correlation to remove +high frequency noise. The flux normalization to obtain these +contrast numbers was computed as explained in Section 4.2, +by using a best fit model of the stellar spectrum to calibrate +contrast. The contrast curves are corrected for algorithmic +throughput, i.e. the throughput loss due to the PSF subtrac- +tion, and for small sample statistics (Mawet et al. 2014). +Figure 10 translates our detection limits from flux/contrast +sensitivities to limits on companion mass. +Three differ- +ent brown dwarf evolution models are considered – Ames- +COND (Baraffe et al. 2003), BEX-HELIOS (Linder et al. +2019), and Sonora-Bobcat (Marley et al. 2021). In each case, +we assume solar metallicity. +While the shortest wavelength observations achieve the +best contrast (in particular F300M), the longer wavelengths +are better at detecting lower mass companions. We find an +overall detection limit of ∼10 MJ. This limit is much higher +than the sub-Jupiter levels that JWST/NIRCam can obtain +for young systems (e.g. Carter et al. 2022), but even for this + +-740 +C14 +C15 +2011.66 +-760 +B20 +780 +M20 +(seu) ++ This work +-800 +ADec +-820 +一840 ++ +2022.61 +-860 +-880 +-1340-1360-1380-1400-1420-1440-1460-1480 +ARA (mas)JWST Filters +This work +Maire+2020 +10-9 +Mesa+2020 +10-10 +10-11 +0000T +15000 +20000 +25000 30000 35000 40000 45000 +50000 +Wavelength (A)Data +Model PsF +0 +2DD +0 +200 +175 +175 +10 +10 +15D +150 +20 +20 +125 +125 +30 +1DD +30 +100 +DN +75 +75 +40 +40 +50 +50 +50 +50 +25 +25 +0 +10 +20 +30 +40 +50 +0 +10 +20 +30 +40 +50 +PixelsJWST OBSERVATIONS OF HD 19467 B +9 +Table 5. NIRCam measurements of HD 19467 B +Separation +Pos. Angle +∆mag +Flux +Filter +(′′) +(deg) +(mag) +(µJy) +F250M +1.597±0.010 +236.9±0.14 +13.67±0.271 +11.96±2.75 +F300M +1.611±0.002 +237.2±0.04 +12.62±0.122 +23.50±2.52 +F360M +1.610±0.003 +236.9±0.07 +11.91±0.127 +31.41±3.58 +F410M +1.604±0.001 +236.8±0.04 +10.45±0.116 +98.74±10.20 +F430M +1.609±0.003 +236.6±0.07 +10.78±0.124 +66.32±7.35 +F460M +1.609±0.003 +236.9±0.07 +11.07±0.121 +41.78±4.64 +NOTE—NIRCam astrometry and photometry from 2022-Aug-12 +NOTE—The astrometric precision for each filter is based solely on the posi- +tional uncertainty relative to the center of the coronagraph mask. The com- +bined astrometry includes an additional term to account for uncertainty in the +stellar position behind the mask (7 mas in each direction). +very old system brown dwarfs are easily detectable outside +of ∼0.4′′ ≃ 10 AU. While the detection limit is relatively in- +dependent of model, it does depend significantly on the age +of the brown dwarf (the system age is discussed in §3.3). +Despite a lack of reference star observations, we are able +to recover the signal of HD 19467 B with two roll angles +and achieve contrasts ∼10−5 at 1–2 arcsec. Regular OPD +measurements enable the use of synthetic PSFs that can aid +PSF subtraction by generating a set of reference PSFs to cap- +ture speckle structure. This suggests that bright companions +could be observed without reference stars, significantly re- +ducing the time spent on the observation. Future work will +investigate the difference between reducing data with and +without reference star observations. Future observations with +a better defined position for the LWB coronagraph should +also provide better contrast close-in. +5. ORBIT OF HD 19467 B +Previous studies estimate the mass of HD 19467 B from +51 to 86 MJ through both model-based estimates and orbital +analyses (Crepp et al. 2014; Maire et al. 2020; Brandt et al. +2021a). We analyze new radial velocities and provide an up- +dated dynamical mass estimate including our new relative as- +trometry and additional RV measurements. +First, we fit the new and previously measured RVs +from HIRES and HARPS (Trifonov et al. 2020) using the +RadVel6 software (Fulton et al. 2018). With the addition +of the new data, we measure a linear slope term of ˙γ = +−0.00412 ± 0.00027 m s−1 d−1 with strong significance. +We attempt to fit for curvature and tentatively detect a curva- +ture term of ¨γ = 1.7+0.81 +−0.78 × 10−7 m s−1 d−2 at 2.1σ. Model +comparison using ∆BIC and ∆AIC (Aikike Information +Criterion, Burnham & Anderson (2002)) show a nearly in- +6 https://radvel.readthedocs.io/en/latest/ +Table 6. Imaging Astrometry +epoch−2450000 +Filter +ρ (mas) +ρerr +PA (deg) +PAerr +Astrometry from Crepp et al. (2014) +5804.1 +K’ +1662.7 +4.9 +243.14 +0.19 +5933.8 +H +1665.7 +7.0 +242.25 +0.26 +5933.8 +K’ +1657.3 +7.2 +242.39 +0.38 +6166.1 +K’ +1661.8 +4.4 +242.19 +0.15 +6205.0 +Ks +1653.1 +4.1 +242.13 +0.14 +Astrometry from Maire et al. (2020) +8032.3 +L’ +1637 +19 +238.68 +0.47 +8061.2 +K1 +1636.7 +1.8 +239.39 +0.13 +8061.2 +K2 +1634.4 +5.0 +239.44 +0.21 +8409.3 +H2 +1631.4 +1.6 +238.88 +0.12 +8409.3 +H3 +1631.4 +1.6 +238.88 +0.12 +New Astrometry (this work; see Table 5) +9803.9 +2.5–4.6µm +1607.6 +7 +236.84 +0.25 +distinguishable model fit to a trend-only and trend plus cur- +vature model. A detection of curvature can place strong con- +straints on the companion orbit, especially for higher eccen- +tricity systems. Figure 11 shows the RV data plotted over the +maximum likelihood model. Appendix 2.2 contains a more +detailed description of the fit comparison and the new radial +velocities used in the analysis. +For the full orbital analysis we include all available RV +measurements from HARPS (Trifonov et al. 2020) and +HIRES (HIRES data including new measurements tabulated +in Appendix B), relative astrometry (Crepp et al. 2014, 2015; +Bowler et al. 2020; Maire et al. 2020) (listed in Table 6), +and absolute astrometry from Hipparcos and Gaia as de- +scribed in Brandt et al. (2021a), which takes advantage of +proper motion anomalies between Hipparcos, Gaia EDR3 +and the Hipparcos-Gaia long-term trend. +We utilize the +cross-calibrated catalog of Hipparcos-Gaia accelerations pre- +sented in Brandt (2021). + +10 +GREENBAUM ET AL. +0 +1 +2 +3 +4 +5 +6 +Separation (arcsec) +16 +17 +18 +19 +20 +21 +22 +5-σ Sensitivities (mag) +F250M +F300M +F360M +F410M +F430M +F460M +10-7 +10-6 +10-5 +10-4 +5-σ Contrast +0 +50 +100 +150 +200 +Separation (AU) +Figure 9. Contrast curves for all filters. Solid lines indicate ADI, and dashed lines indicate ADI and RDI using synthetic PSFs. Data points +indicate the HD 19467 B detections. The use of synthetic PSFs provides the diversity necessary to obtain enhanced contrast at small angular +separations. +We use orvara (Brandt et al. 2021b) to fit orbits to the +radial velocities, absolute astrometry, and relative astrome- +try. orvara is an orbit fitting code that uses ptemcee, a +parallel tempered MCMC scheme (Foreman-Mackey et al. +2013; Vousden et al. 2016). Following the orbital analysis +in Brandt et al. (2021a), we apply a geometric prior to incli- +nation and log-flat priors to semi-major axis and companion +mass. We apply uniform priors to remaining orbital elements. +log-flat priors are applied to RV jitter. We adopt the mass of +M∗ = 0.96 ±0.02 M⊙ based on the analysis in §3 using +asteroseismology, spectroscopy, and Gaia data. +We first predict the position of HD 19467 B in the current +epoch leaving out the new relative astrometry measured with +NIRCam, but including all other data. Figure 12 shows that +our measurement is consistent with the prediction of the best +fit orbits using previous measurements. +Next, we fit for orbital parameters including our new rel- +ative astrometry measurement from NIRCam. Table 7 sum- +marizes the orbit fit results. We infer a mass of 81+14 +−12 MJ. +Our mass estimate for HD 19467 B is within 1-σ of the prior +estimates from in Brandt et al. (2021a) (65.4+5.9 +−4.6 MJ), and +Maire et al. (2020) (74+12 +−9 MJ). We infer an eccentricity of +0.416+0.092 +−0.07 , which is consistent with recent measurements +in Brandt et al. (2021a), 0.54 ± 0.11, Maire et al. (2020), +0.56 ± 0.09, and Bowler et al. (2020), 0.39+0.26 +−0.18. We infer +a period of 386+220 +−108 yr, which is consistent with prior orbital +analyses (Bowler et al. 2020; Maire et al. 2020; Brandt et al. +2021a). The tentative evidence for curvature from the new +radial velocities may indicate that the orbit is close to perias- +tron passage. Given the high eccentricity, it could be a critical +time to monitor this system. +Figure 13 shows a selection of orbits from MCMC poste- +riors overlaid with the relative astrometry used in the fit, and +Figure 14 displays a corner plot of the MCMC posteriors for +orbital parameters. +6. ATMOSPHERE AND EVOLUTION MODEL +COMPARISON +In the following sections we show a preliminary compar- +ison of our near-IR photometry from NIRCam with brown +dwarf atmospheric models, focusing on the Sonora models +(Marley et al. 2021; Karalidi et al. 2021). In our spectral fit- +ting, we only use the model spectral grid with solar carbon- +to-oxygen ratio. +The Sonora-Bobcat cloudless atmospheric models assume +that the atmospheric composition is in thermo-chemical equi- +librium and solve radiative transfer equations for a self- +consistent temperature-pressure profile. The model grid cov- +ers temperatures from 200 to 2400 K, gravity from 10 to 3160 +m s−2, and metallicities from [Fe/H]=-0.5 to 0.5. The model +spectra have a spectral resolution ranging from 0.6 to 20 µm . + +JWST OBSERVATIONS OF HD 19467 B +11 +0 +1 +2 +3 +4 +5 +6 +Separation (arcsec) +10 +100 +Detection limit (MJup) +COND (9.4 Gyr) +F250M +F300M +F360M +F410M +F430M +F460M +0 +50 +100 +150 +200 +Separation (AU) +0 +1 +2 +3 +4 +5 +6 +Separation (arcsec) +10 +100 +Detection limit (MJup) +BEX (9.4 Gyr) +F250M +F300M +F360M +F410M +F430M +F460M +0 +50 +100 +150 +200 +Separation (AU) +0 +1 +2 +3 +4 +5 +6 +Separation (arcsec) +10 +100 +Detection limit (MJup) +Sonora (9.4 Gyr) +F250M +F300M +F360M +F410M +F430M +F460M +0 +50 +100 +150 +200 +Separation (AU) +0 +1 +2 +3 +4 +5 +6 +Separation (arcsec) +10 +100 +Detection limit (MJup) +HD 19467 B +Atmos. Model +COND +BEX +Sonora +0 +50 +100 +150 +200 +Separation (AU) +Figure 10. Detection limits (5−σ) for each filter, in terms of companion mass. The first three panels show limits for three different atmospheric +evolution models (COND, Bex, and Sonora), while the last panel compares the overall detection limit for each of the models. A mass estimate +from atmospheric model fitting (Section 6) is shown as a point in the lower right figure (61 MJ at 1.61′′). +Table 7. Orbit fit results +Parameter +Units +Value +Jitter +m/s +3.17+0.25 +−0.23 +Mpri +M∗ +0.961+0.022 +−0.022 +Msec +MJ +81+14 +−12 +Semi-major Axis +AU +53+19 +−10 +√e sin ω +−0.586+0.068 +−0.061 +√e cos ω +0.08+0.29 +−0.32 +Inclination +deg +127.9+8.1 +−5.7 +Ascending Node +deg +48.9+240 +−7.3 +Mean Longitude +deg +192.7+7.7 +−123 +Parallax +mas +31.226+0.037 +−0.037 +Period +year +382+220 +−108 +Argument of Periastron +deg +278+27 +−29 +Eccentricity +0.416+0.092 +−0.070 +Semi-major Axis +mas +1664+598 +−328 +T0 +JD +2486667+51547 +−4207 +Mass Ratio +0.080+0.014 +−0.012 +Figure 11. Best fit single companion Keplerian orbital model for +HD 19467 to radial velocities measured for HD 19467 by HIRES by +the California Legacy Survey (Rosenthal et al. 2021), new HIRES +observations, and HARPS (Trifonov et al. 2020). The fit slightly +favors a curvature term. +The Sonora-Cholla cloudless models assume chemical dis- +equilibrium. These models assume that atmospheres have so- +lar metallicity (Lodders 2010) with cloud-free atmospheric + +2000 +2010 +2020 +40 +a) +HARPS +30 +HIRES +HIRES pre 2004 +20 +10 +0 +10 +20 +30 +tb) +Residuals +20 +0 +-2012 +GREENBAUM ET AL. +Figure 12. Orvara prediction of relative astrometry at the 2022-08- +12 epoch. The measured astrometry is consistent with the predic- +tion. +structures. +The models are computed using the Picaso +v3.0 atmospheric model (Mukherjee et al. 2022). By in- +cluding an eddy diffusion parameter, Kzz, that ranges from +102 − 107 cm s−2 as an input parameter, the Cholla models +simulate the dynamical mixing that drives various molecular +species like CH4, CO, H2O, and NH3 out of their thermo- +chemical equilibrium abundances. The Cholla models span +over a temperature grid of 500 to 1300 K and a gravity grid +from 56 to 3160 cms−2. +First, we perform a basic χ2 comparison of new JWST +photometry to the Sonora models (Marley et al. 2021; Kara- +lidi et al. 2021). We also perform an MCMC fit of the model +grids with both our new photometry and previously published +medium resolution spectrum obtained with SPHERE-IRDIS +4 +2 +0 +2 +4 + (arcsec) +4 +3 +2 +1 +0 +1 +2 +3 +4 + [arcsec] + 1990 + 2000 + 2010 + 2020 + 2030 +Astrometric Orbits +80 +100 +120 +140 +160 +180 +Mcomp(MJup) +Figure 13. A selection of orbits from the MCMC posteriors. +LSS (Mesa et al. 2020) and ground-based photometry from +SPHERE-IRDIS Maire et al. (2020). From this we derive a +bolometric luminosity and determine model-dependent mass +estimate. +6.1. Model Comparison with NIRCam Photometry Only +We compare the NIRCam photometry with both the +Sonora-Bobcat and Sonora-Cholla model grids to highlight +the broad features of the 2 − 5 µm flux. We allow the ra- +dius scaling to vary when fitting the models to our NIRCam +photometry, which is generally consistent with radii much +smaller than, e.g., those predicted by the Sonora-Bobcat grid +evolutionary tables. +We find that the Bobcat models do not simultaneously cap- +ture both the local flux peak at 3 µm and the steep drop in flux +from 4−5 µm. In general the Bobcat grid favors higher grav- +ity and lower or zero metallicity. However, none of the fits +capture all of the photometry perfectly. A model spectrum +at effective temperature of Teff = 1400K best matches the +data, but does not fully capture the peak flux at ∼ 4 µm with +the drop off at longer wavelength. Gravity and metallicity do +not have a strong effect on the fit. +The Sonora Cholla models, which include effects of dis- +equilibrium chemistry, provide better fits to the photometry, +but suggest a low value for gravity as well as small radius +to scale the flux. The best fit model has Teff = 1200K, +g = 31, and log(Kzz) = 2. Figure 15 (right) shows the +best fitting Cholla model, with a few model grid points that +vary in temperature, eddy diffusion parameter, and gravity. +We note that while lower gravity is favored, the gravity does +not strongly affect the shape of the photometry-only profile, +as can be seen in the third panel. Near-IR spectroscopy with +JWST will be able to further distinguished features of grav- +ity. +Overall, the best fit Sonora-Cholla model provides a bet- +ter fit than the best fit Sonora-Bobcat model. As can be seen +in Figure 15, the Cholla models are better able to simultane- +ously capture the lower flux at shorter wavelengths, the peak +at ∼4 µm, and the subsequent drop in flux, favoring a lower +temperature that is more consistent with previous estimates. +This suggests that modeling disequilibrium chemistry may +be necessary for understanding the atmosphere and evolution +of HD 19467 B. +6.2. Model Fit to 1-5 µm +We fit the Sonora-Cholla cloudless models to the com- +posite dataset that consists of JWST photometry, the IRDIS +spectra (Mesa et al. 2020), and the SPHERE K12 band pho- +tometry (Maire et al. 2020) to constrain the temperature and +gravity of HD19487B. For the IRDIS spectra, we exclude +spectral points with negative flux values. We include an ad- +ditional uncertainty of 7%, similar to the J- and H-band ab- +solute flux uncertainties of HD 19467 (Table 7 of Mesa et al. + +865 +LocationofHD19467.2022.61 +870 +875 +(mas) +880 +885 +890 +895 +900 +905 +1330 +1340 +-1350 +-1360 +Aα (mas)JWST OBSERVATIONS OF HD 19467 B +13 +Mpri (M ) = 0.961+0.022 +0.022 +60 +120 +180 +240 +300 +Msec (MJup) +Msec (MJup) = 81+14 +12 +40 +80 +120 +160 +200 +a (AU) +a (AU) = 53+19 +10 +0.15 +0.30 +0.45 +0.60 +e +e = 0.416+0.092 +0.070 +0.90 +0.93 +0.96 +0.99 +1.02 +Mpri (M ) +105 +120 +135 +150 +i ( ) +60 +120 +180 +240 +300 +Msec (MJup) +40 +80 +120 +160 +200 +a (AU) +0.15 +0.30 +0.45 +0.60 +e +105 +120 +135 +150 +i ( ) +i ( ) = 127.9+8.1 +5.7 +Figure 14. Corner plot showing MCMC posteriors for select orbital parameters. +2020), to account for the possible offset in the absolute flux +levels. +We linearly interpolate the Cholla model spectral grid to +a finer grid with smaller step sizes in the temperature, grav- +ity, and eddy diffusion parameter. In addition to the three +parameters, the other free parameter in the spectral fitting is +the scaling factor, which is the square of the ratio of brown +dwarf radius to the distance (32.03 pc). We use pyphot +7 +to calculate the broadband photometries of model spectra. +7 https://github.com/mfouesneau/pyphot +Our model fitting algorithm minimizes a cost function that +sums the squared difference between the model spectra and +data, weighted by the observational uncertainties, σ, and the +intensities, w, as shown in the following: +cost function = ΣN +i=1{wi +Fλ,i(model) − Fλ,i(data) +σi +}2 +wi = +Fλ,i∆λi +max(Fλ,i∆λi) +(1) +where ∆λi is the wavelength coverage of a datapoint and +the intensity weighting is normalized by the highest intensity +among the datapoints. We include the intensity weighting in +the cost function so that a photometric point with high inten- + +14 +GREENBAUM ET AL. +Figure 15. Left: Measured photometry from NIRCam plotted alongside a few closely matching scenarios from the Sonora-Bobcat models, +letting the radius vary. Right: Measured photometry from NIRCam plotted alongside a few closely matching scenarios from the Sonora-Cholla +models, which include chemical disequilibrium parameterized by the eddy diffusion parameter, log(Kzz). +sity carries more weight in the fitting process than a spectral +point with low intensity even though both could have a simi- +lar signal-to-noise ratio, since the photometric points contain +a larger fraction of total flux measured. We then use emcee +(Foreman-Mackey et al. 2017) to sample the posterior distri- +bution of the fitted parameters with the Markov Chain Monte +Carlo (MCMC) method. We adopt a uniform prior of tem- +perature, logarithmic gravity, and eddy diffusion parame- +ter. We run the MCMC chain with 200 walkers for 20,000 +steps. +Based on the posterior distribution of the MCMC +chains, we derive the best-fit temperature of 1080 ± 22 K, +gravity of log(g) = 4.60+0.2 +−0.1, eddy diffusion parameter of +log Kzz = 3.1+0.6 +−0.4, and radius R of 0.62 ± 0.03RJ. The +value for radius is lower than the expected radius at 9 Gyr, +∼0.8 RJup, according to the evolution model grid at simi- +lar parameters. Prior modeling work has noted a common +discrepancy between the expected radius from evolutionary +models and the radius required to match the flux of a given +temperature that best fits atmospheric models (e.g., Barman +et al. 2011; Marley et al. 2012; Lavie et al. 2017). Maire et al. +(2020) explored radius agreement with evolutionary tracks +with different model atmospheres that included various levels +of clouds in the atmosphere. Future observations, especially +spectroscopy will further help constrain atmospheric models. +We draw 100 parameter sets from the posterior distribution +and plot the corresponding model spectra in Figure 16. Fig- +ure 16 suggests that the model spectra provide a qualitatively +good match to the data but there are some significant residu- +als, especially in the K-band photometries. We remain cau- +tious about the inferred parameters and the seemingly small +uncertainties given the imperfect fit between the data and +model spectra. However, while absolute flux calibration may + +Sonora Bobcat +Best fit: T=1400, logg=5.5, m=0.0, R=0.23Mjup +20 +T=1300, m=+0.5 +LB +T=1400 +→- T=1500, m=-0.5 +16 +→-T=1600, m=-0.5 +14 +12 +1D +0.B +0.6 +0.4 +ADODE +DOSE +40000 +45000 +54000 +55000 +e1 +2D +log(g)=4.5, T=1300 +LB +log(g)=4.75, T=1300 +log(g)=5.0, T=1300 +16 +→log(g)=5.25, T=1300 ++ log(g)=5.5 +14 +12 +LD +0.6 +0.4 +25000 +DODE +DOSE +4500D +5500D +e-1 +2D +0=w +1B +m=-0.5 +→= m=+0.5, log(g)=5.0, T=1300 +16 +14 +12 +LD +0.B +0.6 +0.4 +25000 +ADOE +DOSE +40000 +45000 +55000 +Wavelength (i)Sonora Cholla +le-l +log(9]=3.5. log(Kzz=2) +2D +T=1200K, R=0.44, log(g)=4.0 +LB +-T=1150K,R=0.47 +T=1100K, R=0.51 +16 +T=1050K, R=0.55 +T=1000K, R=0.61 +14 +12 +1D +0.B +0.6 +0.4 +DO5Z +ADODE +DOSE +40000 +45000 +54000 +55000 +e-1 +200 +log(Kzz)=2, log(g)=3.5, T=1100 +175 +log(Kzz)=4, log(g)=3.5, T=1100 +log(Kzz)=7, log(g)=5.5, T=1200 +150 +125 +0.75 +0.50 +0.25 +25000 +DODE +DOSE +40000 +45000 +55000 +e- +log(Kzz)=2 +2D +log(g)=3.5, R=0.51, T=1100 +1B +log(g)=4.3, R=0.47, T=1150 +- +log(g)=4.7, R=0.44, T=1200 +16 +14 +12 +1D +0.B +0.6 +0.4 +25000 +ADOE +ADOSE +40000 +ADOS +DOS +55000 +Wavelength (i)JWST OBSERVATIONS OF HD 19467 B +15 +account for some discrepancy between data sources, the dis- +crepancy alone is not enough to account for inconsistencies +between the best fit atmospheric model and evolutionary grid +predictions. Clouds, which likely drive the rotational mod- +ulation of many T dwarfs (e.g Manjavacas et al. 2019) and +are not included in the Cholla models, could play a key role +in shaping the HD 19467 B emission spectra. Future simul- +taneous observation in the near-IR and mid-IR region will +be useful for testing the role of clouds in the atmosphere of +HD 19467 B with well-constrained age and host star metal- +licity. +6.3. Bolometric Luminosity +The 2-5µm JWST NIRCam broadband photometry, in +combination with the ground-based near-infrared spectra +and photometry, are crucial for pinning down the bolo- +metric luminosity of a ∼1000 K object. We integrate the +flux density over observed data including the previously +published IRDIS-LSS spectra (0.97-1.335 µm & 1.50-1.80 +µm) , ground-based K-band photometry (2.059-2.161 µm & +2.1965-2.3055 µm), and the six JWST NIRCam broad band +photometry. The flux integral in the observed wavelength re- +gions is 3.28 ± 0.2 × 10−6L⊙. +We utilize the fitted model spectra in Section 6 to extrap- +olate flux density beyond the observed wavelength region +and estimate the bolometric luminosity. +We find that the +observational data accounts for around 72% of the bolomet- +ric luminosity. The estimated total bolometric luminosity is +(4.75 ± 0.2) × 10−6L⊙, or log(L/L⊙) = −5.32 ± 0.02. +Based on the combination of JWST NIRCam high-precision +photometry and ground-based data, our results suggest that +we can accurately derive the bolometric luminosity at 4% +precision level. +Based on the independently estimated age and bolometric +luminosity, we then use the Bobcat evolution model to es- +timate mass of HD 19467 B. After linearly interpolating the +mass as a function of age and bolometric luminosity, we de- +rive that the mass is 62 ± 1MJ. Figure 17 shows the Sonora +Bobcat evolution model and where our bolometric luminos- +ity estimate lies. By comparing the mass derived from the age +and Lbol to the dynamical mass, (81+14 +−12MJ), we conclude +that the two masses are consistent with each other within +about two-sigma. +The NIRSpec IFU observations planned for later this year +(PID #1414) will provide a much more complete characteri- +zation of the atmosphere of HD 19467 B, helping to pin down +parameters such as metallicity and Teff. +7. CONCLUSIONS +We have demonstrated the performance of JWST NIRCam +LWB coronagraph on the known binary system HD 19467 B. +Despite missing reference star observations, we are able +to recover the companion with high significance in all 6 +medium NIRCam bands used for the observations. +The main results of this study are as follows: +• The MASKLWB coronagraph works well for sepa- +rations below 1 arcsec (for medium filters excluding +F250M) at contrasts 10−5 and better, even without a +reference star when angular diversity is utilized. This +is expected to improve in the near future when corona- +graphic mask locations are refined through instrument +calibration observations. +• Given the superb stability of JWST, and regular OPD +measurements available, we are able to incorporate +synthetic reference images to further subtract speck- +les, following ADI subtraction, and improve SNR on +the detections. Future observations with reference ob- +servations can be compared with the results presented +in this study. +• We estimate the age of the HD19467 system by com- +bining spectroscopy and Gaia astrometry with astero- +seismic constraints from TESS, finding an age of 9.4± +1.0 Gyr, supporting older estimates of the age. We pro- +vide updated parameters for the host star HD19467. +• We estimate a dynamical mass of HD 19467 B of +81+14 +−12MJ, contributing new relative astrometry from +NIRCam and radial velocities from HIRES, and detect +tentative evidence of curvature in the orbit fit to the +radial velocities. +• A comparison of atmospheric and evolutionary models +to our new 2 − 5 µm photometry favors models that +include disequilibrium chemistry. +• A global fit to the photometry and spectroscopy from +this study and ground-based observations show some +tension between the instrument-to-instrument relative +fluxes and the models. +• The model-derived mass of 62 ± 1 MJ is lower than +the dynamical mass estimate, but within 2-σ. +The NIRCam observations provide the highest fidelity +3−5 µm photometry to date of HD 19467 B, and give an early +test of atmospheric and evolutionary models that JWST will +continue to test throughout the mission. Future observations +with NIRSpec (PID #1414) will further elucidate discrepan- +cies in model spectra and help characterize the chemistry of +HD 19467 B. Improvements in the near future through instru- +ment calibration observations will further refine the perfor- +mance of the coronagraphic mask placement and the perfor- +mance of the MASKLWB mode. + +16 +GREENBAUM ET AL. +Figure 16. The comparison of the fitted Cholla model spectra (light blue lines) to the SPHERE/IRDIS-LSS 1–1.8 µm spectra (gray lines), +SPHERE/IRDIS-K12 photometry (gray diamonds), NaCo L’ band (pink triangle), and JWST NIRCam photometry (golden hexagon) show +that it is challenging for the models to simultaneously explain the near-IR and mid-IR spectra and photometry. The semi-transparent light +blues lines are the 1000 Cholla cloudless model spectra sampled from the posterior distribution of MCMC fitting results. The blue squares are +the effective photometry from the model spectra samples in the corresponding NIRCam filters. The transmission curves of JWST NIRCam +broadband photometry are plotted in colored lines at the bottom of the plot. The y-axis is in unit of intensity (Wm−2). +Figure 17. Based on the bolometric luminosity and age, the Bobcat +evolution models suggest that HD 19467 B has a mass of 62 ± 1 +MJ. The colors indicate the masses predicted by Bobcat evolution +models. White contour lines show the bolometric luminosity and +age with a fixed mass. The uncertainty of HD 19467 B’s bolometric +luminosity is around 4%. +ACKNOWLEDGMENTS +The authors thank the anonymous reviewer for helpful +comments. The authors acknowledge useful discussions with +G. Mirek Brandt and Eric Nielsen. +We also thank Dino +Mesa for providing the data for the near-IR spectrum of +HD 19467 B. +Some of the research described in this publication was +carried out at the Jet Propulsion Laboratory, California In- +stitute of Technology, under a contract with the National +Aeronautics and Space Administration. +D.J. is supported +by NRC Canada and by an NSERC Discovery Grant. +L.E.U.C.’s research was supported by an appointment to +the NASA Postdoctoral Program at the NASA Ames Re- +search Center, administered by Oak Ridge Associated Uni- +versities under contract with NASA. D.R.H. and D.H. ac- +knowledge support from the Research Corporation for Sci- +ence Advancement (Scialog award #26996) and the Na- +tional Science Foundation (AST-2009828). +D.H. also ac- +knowledges support from the Alfred P. Sloan Foundation +and the National Aeronautics and Space Administration +(80NSSC21K0652,80NSSC22K0303). +Some of the data presented in this paper were obtained +from the Mikulski Archive for Space Telescopes (MAST) at + +WeightedChollamodelsfittingresults +fitted models +modelphotometries +SPHERE IRDIS Spectrum +SPHEREK12photometries +JWSTNIRCamphotometries +NaCo L' +10~16 +(zw/M) +5 +10~17 +2 +3 +5 +10 +Wavelength (μm)BobcatEvolutionmode +Mass (Mjup) +4.2 +75 +4.4 +70 +70 +4.6 +4.8 +65 +(L/Lo) +5.0 +60 +5.2 +65 +60 +-5.4 +61 +-5.6 +65 +55 +5.8 +50 +5 +6 +7 +9 +10 +11 +12 +13 +Age (Gyr)JWST OBSERVATIONS OF HD 19467 B +17 +the Space Telescope Science Institute. The specific obser- +vations analyzed can be accessed via 10.17909/v8m9-xk98 +and 10.17909/t9-st5g-3177. +This research has made use of the SIMBAD database, op- +erated at CDS, Strasbourg, France. +Some of the data presented herein were obtained at the W. +M. Keck Observatory, which is operated as a scientific part- +nership among the California Institute of Technology, the +University of California and the National Aeronautics and +Space Administration. The Observatory was made possible +by the generous financial support of the W. M. Keck Foun- +dation. The authors wish to recognize and acknowledge the +very significant cultural role and reverence that the summit +of Maunakea has always had within the indigenous Hawaiian +community. We are most fortunate to have the opportunity to +conduct observations from this mountain. +Software: +SciPy (Virtanen et al. 2020), NumPy (Har- +ris et al. 2020), Matplotlib (Hunter 2007), WebbPSF (Perrin +et al. 2014), PyKLIP (Wang et al. 2015), synphot (Lim & +Hanley 2016), pysynphot (STScI Development Team 2013), +pyphot (Fouesneau 2022) +APPENDIX +A. KLIP FORWARD MODEL OF THE DATA +We compute a forward model of the PSF-subtracted image according to Pueyo (2016) to account for over- and self-subtraction +effects. ADI imaging is particularly susceptible to self-subtraction effects, especially when there is little angular diversity, such +as our data, which contains only two roll angles separated at ∼ 8 degrees. Modeling these effects is essential for appropriately +estimating the properties of the signal (flux, position). Figure 18 displays the comparison between the forward model and the +PSF-subtracted image corresponding to the images displayed in Figure 3. +B. KEPLERIAN ORBIT FIT TO HD 19467 RADIAL VELOCITIES +We fit a Keplerian Orbit model to radial velocities of HD 19467 from HIRES and HARPS instruments, the latter published in +Trifonov et al. (2020), catalog JA+A636A74rvbank table entries DRVmlcnzp and e DRVmlcnzp. HIRES radial velocities, along +with the new measurements are presented in Table 8. Using RadVel (Fulton et al. 2018), we determine that a model with a +curvature term is slightly favored over a model without curvature, however the trend-only and trend plus curvature models are +nearly indistinguishable (Table 9), based on ∆BIC and ∆AIC metrics. This is the first tentative evidence of curvature measured +for HD 19467. Future follow up is needed to further support this finding. +C. THE MCMC POSTERIOR DISTRIBUTION FOR SPECTRAL FITTING +Figure 19 shows the MCMC posterior distribution of the spectral fitting. The spatial structure seen in the posterior distribution +reflects the step size of temperature (10 K), gravity (0.025 dex), and log(Kzz) (0.167) in the interpolated model grid. The sharp +boundaries of radius at 0.5 RJup is reflects the lowest limit of radius range (0.5-1.5 RJup) in the model fitting. +Figure 18. The KLIP forward model that accounts for over subtraction effects compared to the PSF-subtracted data in each of size filters. + +Data +Best-fit Model +Residuals +15 +15 +15 +5 +Counts (DN) +Y (pixels) +一 +10 +10 +10 +国 +0 +5 +5 +5 +0 +0 +0 +0 +10 +0 +10 +0 +10 +X (pixels) +X (pixels) +X (pixels)Data +Best-fit Model +Residuals +20 +20 +20 +10 +Counts (DN) +S +(pixel: +10 +10 +10 +0 +0 +0 +0 +-10 +0 +10 +20 +0 +10 +20 +0 +10 +20 +X (pixels) +X (pixels) +X (pixels)Data +Best-fit Model +Residuals +10 +20 +20 +20 +Counts (DN) +(pixels) +10 +10 +10 +Y +0 +0 +0 +10 +20 +0 +10 +20 +0 +10 +20 +X (pixels) +X (pixels) +X (pixels)Data +Best-fitModel +Residuals +5 +Counts (DN) +S +20 +20 +20 +(pixels +10 +10 +10 +0 +0 +o +0 +0 +10 +20 +0 +10 +20 +0 +10 +20 +X (pixels) +X (pixels) +X (pixels)Data +Best-fit Model +Residuals +15 +15 +15 +Counts (DN) +S +10 +0 +10 +2 +(pixel: +0 +> +5 +5 +2 +0 +0 +0 +0 +10 +0 +10 +0 +10 +X (pixels) +X (pixels) +X (pixels)Data +Best-fit Model +Residuals +20 +20 +20 +Y (pixels) +5 +Counts (DN) +10 +10 +1o +0 +上0 +0 +0 +0 +10 +20 +0 +10 +20 +0 +10 +20 +X (pixels) +X (pixels) +X (pixels)18 +GREENBAUM ET AL. +Table 8. HIRES Radial Velocity Observa- +tions +Instrument +BJDT DB +RV +RV Error +HIRES +2450366.019 +20.64 +1.47 +HIRES +2450418.943 +29.50 +2.17 +HIRES +2450461.84 +34.01 +1.30 +HIRES +2450715.103 +26.13 +4.20 +HIRES +2450716.111 +25.36 +4.31 +HIRES +2450786.847 +23.29 +2.38 +HIRES +2450786.86 +27.42 +1.51 +HIRES +2450806.904 +22.50 +1.50 +HIRES +2450837.744 +14.19 +1.41 +HIRES +2450839.743 +19.41 +1.49 +HIRES +2451012.119 +27.88 +1.40 +HIRES +2451013.12 +19.74 +1.31 +HIRES +2451070.116 +20.60 +4.26 +HIRES +2451072.984 +29.72 +4.36 +HIRES +2451171.777 +26.06 +1.48 +HIRES +2451410.128 +21.00 +1.51 +HIRES +2451543.847 +18.48 +1.60 +HIRES +2451551.792 +19.77 +1.47 +HIRES +2451552.844 +14.65 +1.73 +HIRES +2451582.73 +24.84 +1.70 +HIRES +2451882.806 +29.84 +1.62 +HIRES +2451900.783 +23.36 +1.48 +HIRES +2452134.08 +20.03 +1.58 +HIRES +2452242.908 +19.50 +1.42 +HIRES +2452516.022 +18.34 +1.62 +HIRES +2452575.902 +10.04 +1.72 +HIRES +2452835.128 +19.96 +1.84 +HIRES +2452926.089 +19.92 +4.43 +HIRES +2453240.043 +11.43 +1.16 +HIRES +2453427.785 +8.38 +1.20 +HIRES +2453984.039 +5.01 +1.08 +HIRES +2455807.035 +-3.65 +1.23 +HIRES +2455808.105 +-0.48 +1.42 +HIRES +2455809.088 +1.84 +1.22 +HIRES +2455903.779 +8.54 +1.33 +HIRES +2456152.11 +-2.44 +1.18 +HIRES +2456210.015 +1.14 +1.49 +HIRES +2456519.085 +-0.97 +1.28 +HIRES +2456530.025 +-7.90 +1.20 +HIRES +2456548.035 +-0.47 +1.33 +HIRES +2456586.036 +-2.87 +1.41 +HIRES +2456587.965 +-3.16 +1.42 +HIRES +2456588.997 +4.29 +1.41 +HIRES +2456613.908 +-2.82 +1.38 +HIRES +2456637.791 +-0.19 +1.45 +New data +HIRES +2457245.143 +-0.38 +1.18 +HIRES +2458367.035 +-15.44 +1.76 +HIRES +2459632.706 +-10.11 +1.51 +HIRES +2459649.719 +-13.26 +1.48 +HIRES +2459780.128 +-14.46 +1.29 +HIRES +2459786.103 +-4.38 +1.16 +HIRES +2459787.129 +-15.52 +1.20 + +JWST OBSERVATIONS OF HD 19467 B +19 +Table 9. Model Comparison +AICc Qualitative Comparison +Free Parameters +Nfree +Ndata +RMS +ln L +BIC +AICc +∆AICc +AICc Favored Model +˙γ, ¨γ, σ, γ +8 +128 +3.40 +-330.13 +695.02 +673.41 +0.00 +Nearly Indistinguishable +˙γ, σ, γ +7 +128 +3.40 +-332.43 +692.91 +673.88 +0.47 +Ruled Out +σ, γ +6 +128 +9.50 +-426.33 +872.85 +856.43 +183.02 +Figure 19. The posterior distribution of the MCMC fitting of Cholla models to the SPHERE-IRDIS spectra, SPHERE K12 photometry, and +the JWST NIRCam photometries. + +T (K) = 1081.81187 +-22.48 +log(g) = 4.56±:16 +-0.14 +5.2 +(6)60 +4.8 +4.4 +4.0 +0.66 +4.8 +3.2 +R(Rjup) = 0.62±0:83 +0.75 +-0.03 +R (Rjup) +0.65 +0.55 +.60 +70 +4.4 +5.2 +2.4 +4.0 +O +T (K) +log(g) +Kzz +R (Rjup)20 +GREENBAUM ET AL. +REFERENCES +Aguirre Børsen-Koch, V., Rørsted, J. L., Justesen, A. B., et al. +2022, MNRAS, 509, 4344, doi: 10.1093/mnras/stab2911 +Amara, A., & Quanz, S. P. 2012, MNRAS, 427, 948, +doi: 10.1111/j.1365-2966.2012.21918.x +Baraffe, I., Chabrier, G., Barman, T. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 8800 Greenbelt Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Greenbelt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' MD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' USA (Accepted January 25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2023) ABSTRACT We observed HD 19467 B with JWST’s NIRCam in six filters spanning 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 µm with the Long Wavelength Bar coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The brown dwarf HD 19467 B was initially identified through a long-period trend in the radial velocity of G3V star HD 19467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' HD 19467 B was subsequently detected via coronagraphic imaging and spectroscopy, and characterized as a late-T type brown dwarf with approximate temperature ∼ 1000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We observed HD 19467 B as a part of the NIRCam GTO science program, demonstrating the first use of the NIRCam Long Wavelength Bar coronagraphic mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The object was detected in all 6 filters (contrast levels of 2×10−4 to 2×10−5) at a separation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6′′ using Angular Differential Imaging (ADI) and Synthetic Reference Differential Imaging (SynRDI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Due to a guidestar failure during acquisition of a pre-selected reference star, no reference star data was available for post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' However, RDI was successfully applied using synthetic Point Spread Functions (PSFs) developed from contemporaneous maps of the telescope’s optical configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Additional radial velocity data (from Keck/HIRES) are used to constrain the orbit of the HD 19467 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Photometric data from TESS are used to constrain the properties of the host star, particularly its age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' NIRCam photometry, spectra and photometry from literature, and improved stellar parameters are used in conjunction with recent spectral and evolutionary substellar models to derive physical properties for HD 19467 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Using an age of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='9 Gyr inferred from spectroscopy, Gaia astrometry, and TESS asteroseismology, we obtain a model-derived mass of 62±1 MJ, which is consistent within 2-σ with the dynamically derived mass of 81+14 −12 MJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' INTRODUCTION Brown dwarfs provide a unique testbed for confronting evolutionary and atmospheric models of sub-stellar objects Corresponding author: Alexandra Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Greenbaum azg@ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='edu with well-defined observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Those brown dwarfs which are companions to main sequence stars, as opposed to free- floating, are particularly valuable since they are presumed to inherit observable stellar properties such as metallicity and share similar ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This knowledge constrains many of the free parameters in the comparison of models with observa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='11455v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='SR] 26 Jan 2023 ID2 GREENBAUM ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Low-mass brown dwarf companions to main-sequence stars were initially found through blind imaging searches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' GL229 B (Nakajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 1995), and subsequently as a by-product of planet searches using the radial velocity (RV) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' In the case of HD 19467, Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2014) iden- tified it as a star with a significant RV trend suggestive of a massive brown dwarf companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Coronagraphic imag- ing with Keck NIRC2 first confirmed the presence of the companion (Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This was followed by spec- troscopy with Palomar’s P1640 instrument that characterized HD 19467 B as a brown dwarf with effective temperature of ∼1000 K corresponding to a T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 spectral type (Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' More recently, proper motion measurements from the Hipparcos and Gaia catalogs have been used to identify sys- tems with companions or help characterize them, including HD 19467 (Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Multiple JWST programs will provide imaging and spec- troscopy of HD 19467 B across the near- and mid-IR where brown dwarfs emit most of their energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The program pre- sented here (PID #1189) uses NIRCam (Rieke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' in press) to provide medium and narrow band imaging and photometry of HD 19467 B in 6 bands, spanning 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' At a later date, another JWST program (PID #1414) will use NIRSpec (Jakobsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2022) to obtain high-resolution (R ≃ 2700) 3–5 µm spectra of HD 19467 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' JWST observations of the G3V star HD 19467 with its T5+ brown dwarf companion, HD 19467 B (Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2014), represent one of the earliest exercises of the NIRCam Coro- nagraphic LW Bar (Krist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Beichman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Girard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2022), providing an opportunity for an early sci- entific result and a demonstration of the capabilities of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The NIRCam observations presented in this study are de- signed to accomplish three main goals: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Provide an early dataset that exercises the bar mask on NIRCam, especially without a reference star (§2 & 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Refine the orbital parameters of HD 19467 B with a new imaging data point along with new RV data from Keck/HIRES (§5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Add additional photometric measurements to better constrain the physical properties of the brown dwarf (§6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We also include new analysis of TESS observations to con- strain properties of the host star (§3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' OBSERVATIONS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' NIRCam Observations NIRCam observed HD 19467 on 2022-Aug-12 with the long-wavelength bar (LWB) coronagraphic mask in subar- ray mode with six filters: F250M, F300M, F360M, F410M, F430M, and F460M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The target star was observed at two tele- scope roll angles separated by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='72 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Table 1 shows a summary of the observations and settings per filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Observa- tions of HD 19467 were taken with the long-wavelength bar (MASKLWB) coronagraph, providing a test of NIRCam’s capabilities at smaller inner working angles than are possi- ble with the round masks (4λ/D for MASKLWB vs 6λ/D for MASK210R, MASK335R, and MASK430R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Krist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' At the time of these observations, the MASKLWB positions were not well-defined, with a y-offset ∼70 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Future use of this mode with updated position definition will improve the ability to center the star on the mask and therefore contrast performance, especially close in to the mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' These observations represent one of the first post- commissioning uses of the bar coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The observation plan was initially scheduled to include se- quential observations of the reference star HD 19096 in or- der to perform PSF subtraction using Reference Differential Imaging (RDI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' However, the reference observations were unsuccessful because the telescope failed to acquire a guide star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Instead, we performed post-processing using only an- gular diversity along with models of the telescope and in- strument’s optical performance enabled by regular measure- ments of the telescope wavefront error, simulating Reference Star Differential Imaging but without the actual observation of a reference star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' A very similar approach has been applied to enable high contrast imaging with the Hubble Space Tele- scope by modeling the instrument PSF (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Krist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' JWST’s stability and regular measurements of the wavefront further enable this technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' High contrast observations with only angular diversity can significantly reduce the ob- servation time and overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We demonstrate that this can be an appropriate strategy for bright and widely separated companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Radial Velocity Observations New radial velocity measurements of HD 19467 were ob- tained in July through August 2022 using the High Resolu- tion spectrometer (HIRES) on the Keck I Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The new RV measurements are processed using standard data reduc- tion techniques described in Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (1996) and Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The majority of the RVs come from Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2021), where the reduction techniques are described in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' In brief, the HIRES RV values are measured using an iodine cell-based design in order to wavelength cal- ibrate the stellar spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The spectral region from 5000- 6200 ˚A is used for measuring the radial velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We com- bine the new observations with previous measurements for a total of 53 RV measurements spanning 25 years for the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The data including the new measurements are listed in Table 8 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' TESS Observations JWST OBSERVATIONS OF HD 19467 B 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' NIRCam Observing Parameters (PID:#1189) Target Filter Readout Groups/Int Ints/Exp Dithers Exp Time (s) Subarray SUB320;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Roll 1 HD 19467 F250M MEDIUM2 10 10 1 983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='517 HD 19467 F300M MEDIUM2 10 5 1 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='758 HD 19467 F360M MEDIUM2 10 5 1 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='758 HD 19467 F410M MEDIUM2 10 5 1 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='758 HD 19467 F430M MEDIUM2 10 5 1 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='758 HD 19467 F460M MEDIUM2 10 5 1 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='758 Subarray SUB320;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Roll 2 HD 19467 F250M MEDIUM2 10 10 1 983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='517 HD 19467 F300M MEDIUM2 10 5 1 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='758 HD 19467 F360M MEDIUM2 10 5 1 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='758 HD 19467 F410M MEDIUM2 10 5 1 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='758 HD 19467 F430M MEDIUM2 10 5 1 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='758 HD 19467 F460M MEDIUM2 10 5 1 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='758 NOTE—Observations of reference star HD19096 were not executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' NOTE—Total Time refers to the effective exposure time reported in the data headers, keyword XPOSURE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' HD 19467 was observed by the TESS spacecraft (Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2015) in Sectors 4 and 31, resulting in ≈ 60 days of high-precision optical photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Sector 31 includes data obtained with 20-second cadence, a new observing mode in- troduced in the TESS extended mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' TESS 20-second data shows improved photometric precision for bright stars such as HD 19467 (Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2022), and we therefore fo- cus on 20-second data here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We used the PDC-MAP light curves provided by the Science Processing Operations Cen- ter (SPOC, Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2016), which have been optimized to remove instrumental variability (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Stumpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2012), and remove all data with quality flags not equal to zero which yields the best precision for 20-second data (Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' HOST STAR PROPERTIES HD 19467 is a slightly metal poor G3 main sequence star (Gomes da Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021) as summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' In some cases multiple values are given for key parame- ters to give an idea of their spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The biggest discrep- ancy concerns the age estimates which range from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='41+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='34 to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='88±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='56 Gyr (Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Gomes da Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020) apply different approaches to age determination and provide a thorough discussion of their merits and drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Their work generally suggests older ages than the initial Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2014) estimation but they note that the chem- ical abundance and kinematics likely place HD 19467 in the thin disk population, suggesting an age younger than 10 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We discuss our choice of age in more detail below, including new asteroseismology data from TESS, which favor an older age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Analysis of TESS photometry The top panel of Figure 1 shows the TESS 20-second ca- dence light curve for HD 19467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We observe no signifi- cant long-term variability, with an RMS of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 ppm over 6 hour timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' To search for high-frequency variability, we used the established asteroseismic tools pySYD (Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Chontos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021) and FAMED (Corsaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020), which analyze the data in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Both meth- ods detected a significant power excess near ≈ 2200 µHz, consistent with the expected ≈7 minute timescale of solar- like oscillations (Bedding 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Garc´ıa & Ballot 2019) based on the spectroscopic temperature and surface gravity (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We also analyzed the data in the time-domain using a Gaussian Process (GP) model with a stochastically driven damped harmonic oscillator (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2017), which has been demonstrated to outperform traditional fre- quency analysis tools in recovering low S/N oscillations (Hey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=', in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The GP analysis strongly favored a model with an oscillating component with a ∆BIC=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The bottom panel of Figure 1 shows the power spectrum of the 20-second light curve centered on the power excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Solar-like oscillations are described by a frequency of max- imum power (νmax) and a large frequency separation (∆ν), which approximately scale with log g and the mean stellar density, respectively (Ulrich 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We derive νmax = 2180 ± 100 µHz, with the central value taken from the median of three solutions (pySYD, FAMED, GP), and uncertainties calculated from the scatter over individual methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The low S/N of the de- tection precludes an unambiguous detection of ∆ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Visual inspection of an echelle diagram indicates ∆ν ≈ 101µHz, consistent with the derived νmaxvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Physical Properties of HD 19467 We adopted the effective temperature (Teff) and metallic- ity ([M/H]) from Brewer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2016), derived from a line- by-line analysis of a Keck/HIRES spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Literature val- ues from spectroscopy and Gaia color-temperature relations (Casagrande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021) are highly consistent, with a range of 40 K in Teff and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='04 dex in iron abundance (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We used these ranges as an estimate for uncertainties, resulting in Teff = 5747 ± 40 K and [M/H] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='04 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' These uncertainties are smaller than those recommended by Tayar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2022), which is justified by the fact that star has properties similar to the Sun and thus suffers from smaller systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We then combined the asteroseismic νmax measurement, Gaia DR3 parallax, 2MASS K-band magnitude, Teff and [M/H] with isoclassify (Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2017) and BASTA (Aguirre Børsen-Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2022), which perform Bayesian inference of stellar parameters given input observables us- ing the stellar evolution models MIST (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2016) and 4 GREENBAUM ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2145 2150 2155 2160 2165 Time [BTJD] −1500 −1000 −500 0 500 1000 1500 2000 Flux [ppm] TESS 20s Binned 1500 1750 2000 2250 2500 2750 3000 Frequency [µHz] 0 20 40 60 80 100 120 140 160 PSD [ppm2/µHz] Smoothed Model Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Top: TESS Sector 31 light curve of HD 19467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Black points show the original 20-second cadence data, red points show the data binned to a timescale of 6 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Bottom: Power spec- trum of the data centered on the detected power excess near ≈ 2200 µHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The blue line and filled area shows the median and standard deviation of the GP model posterior BASTI (Pietrinferni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2004), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Importantly, νmax tightly constrains the surface gravity to log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='28, which combined with the radius constraint from the Gaia par- allax provides a tight constraint on stellar mass, which in turn constrains stellar age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Both tools consistently imply a mass of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='95 M⊙, which, given that the star has slightly evolved off the main-sequence (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 R⊙), implies an old age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 2 shows the age posteriors from both evolutionary models and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The final stellar parameters adopted in our study are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We adopt the self-consistent solution derived from isoclassify, but add in quadrature the difference to the BASTA results to account for systematic errors due to differ- ent model grids (Tayar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Observations of the Host Star HD 19467 Property Value Units Comments Spectral Type G3V Gomes da Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2021) Teff 5720±10 K Gomes da Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2021) Teff 5747±25 K Brewer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2016) Teff 5770±80 K Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020) Teff 5742±10 K Nissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020) Mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='953±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='022 M⊙ Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020) Mass 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='960±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='02 M⊙ This work (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2) Age 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='41+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='34 Gyr Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2021a) Age 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 Gyr Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020, Table 1) Age 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='06+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='82 Gyr Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2019) Age 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='882±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='564 Gyr Gomes da Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2021) Age 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 Gyr This work (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3) [Fe/H] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='01 dex Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020) [Fe/H] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='04 dex This work (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2) log(g) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='06 cgs Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020) log(g) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='04 cgs This work (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (Eq 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Ep 2000) 03h07m18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='570s Gaia DR3 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (Eq 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Ep 2000) −13o45′42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='419′′ Gaia DR3 Distance 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='03 pc Gaia DR3 Proper Motion (µα, µδ) (−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='694, −240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='64) mas/yr Gaia DR3 RUWE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0566 Gaia DR3 G 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='814±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='003 mag Gaia DR3 H 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='447±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='033 mag 2MASS W1 [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 µm] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='16 mag WISE W2 [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 µm] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='06 mag WISE 6 8 10 12 14 Age [Gyr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10 Density Isoclassify BASTA Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Posterior distributions for the age of HD 19467 based on isochrone modeling with isoclassify (blue) and BASTA (orange) using constraints from asteroseismology, spectroscopy and Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The Age of HD 19467 The age of HD 19467 is important for interpreting the mass and atmospheric composition of the brown dwarf companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' As already mentioned, literature estimates have a significant spread, ranging from 5–12 Gyr (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Younger, Sun-like JWST OBSERVATIONS OF HD 19467 B 5 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Adopted Stellar Parameters for HD 19467A Effective temperature, Teff (K) 5747 ± 40 Metallicity, [M/H] (dex) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='04 Luminosity, L ( L⊙) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='06 Stellar radius, R⋆ ( R⊙) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='03 Stellar mass, M⋆ ( M⊙) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='02 Stellar density, ρ⋆ (cgs) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='03 Surface gravity, log g (cgs) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='04 Age, t (Gyr) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 NOTE—Teff and [M/H] are adopted from Brewer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2016), with uncertainties accounting for the spread in literature results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' All other properties are derived from the combination of constraints from asteroseismology, spectroscopy and Gaia (see §3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' ages come from stellar rotation (Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020) and ac- tivity (Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021a), while older ages are preferred by isochrone fitting (Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The rotation-age is based on the detection of photometric pe- riod of ≈ 29 days with an amplitude of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5% from ground- based ASAS data (Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The high-precision of TESS light curve in Figure 1 rules out rotational modu- lation at the level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5% over 25 day timescale suggested by the ASAS data, which implies that the rotation period for HD 19467 is undetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This is consistent with results from the Kepler Mission, which demonstrated that typical ro- tational amplitudes in mature Sun-like stars are on the order of a few hundred ppm (McQuillan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021) and thus are generally not detectable using ground- based photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Chromospheric activity-based ages also become more challenging for stars with Sun-like and older ages due the flattening of the age-activity relation, making age constraints sensitive to small changes in R′ HK measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' While some literature values for R′ HK favor near so- lar values (and thus ages) for HD 19467, others are consis- tent with older, isochrone-based ages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' R′ HK = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3 Gyr, Lorenzo-Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The asteroseismic detection from TESS supports an older age for HD 19467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' While the low S/N precludes a direct age from a measurement of individual oscillation frequencies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Mathur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Metcalfe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Silva Aguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2017), the νmax measurement precisely constrains log g and thus stellar mass independent of stellar evolutionary models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' With a mass similar to solar (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='02 M⊙), HD 19467 must have an age significantly older than the Sun to reach a radius of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 R⊙1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' As discussed by Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020), an age of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 Gyr is compatible with the slight enhancements in alpha elements and sub-solar metallicity, 1 This analogy is only slightly affected by the sub-stellar metallicity of HD 19467;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' a solar-mass star with −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='04 has an age of ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 Gyr at solar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' placing the star in the transition region between chemical “thin-disk” and “thick-disk” stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Overall, we conclude that HD 19467 is an ≈4-5 Gyr older analog to our Sun and adopt an age of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 Gyr (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' NIRCAM DATA REDUCTION AND POST PROCESSING We use the processed images retrieved from the Mikulski Archive for Space Telescopes (MAST)2 that have been cor- rected for bad pixels, flat-fielding, and background subtrac- tion with the jwst pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The product we use are the calints files which result from Stage 2 of the pipeline and have been through a photometric calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The data were processed with calibrations software version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3 and cali- bration reference data context jwst 0943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='pmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' In addi- tion to these data, we take advantage of wavefront informa- tion provided by Optical Path Difference (OPD) maps taken by the NIRCam wavefront sensing team3 to generate a NIR- Cam PSF model close in time to our science observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' For synthetic PSFs, we utilize the OPD from 2022-08-11, R2022081102-NRCA3 FP1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='fits, the closest in time preced- ing our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' PSF Subtraction We apply principal component analysis (PCA) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Lafreni`ere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Amara & Quanz 2012) via Karhunen Lo´eve Image Projection (KLIP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Soummer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2012), to subtract the residual stellar intensity from the science frames using the images taken in two roll angles for angular di- versity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We perform PSF subtraction using the open source Python package pyKLIP (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2015) using Angular Differential Imaging (ADI) and Reference Differential Imag- ing (RDI) using a synthetic reference PSF, as described be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The results of the PCA reduction for all filters is dis- played in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' For all filters except filter F250M, the data contained in the two rolls suffices to obtain an unambiguous detection of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' For the F250M case, although the companion’s signal is visible using only ADI, we resorted to using RDI with a set of synthetic PSFs in order to confirm the signal is indeed from the companion and not due to residual speckles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' In Figure 4 we show the comparison, for the F250M filter, between only using the roll frames for the PCA reduction (ADI), and assisting the PCA reduction with a set of synthetic PSFs (ADI+SynRDI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The grid of synthetic stellar PSFs is generated us- ing WebbPSF (Perrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2014) and tools from webbpsf ext4 at offset locations with respect to the coro- 2 https://mast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='edu/ 3 https://webbpsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='io/en/latest/available opds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='html 4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='com/JarronL/webbpsf ext 6 GREENBAUM ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Post-processed images of HD 19467 in the six NIRCam filters observed in this program, rotated so that North is up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Images were reduced using the pyKLIP algorithm as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Arrows indicate the detected companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Top: Comparison between using only ADI for the PCA reduction (left), and using RDI with synthetic PSFs generated with WebbPSF (right) for the the F250M data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The addition of RDI reduces the speckle noise in the PSF-subtracted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The data are oriented so that North is up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Bottom: The forward model compared with the PSF-subtracted data for F250M, using the sythetic PSFs as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' F250M F300M F360M 2 2 [arcsec] 0 0 0 ec + 0 2 2 2 i 0 1 2 2 1 0 1 2 2 1 0 1 2 F410M F430M F460M 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 2 20 2 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 10 15 1- 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 ((mly) 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 0- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 0 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 1- 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 2 2 1D 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 2 1 0 1 2 1 0 2 2 i 0 1 RA [arcsec]ADI Only ADI + SynRDI 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 (Aw) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 Flux 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5Data Best-fit Model Residuals 15 15 15 Counts (DN) S 10 0 10 2 (pixel: 0 > 5 5 2 0 0 0 0 10 0 10 0 10 X (pixels) X (pixels) X (pixels)JWST OBSERVATIONS OF HD 19467 B 7 nagraph focal plane mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We generate simulated PSFs in different sets of 9-point grid pattern at even spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We simulate spacings of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5, 7, 15, 25, 40 mas, in addition to a set of rotations of the coronagraphic-PSF with respect to the detector of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This aims to emulate the speckles present in the data frames, and assists the PCA reduction with the diversity in speckle structure needed to perform a more optimal reference subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' As mentioned above, for filters F300M, F360M, F410M, F430M, and F460M, ADI suffices for a clear detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' As a second step we use RDI with the synthetic PSFs to further subtract the unwanted starlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This is motivated by the fact that the companion PSF’s northern lobe falls near the diffrac- tion speckles caused by the bar coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The num- ber of Karhunen Lo´eve (KL) modes determines how much of the synthetic PSFs are used for the subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Since these have been generated with arbitrary offsets, there is a risk of subtracting the light from the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We use 15 KL modes, which minimizes over-subtraction and clears out slightly more of the residual starlight around the northern lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This was done by visual inspection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' a more in-depth analysis on how to optimally use synthetic PSFs will be ex- plored in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Photometry To accurately extract the flux and position of HD 19467 B, we account for over-subtraction effects on the PSF that arise during the reduction process (described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1) with a forward model based on the method described by Pueyo (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We make use of its implementation on pyKLIP (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The companion PSF is modeled using WebbPSF (Perrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2014) for each filter and accounting for its position with respect to the bar focal plane mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' An accurate position of the simulated PSF is particularly impor- tant in the case of the shorter wavelength filters: poorer spa- tial sampling of the pixels compared to the diffraction limit at smaller wavelengths (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 µm is sub-Nyquist) results in an acute sensitivity of the PSF structure as seen in the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' An accurate positioning for the case of F250M was done by trial and error simulating a grid of PSF offsets and selecting the best fit by least square difference between the simulation and the coadded science frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The model PSF are sim- ulated using the OPD map closest in time and prior to the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The flux and position of the companion are extracted with pyKLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We fit a model of its photometry and astrometry to the reduced data using an MCMC approach (emcee;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 5 shows the PSF model fit to the reduced data for filter F360M, the filter in which we obtain the highest SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Appendix A contains the full gallery of forward model comparisons with the PSF- subtracted images in each band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The best fit model to the PSF-subtracted signal used to measure the photometry and astrometry of HD 19467 B in filter F360M, where the highest SNR was achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The residuals show the companion is fit well by the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Adopted Photometry For HD 19467 A Filter Flux (Jy) F250M flux (Jy) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='07 F300M flux (Jy) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='63±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='05 F335M flux (Jy) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='04 F360M flux (Jy) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='04 F410M flux (Jy) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='49±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='03 F430M flux (Jy) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='03 F460M flux (Jy) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='02 NOTE—Predicted fluxes in JWST wavebands are based on BOSZ stellar models (Bohlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The flux calibration of the signal is determined based on the jwst stage 2 pipeline photometric calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We apply a flux correction to the photometry based on measured atten- uation factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='92 of the Bar mask Lyot stop at ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We fit a G3V stellar photosphere model to 1-5 µm photom- etry from 2MASS (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2003) and WISE (Cutri & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We also find a ∼ 2% error in fitting the stellar model to the IR measurements, and apply this error to con- trast reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Table 4 shows the estimated stellar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Com- parison of the calibrated flux measured from the acquisition and astrometric confirmation images, both taken through the neutral density square, produced from the stage 2 pipeline is consistent with the estimated stellar spectrum within ∼ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We therefore apply a 10% uncertainty to reported absolute photometry of HD 19467 B in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 6 shows the measured photometry in each NIR- Cam band alongside recent measurements and limits from the ground (Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The F460M flux is consistent with the M band upper limit obtained with VLT NaCo Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020), however there appears to be some tension with the NaCo L’ flux compared with F360M and F410M photometry measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2022) also noted a discrepancy in measurements from NaCo L′ and JWST NIRCam photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The difference in passband on Data Best-fit Model Residuals 20 20 20 Y (pixels) 5 Counts (DN) 10 10 1o 0 上0 0 0 0 10 20 0 10 20 0 10 20 X (pixels) X (pixels) X (pixels)8 GREENBAUM ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' New NIRCam photometry (blue stars) compared with recent ground-based measurements from VLT-SPHERE and VLT- NACO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Left: An example of the raw MASKLWB coronagraph data for an image in the F360M filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Right: The best fit model PSF simulated using the most recent preceding OPD map, used to measure the centroid of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' a steeply rising part of the spectrum, possible water vapor effects, as well as calibration uncertainties may account for this discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Continued refinement of JWST photomet- ric calibrations will help identify any biases in photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' For this study, we do not incorporate NaCo photometry into the analysis, but rather present the measurement comparison for future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Table 5 shows our measured photometry and relative as- trometry for HD 19467 B (see next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Relative Astrometry A major challenge of obtaining relative astrometry of a companion in coronagraphic imaging is that the primary star is occulted by the focal plane mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Knowledge of the wave- front from published OPD maps, and a highly structured PSF enable a forward model based cross-correlation with the data to fit for the centroid of the star behind the mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We per- form a cross-correlation of model PSFs with the data using the chi2 shift in the image-registration Python package5 to measure the best fit position of the star behind the 5 https://image-registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='io/ Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Astrometric position of HD 19467 B relative to its parent star, compared with previous measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' mask (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We obtain a centroiding error ∼7 mas, con- sistent with the measured sensitivity in Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The companion position is recovered with the joint astrom- etry and photometry model fit to the reduced data as de- scribed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The model fit errors provide the un- certainty in the relative position to the measured star position on the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We add the star position uncertainty to the reported errors (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 8 shows the new astrometric measurement compared to previous relative astrometry mea- surements of HD 19467 B (Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2014, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Performance and Sensitivity In Figure 9 we show the contrast curves for the reduced images after PSF subraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The contrast is measured with PyKLIP by computing the noise in an azimuthal annulus at each separation, using a Gaussian cross correlation to remove high frequency noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The flux normalization to obtain these contrast numbers was computed as explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2, by using a best fit model of the stellar spectrum to calibrate contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The contrast curves are corrected for algorithmic throughput, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' the throughput loss due to the PSF subtrac- tion, and for small sample statistics (Mawet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 10 translates our detection limits from flux/contrast sensitivities to limits on companion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Three differ- ent brown dwarf evolution models are considered – Ames- COND (Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2003), BEX-HELIOS (Linder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2019), and Sonora-Bobcat (Marley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' In each case, we assume solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' While the shortest wavelength observations achieve the best contrast (in particular F300M), the longer wavelengths are better at detecting lower mass companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We find an overall detection limit of ∼10 MJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This limit is much higher than the sub-Jupiter levels that JWST/NIRCam can obtain for young systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2022), but even for this 740 C14 C15 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='66 760 B20 780 M20 (seu) + This work 800 ADec 820 一840 + 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='61 860 880 1340-1360-1380-1400-1420-1440-1460-1480 ARA (mas)JWST Filters This work Maire+2020 10-9 Mesa+2020 10-10 10-11 0000T 15000 20000 25000 30000 35000 40000 45000 50000 Wavelength (A)Data Model PsF 0 2DD 0 200 175 175 10 10 15D 150 20 20 125 125 30 1DD 30 100 DN 75 75 40 40 50 50 50 50 25 25 0 10 20 30 40 50 0 10 20 30 40 50 PixelsJWST OBSERVATIONS OF HD 19467 B 9 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' NIRCam measurements of HD 19467 B Separation Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Angle ∆mag Flux Filter (′′) (deg) (mag) (µJy) F250M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='597±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='010 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='14 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='271 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='96±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='75 F300M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='611±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='002 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='04 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='62±0.' metadata={'source': 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2022-Aug-12 NOTE—The astrometric precision for each filter is based solely on the posi- tional uncertainty relative to the center of the coronagraph mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The com- bined astrometry includes an additional term to account for uncertainty in the stellar position behind the mask (7 mas in each direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' very old system brown dwarfs are easily detectable outside of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4′′ ≃ 10 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' While the detection limit is relatively in- dependent of model, it does depend significantly on the age of the brown dwarf (the system age is discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Despite a lack of reference star observations, we are able to recover the signal of HD 19467 B with two roll angles and achieve contrasts ∼10−5 at 1–2 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Regular OPD measurements enable the use of synthetic PSFs that can aid PSF subtraction by generating a set of reference PSFs to cap- ture speckle structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This suggests that bright companions could be observed without reference stars, significantly re- ducing the time spent on the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Future work will investigate the difference between reducing data with and without reference star observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Future observations with a better defined position for the LWB coronagraph should also provide better contrast close-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' ORBIT OF HD 19467 B Previous studies estimate the mass of HD 19467 B from 51 to 86 MJ through both model-based estimates and orbital analyses (Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We analyze new radial velocities and provide an up- dated dynamical mass estimate including our new relative as- trometry and additional RV measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' First, we fit the new and previously measured RVs from HIRES and HARPS (Trifonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020) using the RadVel6 software (Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' With the addition of the new data, we measure a linear slope term of ˙γ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='00412 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='00027 m s−1 d−1 with strong significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We attempt to fit for curvature and tentatively detect a curva- ture term of ¨γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='78 × 10−7 m s−1 d−2 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Model comparison using ∆BIC and ∆AIC (Aikike Information Criterion, Burnham & Anderson (2002)) show a nearly in- 6 https://radvel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='io/en/latest/ Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Imaging Astrometry epoch−2450000 Filter ρ (mas) ρerr PA (deg) PAerr Astrometry from Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2014) 5804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1 K’ 1662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='9 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='19 5933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 H 1665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='7 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='38 6166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1 K’ 1661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='15 6205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 Ks 1653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='14 Astrometry from Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020) 8032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3 L’ 1637 19 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='47 8061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 K1 1636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='13 8061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 K2 1634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='21 8409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3 H2 1631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='12 8409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3 H3 1631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='12 New Astrometry (this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' see Table 5) 9803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6µm 1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 7 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='25 distinguishable model fit to a trend-only and trend plus cur- vature model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' A detection of curvature can place strong con- straints on the companion orbit, especially for higher eccen- tricity systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 11 shows the RV data plotted over the maximum likelihood model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Appendix 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 contains a more detailed description of the fit comparison and the new radial velocities used in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' For the full orbital analysis we include all available RV measurements from HARPS (Trifonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020) and HIRES (HIRES data including new measurements tabulated in Appendix B), relative astrometry (Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2014, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020) (listed in Table 6), and absolute astrometry from Hipparcos and Gaia as de- scribed in Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2021a), which takes advantage of proper motion anomalies between Hipparcos, Gaia EDR3 and the Hipparcos-Gaia long-term trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We utilize the cross-calibrated catalog of Hipparcos-Gaia accelerations pre- sented in Brandt (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 10 GREENBAUM ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 0 1 2 3 4 5 6 Separation (arcsec) 16 17 18 19 20 21 22 5-σ Sensitivities (mag) F250M F300M F360M F410M F430M F460M 10-7 10-6 10-5 10-4 5-σ Contrast 0 50 100 150 200 Separation (AU) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Contrast curves for all filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Solid lines indicate ADI, and dashed lines indicate ADI and RDI using synthetic PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Data points indicate the HD 19467 B detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The use of synthetic PSFs provides the diversity necessary to obtain enhanced contrast at small angular separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We use orvara (Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021b) to fit orbits to the radial velocities, absolute astrometry, and relative astrome- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' orvara is an orbit fitting code that uses ptemcee, a parallel tempered MCMC scheme (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Vousden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Following the orbital analysis in Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2021a), we apply a geometric prior to incli- nation and log-flat priors to semi-major axis and companion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We apply uniform priors to remaining orbital elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' log-flat priors are applied to RV jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We adopt the mass of M∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='96 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='02 M⊙ based on the analysis in §3 using asteroseismology, spectroscopy, and Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We first predict the position of HD 19467 B in the current epoch leaving out the new relative astrometry measured with NIRCam, but including all other data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 12 shows that our measurement is consistent with the prediction of the best fit orbits using previous measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Next, we fit for orbital parameters including our new rel- ative astrometry measurement from NIRCam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Table 7 sum- marizes the orbit fit results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We infer a mass of 81+14 −12 MJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Our mass estimate for HD 19467 B is within 1-σ of the prior estimates from in Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2021a) (65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='9 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 MJ), and Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020) (74+12 −9 MJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We infer an eccentricity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='416+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='092 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='07 , which is consistent with recent measurements in Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2021a), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='11, Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='09, and Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We infer a period of 386+220 −108 yr, which is consistent with prior orbital analyses (Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The tentative evidence for curvature from the new radial velocities may indicate that the orbit is close to perias- tron passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Given the high eccentricity, it could be a critical time to monitor this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 13 shows a selection of orbits from MCMC poste- riors overlaid with the relative astrometry used in the fit, and Figure 14 displays a corner plot of the MCMC posteriors for orbital parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' ATMOSPHERE AND EVOLUTION MODEL COMPARISON In the following sections we show a preliminary compar- ison of our near-IR photometry from NIRCam with brown dwarf atmospheric models, focusing on the Sonora models (Marley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Karalidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' In our spectral fit- ting, we only use the model spectral grid with solar carbon- to-oxygen ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The Sonora-Bobcat cloudless atmospheric models assume that the atmospheric composition is in thermo-chemical equi- librium and solve radiative transfer equations for a self- consistent temperature-pressure profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The model grid cov- ers temperatures from 200 to 2400 K, gravity from 10 to 3160 m s−2, and metallicities from [Fe/H]=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The model spectra have a spectral resolution ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 to 20 µm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' JWST OBSERVATIONS OF HD 19467 B 11 0 1 2 3 4 5 6 Separation (arcsec) 10 100 Detection limit (MJup) COND (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 Gyr) F250M F300M F360M F410M F430M F460M 0 50 100 150 200 Separation (AU) 0 1 2 3 4 5 6 Separation (arcsec) 10 100 Detection limit (MJup) BEX (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 Gyr) F250M F300M F360M F410M F430M F460M 0 50 100 150 200 Separation (AU) 0 1 2 3 4 5 6 Separation (arcsec) 10 100 Detection limit (MJup) Sonora (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 Gyr) F250M F300M F360M F410M F430M F460M 0 50 100 150 200 Separation (AU) 0 1 2 3 4 5 6 Separation (arcsec) 10 100 Detection limit (MJup) HD 19467 B Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Model COND BEX Sonora 0 50 100 150 200 Separation (AU) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Detection limits (5−σ) for each filter, in terms of companion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The first three panels show limits for three different atmospheric evolution models (COND, Bex, and Sonora), while the last panel compares the overall detection limit for each of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' A mass estimate from atmospheric model fitting (Section 6) is shown as a point in the lower right figure (61 MJ at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='61′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Orbit fit results Parameter Units Value Jitter m/s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='17+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='23 Mpri M∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='961+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='022 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='022 Msec MJ 81+14 −12 Semi-major Axis AU 53+19 −10 √e sin ω −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='586+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='068 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='061 √e cos ω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='08+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='32 Inclination deg 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='9+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='7 Ascending Node deg 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='9+240 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3 Mean Longitude deg 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='7+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='7 −123 Parallax mas 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='226+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='037 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='037 Period year 382+220 −108 Argument of Periastron deg 278+27 −29 Eccentricity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='416+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='092 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='070 Semi-major Axis mas 1664+598 −328 T0 JD 2486667+51547 −4207 Mass Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='080+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='014 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='012 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Best fit single companion Keplerian orbital model for HD 19467 to radial velocities measured for HD 19467 by HIRES by the California Legacy Survey (Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021), new HIRES observations, and HARPS (Trifonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The fit slightly favors a curvature term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The Sonora-Cholla cloudless models assume chemical dis- equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' These models assume that atmospheres have so- lar metallicity (Lodders 2010) with cloud-free atmospheric 2000 2010 2020 40 a) HARPS 30 HIRES HIRES pre 2004 20 10 0 10 20 30 tb) Residuals 20 0 2012 GREENBAUM ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Orvara prediction of relative astrometry at the 2022-08- 12 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The measured astrometry is consistent with the predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The models are computed using the Picaso v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 atmospheric model (Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' By in- cluding an eddy diffusion parameter, Kzz, that ranges from 102 − 107 cm s−2 as an input parameter, the Cholla models simulate the dynamical mixing that drives various molecular species like CH4, CO, H2O, and NH3 out of their thermo- chemical equilibrium abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The Cholla models span over a temperature grid of 500 to 1300 K and a gravity grid from 56 to 3160 cms−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' First, we perform a basic χ2 comparison of new JWST photometry to the Sonora models (Marley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Kara- lidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We also perform an MCMC fit of the model grids with both our new photometry and previously published medium resolution spectrum obtained with SPHERE-IRDIS 4 2 0 2 4 (arcsec) 4 3 2 1 0 1 2 3 4 [arcsec] 1990 2000 2010 2020 2030 Astrometric Orbits 80 100 120 140 160 180 Mcomp(MJup) Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' A selection of orbits from the MCMC posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' LSS (Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020) and ground-based photometry from SPHERE-IRDIS Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' From this we derive a bolometric luminosity and determine model-dependent mass estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Model Comparison with NIRCam Photometry Only We compare the NIRCam photometry with both the Sonora-Bobcat and Sonora-Cholla model grids to highlight the broad features of the 2 − 5 µm flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We allow the ra- dius scaling to vary when fitting the models to our NIRCam photometry, which is generally consistent with radii much smaller than, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=', those predicted by the Sonora-Bobcat grid evolutionary tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We find that the Bobcat models do not simultaneously cap- ture both the local flux peak at 3 µm and the steep drop in flux from 4−5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' In general the Bobcat grid favors higher grav- ity and lower or zero metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' However, none of the fits capture all of the photometry perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' A model spectrum at effective temperature of Teff = 1400K best matches the data, but does not fully capture the peak flux at ∼ 4 µm with the drop off at longer wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Gravity and metallicity do not have a strong effect on the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The Sonora Cholla models, which include effects of dis- equilibrium chemistry, provide better fits to the photometry, but suggest a low value for gravity as well as small radius to scale the flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The best fit model has Teff = 1200K, g = 31, and log(Kzz) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 15 (right) shows the best fitting Cholla model, with a few model grid points that vary in temperature, eddy diffusion parameter, and gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We note that while lower gravity is favored, the gravity does not strongly affect the shape of the photometry-only profile, as can be seen in the third panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Near-IR spectroscopy with JWST will be able to further distinguished features of grav- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Overall, the best fit Sonora-Cholla model provides a bet- ter fit than the best fit Sonora-Bobcat model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' As can be seen in Figure 15, the Cholla models are better able to simultane- ously capture the lower flux at shorter wavelengths, the peak at ∼4 µm, and the subsequent drop in flux, favoring a lower temperature that is more consistent with previous estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This suggests that modeling disequilibrium chemistry may be necessary for understanding the atmosphere and evolution of HD 19467 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Model Fit to 1-5 µm We fit the Sonora-Cholla cloudless models to the com- posite dataset that consists of JWST photometry, the IRDIS spectra (Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020), and the SPHERE K12 band pho- tometry (Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020) to constrain the temperature and gravity of HD19487B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' For the IRDIS spectra, we exclude spectral points with negative flux values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We include an ad- ditional uncertainty of 7%, similar to the J- and H-band ab- solute flux uncertainties of HD 19467 (Table 7 of Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 865 LocationofHD19467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='61 870 875 (mas) 880 885 890 895 900 905 1330 1340 1350 1360 Aα (mas)JWST OBSERVATIONS OF HD 19467 B 13 Mpri (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='961+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='022 60 120 180 240 300 Msec (MJup) Msec (MJup) = 81+14 12 40 80 120 160 200 a (AU) a (AU) = 53+19 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='60 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='416+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='02 Mpri (M ) 105 120 135 150 i ( ) 60 120 180 240 300 Msec (MJup) 40 80 120 160 200 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='60 e 105 120 135 150 i ( ) i ( ) = 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='9+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='7 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Corner plot showing MCMC posteriors for select orbital parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020), to account for the possible offset in the absolute flux levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We linearly interpolate the Cholla model spectral grid to a finer grid with smaller step sizes in the temperature, grav- ity, and eddy diffusion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' In addition to the three parameters, the other free parameter in the spectral fitting is the scaling factor, which is the square of the ratio of brown dwarf radius to the distance (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='03 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We use pyphot 7 to calculate the broadband photometries of model spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 7 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='com/mfouesneau/pyphot Our model fitting algorithm minimizes a cost function that sums the squared difference between the model spectra and data, weighted by the observational uncertainties, σ, and the intensities, w, as shown in the following: cost function = ΣN i=1{wi Fλ,i(model) − Fλ,i(data) σi }2 wi = Fλ,i∆λi max(Fλ,i∆λi) (1) where ∆λi is the wavelength coverage of a datapoint and the intensity weighting is normalized by the highest intensity among the datapoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We include the intensity weighting in the cost function so that a photometric point with high inten- 14 GREENBAUM ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Left: Measured photometry from NIRCam plotted alongside a few closely matching scenarios from the Sonora-Bobcat models, letting the radius vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Right: Measured photometry from NIRCam plotted alongside a few closely matching scenarios from the Sonora-Cholla models, which include chemical disequilibrium parameterized by the eddy diffusion parameter, log(Kzz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' sity carries more weight in the fitting process than a spectral point with low intensity even though both could have a simi- lar signal-to-noise ratio, since the photometric points contain a larger fraction of total flux measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We then use emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2017) to sample the posterior distri- bution of the fitted parameters with the Markov Chain Monte Carlo (MCMC) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We adopt a uniform prior of tem- perature, logarithmic gravity, and eddy diffusion parame- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We run the MCMC chain with 200 walkers for 20,000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Based on the posterior distribution of the MCMC chains, we derive the best-fit temperature of 1080 ± 22 K, gravity of log(g) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='60+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1, eddy diffusion parameter of log Kzz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4, and radius R of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='03RJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The value for radius is lower than the expected radius at 9 Gyr, ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 RJup, according to the evolution model grid at simi- lar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Prior modeling work has noted a common discrepancy between the expected radius from evolutionary models and the radius required to match the flux of a given temperature that best fits atmospheric models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=', Barman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Marley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Lavie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020) explored radius agreement with evolutionary tracks with different model atmospheres that included various levels of clouds in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Future observations, especially spectroscopy will further help constrain atmospheric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We draw 100 parameter sets from the posterior distribution and plot the corresponding model spectra in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Fig- ure 16 suggests that the model spectra provide a qualitatively good match to the data but there are some significant residu- als, especially in the K-band photometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We remain cau- tious about the inferred parameters and the seemingly small uncertainties given the imperfect fit between the data and model spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' However, while absolute flux calibration may Sonora Bobcat Best fit: T=1400, logg=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5, m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0, R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='23Mjup 20 T=1300, m=+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 LB T=1400 →- T=1500, m=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 16 →-T=1600, m=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 14 12 1D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 ADODE DOSE 40000 45000 54000 55000 e1 2D log(g)=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5, T=1300 LB log(g)=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='75, T=1300 log(g)=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0, T=1300 16 →log(g)=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='25, T=1300 + log(g)=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 14 12 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 25000 DODE DOSE 4500D 5500D e-1 2D 0=w 1B m=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 →= m=+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5, log(g)=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0, T=1300 16 14 12 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 25000 ADOE DOSE 40000 45000 55000 Wavelength (i)Sonora Cholla le-l log(9]=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' log(Kzz=2) 2D T=1200K, R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='44, log(g)=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 LB T=1150K,R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='47 T=1100K, R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='51 16 T=1050K, R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='55 T=1000K, R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='61 14 12 1D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 DO5Z ADODE DOSE 40000 45000 54000 55000 e-1 200 log(Kzz)=2, log(g)=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5, T=1100 175 log(Kzz)=4, log(g)=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5, T=1100 log(Kzz)=7, log(g)=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5, T=1200 150 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='25 25000 DODE DOSE 40000 45000 55000 e- log(Kzz)=2 2D log(g)=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5, R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='51, T=1100 1B log(g)=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3, R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='47, T=1150 log(g)=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='7, R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='44, T=1200 16 14 12 1D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 25000 ADOE ADOSE 40000 ADOS DOS 55000 Wavelength (i)JWST OBSERVATIONS OF HD 19467 B 15 account for some discrepancy between data sources, the dis- crepancy alone is not enough to account for inconsistencies between the best fit atmospheric model and evolutionary grid predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Clouds, which likely drive the rotational mod- ulation of many T dwarfs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='g Manjavacas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2019) and are not included in the Cholla models, could play a key role in shaping the HD 19467 B emission spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Future simul- taneous observation in the near-IR and mid-IR region will be useful for testing the role of clouds in the atmosphere of HD 19467 B with well-constrained age and host star metal- licity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Bolometric Luminosity The 2-5µm JWST NIRCam broadband photometry, in combination with the ground-based near-infrared spectra and photometry, are crucial for pinning down the bolo- metric luminosity of a ∼1000 K object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We integrate the flux density over observed data including the previously published IRDIS-LSS spectra (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='97-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='335 µm & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='50-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='80 µm) , ground-based K-band photometry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='059-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='161 µm & 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1965-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='3055 µm), and the six JWST NIRCam broad band photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The flux integral in the observed wavelength re- gions is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 × 10−6L⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We utilize the fitted model spectra in Section 6 to extrap- olate flux density beyond the observed wavelength region and estimate the bolometric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We find that the observational data accounts for around 72% of the bolomet- ric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The estimated total bolometric luminosity is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2) × 10−6L⊙, or log(L/L⊙) = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Based on the combination of JWST NIRCam high-precision photometry and ground-based data, our results suggest that we can accurately derive the bolometric luminosity at 4% precision level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Based on the independently estimated age and bolometric luminosity, we then use the Bobcat evolution model to es- timate mass of HD 19467 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' After linearly interpolating the mass as a function of age and bolometric luminosity, we de- rive that the mass is 62 ± 1MJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 17 shows the Sonora Bobcat evolution model and where our bolometric luminos- ity estimate lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' By comparing the mass derived from the age and Lbol to the dynamical mass, (81+14 −12MJ), we conclude that the two masses are consistent with each other within about two-sigma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The NIRSpec IFU observations planned for later this year (PID #1414) will provide a much more complete characteri- zation of the atmosphere of HD 19467 B, helping to pin down parameters such as metallicity and Teff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' CONCLUSIONS We have demonstrated the performance of JWST NIRCam LWB coronagraph on the known binary system HD 19467 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Despite missing reference star observations, we are able to recover the companion with high significance in all 6 medium NIRCam bands used for the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The main results of this study are as follows: The MASKLWB coronagraph works well for sepa- rations below 1 arcsec (for medium filters excluding F250M) at contrasts 10−5 and better, even without a reference star when angular diversity is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This is expected to improve in the near future when corona- graphic mask locations are refined through instrument calibration observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Given the superb stability of JWST, and regular OPD measurements available, we are able to incorporate synthetic reference images to further subtract speck- les, following ADI subtraction, and improve SNR on the detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Future observations with reference ob- servations can be compared with the results presented in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We estimate the age of the HD19467 system by com- bining spectroscopy and Gaia astrometry with astero- seismic constraints from TESS, finding an age of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 Gyr, supporting older estimates of the age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We pro- vide updated parameters for the host star HD19467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We estimate a dynamical mass of HD 19467 B of 81+14 −12MJ, contributing new relative astrometry from NIRCam and radial velocities from HIRES, and detect tentative evidence of curvature in the orbit fit to the radial velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' A comparison of atmospheric and evolutionary models to our new 2 − 5 µm photometry favors models that include disequilibrium chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' A global fit to the photometry and spectroscopy from this study and ground-based observations show some tension between the instrument-to-instrument relative fluxes and the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The model-derived mass of 62 ± 1 MJ is lower than the dynamical mass estimate, but within 2-σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The NIRCam observations provide the highest fidelity 3−5 µm photometry to date of HD 19467 B, and give an early test of atmospheric and evolutionary models that JWST will continue to test throughout the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Future observations with NIRSpec (PID #1414) will further elucidate discrepan- cies in model spectra and help characterize the chemistry of HD 19467 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Improvements in the near future through instru- ment calibration observations will further refine the perfor- mance of the coronagraphic mask placement and the perfor- mance of the MASKLWB mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 16 GREENBAUM ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The comparison of the fitted Cholla model spectra (light blue lines) to the SPHERE/IRDIS-LSS 1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 µm spectra (gray lines), SPHERE/IRDIS-K12 photometry (gray diamonds), NaCo L’ band (pink triangle), and JWST NIRCam photometry (golden hexagon) show that it is challenging for the models to simultaneously explain the near-IR and mid-IR spectra and photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The semi-transparent light blues lines are the 1000 Cholla cloudless model spectra sampled from the posterior distribution of MCMC fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The blue squares are the effective photometry from the model spectra samples in the corresponding NIRCam filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The transmission curves of JWST NIRCam broadband photometry are plotted in colored lines at the bottom of the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The y-axis is in unit of intensity (Wm−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Based on the bolometric luminosity and age, the Bobcat evolution models suggest that HD 19467 B has a mass of 62 ± 1 MJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The colors indicate the masses predicted by Bobcat evolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' White contour lines show the bolometric luminosity and age with a fixed mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The uncertainty of HD 19467 B’s bolometric luminosity is around 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank the anonymous reviewer for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The authors acknowledge useful discussions with G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Mirek Brandt and Eric Nielsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We also thank Dino Mesa for providing the data for the near-IR spectrum of HD 19467 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Some of the research described in this publication was carried out at the Jet Propulsion Laboratory, California In- stitute of Technology, under a contract with the National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' is supported by NRC Canada and by an NSERC Discovery Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='C.’s research was supported by an appointment to the NASA Postdoctoral Program at the NASA Ames Re- search Center, administered by Oak Ridge Associated Uni- versities under contract with NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' ac- knowledge support from the Research Corporation for Sci- ence Advancement (Scialog award #26996) and the Na- tional Science Foundation (AST-2009828).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' also ac- knowledges support from the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Sloan Foundation and the National Aeronautics and Space Administration (80NSSC21K0652,80NSSC22K0303).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=" Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at WeightedChollamodelsfittingresults fitted models modelphotometries SPHERE IRDIS Spectrum SPHEREK12photometries JWSTNIRCamphotometries NaCo L' 10~16 (zw/M) 5 10~17 2 3 5 10 Wavelength (μm)BobcatEvolutionmode Mass (Mjup) 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 70 70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 65 (L/Lo) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 60 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 65 60 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 61 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='6 65 55 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 50 5 6 7 9 10 11 12 13 Age (Gyr)JWST OBSERVATIONS OF HD 19467 B 17 the Space Telescope Science Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The specific obser- vations analyzed can be accessed via 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='17909/v8m9-xk98 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='17909/t9-st5g-3177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This research has made use of the SIMBAD database, op- erated at CDS, Strasbourg, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Some of the data presented herein were obtained at the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Keck Observatory, which is operated as a scientific part- nership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The Observatory was made possible by the generous financial support of the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Keck Foun- dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' We are most fortunate to have the opportunity to conduct observations from this mountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Software: SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020), NumPy (Har- ris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2020), Matplotlib (Hunter 2007), WebbPSF (Perrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2014), PyKLIP (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2015), synphot (Lim & Hanley 2016), pysynphot (STScI Development Team 2013), pyphot (Fouesneau 2022) APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' KLIP FORWARD MODEL OF THE DATA We compute a forward model of the PSF-subtracted image according to Pueyo (2016) to account for over- and self-subtraction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' ADI imaging is particularly susceptible to self-subtraction effects, especially when there is little angular diversity, such as our data, which contains only two roll angles separated at ∼ 8 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Modeling these effects is essential for appropriately estimating the properties of the signal (flux, position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 18 displays the comparison between the forward model and the PSF-subtracted image corresponding to the images displayed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' KEPLERIAN ORBIT FIT TO HD 19467 RADIAL VELOCITIES We fit a Keplerian Orbit model to radial velocities of HD 19467 from HIRES and HARPS instruments, the latter published in Trifonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' (2020), catalog JA+A636A74rvbank table entries DRVmlcnzp and e DRVmlcnzp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' HIRES radial velocities, along with the new measurements are presented in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Using RadVel (Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' 2018), we determine that a model with a curvature term is slightly favored over a model without curvature, however the trend-only and trend plus curvature models are nearly indistinguishable (Table 9), based on ∆BIC and ∆AIC metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' This is the first tentative evidence of curvature measured for HD 19467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Future follow up is needed to further support this finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' THE MCMC POSTERIOR DISTRIBUTION FOR SPECTRAL FITTING Figure 19 shows the MCMC posterior distribution of the spectral fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The spatial structure seen in the posterior distribution reflects the step size of temperature (10 K), gravity (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='025 dex), and log(Kzz) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='167) in the interpolated model grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The sharp boundaries of radius at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 RJup is reflects the lowest limit of radius range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 RJup) in the model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The KLIP forward model that accounts for over subtraction effects compared to the PSF-subtracted data in each of size filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='Best-fit Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='Residuals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='15 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='Counts (DN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='(pixel: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='> ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='X (pixels) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='X (pixels) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='X (pixels)Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='Best-fit Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='Residuals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='Y (pixels) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='Counts (DN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='1o ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='上0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='X (pixels) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='X (pixels) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='X (pixels)18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='GREENBAUM ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' HIRES Radial Velocity Observa- tions Instrument BJDT DB RV RV Error HIRES 2450366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='019 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='47 HIRES 2450418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='943 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='17 HIRES 2450461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='84 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='30 HIRES 2450715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='103 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='20 HIRES 2450716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='111 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='31 HIRES 2450786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='847 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='38 HIRES 2450786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='86 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='51 HIRES 2450806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='904 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='50 HIRES 2450837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='744 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='41 HIRES 2450839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='743 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='49 HIRES 2451012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='119 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='40 HIRES 2451013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='12 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='31 HIRES 2451070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='116 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='26 HIRES 2451072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='984 29.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='103 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='16 HIRES 2459787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='129 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='20 JWST OBSERVATIONS OF HD 19467 B 19 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' Model Comparison AICc Qualitative Comparison Free Parameters Nfree Ndata RMS ln L BIC AICc ∆AICc AICc Favored Model ˙γ, ¨γ, σ, γ 8 128 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='40 330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='13 695.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='02 673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='00 Nearly Indistinguishable ˙γ, σ, γ 7 128 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='40 332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='43 692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='91 673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='47 Ruled Out σ, γ 6 128 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='50 426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='33 872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='85 856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='43 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='02 Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' The posterior distribution of the MCMC fitting of Cholla models to the SPHERE-IRDIS spectra, SPHERE K12 photometry, and the JWST NIRCam photometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content=' T (K) = 1081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='81187 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='48 log(g) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='56±:16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 (6)60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 R(Rjup) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='62±0:83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='03 R (Rjup) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='55 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='60 70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FJT4oBgHgl3EQfISxO/content/2301.11455v1.pdf'} +page_content='0 O T (K) log(g) Kzz R (Rjup)20 GREENBAUM ET AL.' metadata={'source': 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Agouri1, A. Waqdim1, A. Abbassi1,∗, M. Ouali1, S. Taj1, B. Manaut1,†, M. Driouich1 +1 Laboratory of Research in Physics and Engineering Sciences, +Sultan Moulay Slimane University, Polydisciplinary Faculty, Beni Mellal, 23000, Morocco. +February 1, 2023 +Abstract +In the present paper, the electronic, structural, thermodynamic, and elastic properties of cubic +PbGeO3 perovskite oxide are presented and computed using the WIEN2k code. The structural +properties have been evaluated and they are in good agreement with the theoretical and exper- +imental data. A phonon dispersion is made and it reveals that the cubic PbGeO3 perovskite is +dynamically stable. In addition, the electronic properties of PbGeO3 shows an opening gap energy, +meaning a semiconductor behavior with an indirect band gap equal to 1.67 eV . Moreover, the +obtained elastic constants of cubic PbGeO3 satisfy Born’s mechanical stability criteria, and this in- +dicates that our compound behaves as a stable ductile material. The temperature-pressure effects +on thermodynamic parameters are investigated using the Gibbs2 package. Finally, based on the +obtained results about the cubic PbGeO3 perovskite properties, we assume that this compound +will have potential applications. +Keywords: DFT, Elastic, Thermodynamic, Perovskite, WIEN2K, mBJ. +∗Corresponding author, E-mail: abbassi.abder@gmail.com +†Corresponding author, E-mail: b.manaut@usms.ma +1 +arXiv:2301.13493v1 [cond-mat.mtrl-sci] 31 Jan 2023 + +2 +1. +Introduction +Since the discovery of CaTiO3 [1], the perovskite oxides family has been a major subject of interest, +and this is mainly due to its multi-functional character [2,3]. It has received great attention for its +exploitation in many applications such as solar cells [4], spintronic and optoelectronic [5, 6]. The +exploitation of these materials mainly depends on their flexible structure, variable formula, and also +their various properties [7,8]. Consequently, a lot of experimental and theoretical studies have proved +that the perovskite oxides family has unique physical properties like photoelectric [9], magnetic [10], +ferroelectric [11],.. etc. +Pb-based perovskite oxides have long been investigated for their rich and interesting properties. +They are considered proper candidates in energy conversion such as in piezoelectric and ferroelec- +tric devices [12]. +Recently, several experimental and theoretical studies have been interested on +Lead-based perovskite oxides. The unit cell compounds of PbXO3 (X = Ti, V ) were synthesized, +and their crystal structures are determined using Neutron and X-ray diffraction [13, 14]. +In ad- +dition, these compounds have largely been used in ferroelectric and optical sensors [15, 16]. The +PbZrxTi1−xO3 (x = 0, 0.4, 0.6, 1) were prepared and investigated, proving the exploitation of +these compounds in optical applications [17]. The structural, magnetic, and electronic properties +of tetragonal structures (Pbmm and P4/mmm) of PbMnO3 were calculated theoretically using the +density functional theory (DFT) approach [18]. Besides, J.B. Goodenough et al., showed in [19] the +varied roles of Pb in transition-metal PbTMO3 (TM = V, Mn, Ni, Mn, Ti, Fe, Ru) perovskites. +Due to its excellent and unique physical properties, Cubic PbGeO3 perovskite oxide have received +more attention in both experimental and theoretical physics. In addition, the PbGeO3 perovskite +crystallizes in cubic structure which is reported in experimental and theoretical works [20]. +Us- +ing X-ray photoelectron spectroscopy, they have calculated the binding energy of the PbGeO3 and +Pb5Ge3O11 phases and showed their optical transmission characteristics [11]. +Other researchers +showed that PbGeO3 is considered an interesting choice for lithium batteries [21,22]. Theoretically, +optoelectronic and thermoelectric properties of cubic PbGeO3 were evaluated within the DFT ap- +proach [23]. +In this paper, we investigate the electronic, structural, thermodynamic and elastic properties of cu- +bic PbGeO3 perovskite oxide using Full Potential Linearised Augmented Plane Wave (FP-LAPW) +method within the DFT approach. Our report is structured as follows: We start with computa- +tional procedures, and then we analyze and discuss the obtained results about the studied physical + +3 +properties of PbGeO3 perovskite. Finally, a conclusion of the main results is given in the last section. +2. +Computational details +In this paper, we investigate the physical properties of cubic PbGeO3 perovskite oxide by using the +FP-LAPW method within DFT approach [24] as implemented in the WIEN2k code [25]. Based on +the Perdew-Burke-Ernzerhof approximation (PBE-GGA) [26] and modified Becke-Johnson (mBJ) +exchange potential [27], we have studied the structural and electronic properties of the PbGeO3 +perovskite oxide. The elastic parameters have been determined using ElaStic-1.1 package [28]. The +separation energy between core and valence electrons is −10.0 Ry. The number of plane waves is +limited by RMT ×Kmax = 8. The lmax parameter is taken to be 10 and the Fourier expanded change +density is Gmax = 12. The integration of first Brillouin zone is performed with (6 × 6 × 6) k-points +grid in reciprocal space. The crystal structure is designed using VESTA program [29]. +The thermodynamic parameters of the cubic PbGeO3 are determined by using the quasi-harmonic +Debye model [30, 31]. The non-equilibrium Gibbs function G∗(P, V, T) is defined by the following +equation : +G∗(P, V, T) = E(V ) + PV + Hvibration[ΘD(V ), T], +(1) +where PV represents the constant hydrostatic pressure condition and E(V ) is the equilibrium energy +per unit cell. +The Hvibration[ΘD(V ), T] denotes the vibrational term, which can be written as: +Hvibration (ΘD(V ), T) = mKBT +�9ΘD +8T ++ 3 ln +� +1 − e−ΘD/T � +− D +�ΘD +T +�� +, +(2) +where m represents the number of atoms per formula and D +� +ΘD +T +� +is the Debye integral. +The Debye temperature ΘD is expressed as : +ΘD = +ℏ +KB +� +6π2V 1/2m +�1/3 +f(σ) +� +Bs +M , +(3) +where M stands for the molecular mass per unit cell. +The adiabatic bulk modulus BT is approximately defined as the static compressibility : +BT ≈ B(V ) = V d2E(V ) +dV 2 +. +(4) +The G∗(P, V, T) function is minimized with respect to the volume V as: +�dG∗(P, V, T) +dV +� +P,T +. +(5) + +4 +By solving the equation (5), we find the thermal equation of states V (P, T). +The thermal expansion α, bulk modulus B and the heat capacity (at volume constant) CV are +successively given by the following equations [30]: +α = γCV +BT V , +(6) +BT = V +�d2G∗(P, V, T) +d2V +� +P,T +, +(7) +CV = 3mk +� +4D +�Θ +T +� +− +3Θ/T +eΘ/T − 1 +� +. +(8) +3. +Results and discussions +3.1. +Structure and stability +Perovskite oxide PbGeO3 was first optimized based on the experimental lattice parameter. PbGeO3 +has an ideal cubic phase with a space group Pm3m. The atomic coordinates of the primitive cell +of cubic PbGeO3 are defined as Pb : (0, 0, 0), Ge : (1/2, 1/2, 1/2) and O : (0, 1/2, 1/2). Figure 1 +shows the variation of the total energy as a function of the unit cell volume in addition to the flexible +structure of cubic PbGeO3. +Figure 1: The crystal structure and the optimization plot of PbGeO3. +The calculated values of lattice constant (a) and bulk modulus (B) of our compound are sum- +marized in table 1. We notice that the obtained results are in good agreement with the theoretical +and experimental works [20,23]. + +46506.23 +46506,24 +46506,25 +Pb +PL +Pb +P +Energy (Ry) +46506.26 +Pb +46506.27 +Pb +Pb +46506.28 +46506.29 +46506.30 +360 +380 +400 +420 +440 +460 +4805 +Compound +a(˚A) +B(GPa) +Methods +PbGeO3 +3.8984 +152.5279 +Our work +3.9680 +Exp [20] +3.8320, 3.8420, 3.9002, +198, 157.8647, 180.7519, +Theory [20,23] +3.8404, 3.8536, 3.8150. +181.9745, 201.1913. +Table 1: Calculated lattice constant (a) and Bulk modulus (B) of PbGeO3 compound. +In order to examine the dynamic stability of cubic PbGeO3, we have calculated the phonon +dispersion using the supercell method within phonopy code [32]. +Figure 2 presents the phonon +dispersion of our material PbGeO3. According to this figure, the phonon dispersion curve of our +compound shows positive frequencies along the high symmetry directions, indicating the dynamic +stability of the cubic PbGeO3 phase. +Figure 2: Phonon dispersion curve of PbGeO3. +4. +Electronic properties +Regarding the electronic properties of cubic PbGeO3, we have investigated the band structure, partial +and total density of state using PBE + TB-mBJ exchange-correlation potential. We mention that, + +25 +20 +15 +Frequency +10 +5 +0 +G +M +x +R +G6 +the mBJ approach gives a large band gap energy, and it solves the problem of underestimation band +gap energy [27]. +Figure 3 shows the obtained electronic band structure of PbGeO3 along the high symmetry directions +using the mBJ approach. We can see from this figure that PbGeO3 shows a semiconductor behavior. +In addition, we note that the valence band maximum (VBM) and the conduction band minimum +(CBM) are placed, respectively, at X and Γ points. This means that PbGeO3 has an indirect band +gap equal to 1.67 eV (Γ-X). Moreover, our result is consistent with other theoretical calculation +(A.Day et al. [23]). +Figure 3: Band structure of PbGeO3 using mBJ approach. +For a better illustration of the contribution of different band energies in the band structure, we +have calculated the partial and total density of states as presented in figure 4. We note that the +lower region of the valence band consists of all orbitals such as s − Pb, sp − Ge and p of Oxygen +with hybridization between Pb and Oxygen in TDOS. Near the Fermi level, the p−Oxygen is mixed +with s − Pb which represents a strong hybridization of p and s of Oxygen and Lead, respectively. +The gap energy is clearly shown due to the contribution of the s − Ge and p orbitals of Oxygen. For +the conduction band (CB), a large dispersion of s orbital of Ge is observed, and it forms a covalent +chemical bond with the p orbital of oxygen. Similar results are shown in the case of SrGeO3 [33]. +Besides, we observe a hybrid state between the p orbitals of Pb and Oxygen around 5 eV . + +8 +6 +4- +2 - +Energy (ev) +2 +4 +-6- +-8- +R +x +M7 +Figure 4: The partial and total density of the states of PbGeO3 using the mBJ approach. +4.1. +Elastic properties +Elastic constants play a significant role in material engineering as it provides important information +about the ductility, brittleness, stillness, and also the mechanical stability of materials [34]. The +elastic properties of cubic PbGeO3 are investigated using the ElaStic1-1 package [28]. PbGeO3 is +a cubic crystalline structure, which has three independent elastic constants (C11, C12 and C44). +Table 2 shows the calculated elastic constants of our compound using the modified Becke Johnson +(mBJ) approximation. We note that C11 represents the longitudinal distortion (compression) and +describes the hardness of the material, C12 is based on the transverse distortion (compression) and +C44 represents the resistance to shear deformation, which is based on the shear modulus [34]. +The mechanical stability is verified using Born’s criteria that are given by [35]: +� +� +� +� +� +� +� +� +� +C11 − C12 > 0, +C44 > 0, +C11 + 2C12 > 0. +(9) +From Table 2, the calculated elastic constants respect Born’s criteria, and this clearly means that +cubic PbGeO3 is mechanically stable. From table 2, one can see that the longitudinal distortion + +1.88 +E +s +1.41 +0.94 +0.47 +0.00 +Ge +0.14 +0.10 +DOs(States/eV) +0.05 +00'0 +Pb +66'0 +0.66 +0.33 +0.,00 +TDOS +Pb +6.00 +Ge +0 +4.00 +2.00 +0.00 +-10 +.5 +0 +5 +Enerov(ev)8 +value is higher than C44 value, meaning a weak resistance to the shear modulus compared to the +longitudinal distortion [36]. +Elastic Coefficients +Coefficient +C11 +C12 +C44 +PbGeO3 +153.4 +141.1 +115.5 +Table 2: The obtained elastic constants of the PbGeO3 compound. +Based on elastic constant Cαβ, The mechanical constants such as shear modulus (G), bulk modu- +lus (B), Cauchy pressure (C”), Pugh’s ratio (B/G), Poisson’s ratio (ν), anisotropy (A) are calculated +through the VRH (Voigt-Reuss-Hill) approximation by using the following formulas [37,38]: +B = 2C12 + C11 +3 +, +G = GR + GV +2 +, +Y = +9BG +3B + G, +ν = 3B − 2G +2(G + 3B), +(10) +where GR and GV are successively the shear modulus of Reuss and Voight approaches such that: +GV = C11 + 3C44 − C12 +5 +, +GR = +5C44(C11 − C12) +3(C11 + C12 + 4C44). +(11) +The obtained results of mechanical parameters are listed in table 3. The bulk and shear modulus +can be used to measure the rigidity of materials [37]. The calculated G and B values are 43.03 GPa +and 145.18 GPa, respectively. We notice that the B value found from elastic constants is nearer to +that obtained by fitting the Birch-Murnaghan equation of state. This comparison ensures that our +computed elastic constants are correct. In addition, Young’s modulus Y also deals with the hardness +or stillness of materials [39]. The obtained value of Y is equal to 117.49 GPa, which is important as +it is greater than 100. Therefore, we assume that PbGeO3 behaves as a hard material. Anisotropy +(A) is also an important parameter in industrial science for detecting micro-cracks in materials [40]. +From table 3 the calculated value of A is 0.6, meaning that PbGeO3 shows an anisotropy aspect. +The Poisson’s ratio (ν), Cauchy pressure (C”) and Pugh’s ratio (B/G) reveal the ductile or brittle +aspect of materials. It is well known that the critical value of Pugh’s ratio (B/G) which separates +the ductile/brittle aspect is found to be 1.75. Consequently, a material will show a ductile aspect +for values of B/G higher than 1.75, while it shows a brittle nature for values of B/G less than this +critical value [41]. Another index of brittleness/ductility is the Cauchy pressure (C”). The positive +and negative values of C” are, respectively, related to the ductility and brittleness nature [42]. The +parameter of Poisson’s ratio (ν) is also a significant factor to distinguish the ductility/brittleness of +materials with its critical value which is 0.26. The material will be ductile (brittle) when the Poisson’s +ratio is higher (less) than 0.26 [43]. Based on these roles and the obtained results, we conclude that +PbGeO3 shows a ductility aspect. + +9 +Mechanical Constants +Constant +B(GPa) +G(GPa) +B/G +Y (GPa) +C”(GPa) +ν +A +PbGeO3 +145.18 +43.03 +3.37 +117.49 +25.6 +0.37 +0.67 +Table 3: The obtained values of shear modulus G, bulk modulus B, Pugh’s ratio B/G, Young’s +modulus Y , Poisson’s ratio ν, Cauchy pressure C +′′ and elastic anisotropic factor A of PbGeO3 +compound. +4.2. +Thermodynamic properties +The study of thermodynamic properties depending on temperature and pressure provides detailed +information about material applications and offers a point of view on their fabrication [44]. +The thermodynamic parameters such as bulk modulus (B), volume (V ), thermal expansion (α), +Deybe temperature (θD) and heat capacity (CV ) have been calculated using the Gibbs2 package +within the quasi-harmonic approach [30]. The temperature-pressure effects on thermodynamic pa- +rameters of cubic PbGeO3 perovskite oxide have been studied in the range of 0 to 1000 K for +temperature and 0 to 30 GPa for pressure. +Figure 5: Unit cell volume versus temperature of cubic PbGeO3 perovskite oxide at different pres- +sures. + +430 +0 GPa +10 GPa +420 +20 GPa +30 GPa +410 +400 +V(bohr3) +390 +380 +370 +360 +350 +340 +0 +100 +200 +300 +400　500 +600 +700 +800 +900 +1000 +Temperature (K)10 +The effect of both pressure and temperature on the unit cell volume of cubic PbGeO3 perovskite +oxide is illustrated in figure 5. We notice that the volume increases with increasing temperature (ex- +pansion), and decreases with increasing pressure (compression). We also remark that the calculated +value of volume at 0 GPa and 0 K is in good agreement with what we found in the structural data. +The variation of bulk modulus B of PbGeO3 versus temperature at certain pressures is shown in +figure 6. We notice that the effect of temperature and pressure on the bulk modulus B is opposite +to their effect on the volume curve. Indeed, the bulk modulus B increases with increasing pressure, +and it decreases when we increase the temperature. Therefore, the compressibility decreases with +pressure at a certain temperature, while it increases with increasing temperature at a particular +pressure [45]. +Figure 6: Bulk modulus versus temperature of cubic PbGeO3 perovskite oxide at different pressures. +4.2..1 +Thermal expansion +The thermal expansion coefficient (α) of cubic PbGeO3 is calculated with respect to temperature +at certain pressures as displayed in figure 7. It is a significant parameter that provides information +about the inter-atomic forces of materials, and it is also linked to the anharmonicity of the lattice +interaction potential [46]. +From figure 7, it can be seen that the expansion coefficient α increases rapidly up to 300 K, then +we observe a nearly linear increase at higher temperatures. On the other side, increasing pressure at + +300 +0 GPa +10 GPa +280 +20 GPa +30 GPa +260 +Bulk Modulus (GPa) +240 +220 +200 +180 +160 +140 +120 +0 +100 +200 +400　500 +300 +600 +700 +800 +900 +1000 +Temperature (K)11 +a particular temperature leads to a decrease in the thermal expansion coefficient. This result indi- +cates that the PbGeO3 exhibits excellent volume invariance under high pressure [47]. Consequently, +increasing pressure and temperature have opposite effects on thermal expansion. +Figure 7: Thermal expansion (α) as a function of temperature of cubic PbGeO3 perovskite oxide at +different pressures. +4.2..2 +Debye temperatures +The Debye temperature is a special temperature of solids which exhibits the temperature at which the +atomic vibrations of material reach their maximum of possible modes [48]. It is a proper estimation +of the rigidity of solids [49]. +Figure 8 illustrates the Debye temperature versus temperature for PbGeO3 at given pressures. It is +obvious that the θD curve is approximately constant in the range of 0 to 100 K for all considered +values of pressure. This result indicates that the crystal experiences a weak anharmonicity and a slight +expansion in this temperature range. Beyond 120 K, the Debye temperature θD is reduced gradually +with increasing temperature, and this indicates a variation in the atomic vibration spectrum [50]. + +6 +0 GPa +10 GPa +20 GPa +30 GPa +5 +0 +0 +100 +200 +300 +400500 +600 +700 +800 +900 +1000 +Temperature (K)12 +Figure 8: Debye temperature (θD) as a function of temperature of cubic PbGeO3 perovskite oxide +at different pressures. +4.2..3 +Heat capacity +The heat capacity is not only a significant feature that provides the necessary information about the +vibrational characteristics of the lattice but also an obligatory parameter for many applications [51]. +The variation of the heat capacity CV of our compound as a function of temperature is shown +in figure 9. We notice that the heat capacity presents a similar behavior with different pressures. +For a particular pressure, it can be seen that CV obeys T 3 at low temperatures, which accords +with the simple Debye model [52]. Then, the CV increases slowly with increasing temperature, and +it approaches the classical Dulong-Petit limit (3R × 5 atoms = 15R = 124.71 J/K.mol) at high +temperatures. [52]. + +0 GPa +10 GPa +600 +20 GPa +30 GPa +Debye temperature +550 +500 +450 +400 +0 +100 +200 +300 +400500　 +600　700 +800 +900 +1000 +Temperature (K)13 +Figure 9: Heat capacity (CV ) with respect to temperature of cubic PbGeO3 perovskite oxide at +different pressures. +5. +Conclusion +In this paper, we have studied the electronic, elastic, structural, and thermodynamic properties of cu- +bic PbGeO3 based on the FP-LAPW method. The mechanical and elastic parameters are evaluated, +and they prove that the cubic PbGeO3 is mechanically stable. In addition, the phonon dispersion +shows positive frequencies, confirming that the PbGeO3 compound is dynamically stable. Next, we +analyze the electronic properties which reveal that cubic PbGeO3 is a p-type semiconductor with an +indirect band gap. 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Molina-Luna, Computational Materials Science 197, 110609 (2021), DOI:10.1016/j. +commatsci.2021.110609. + diff --git a/PdFRT4oBgHgl3EQfIzcw/content/tmp_files/load_file.txt b/PdFRT4oBgHgl3EQfIzcw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..557838ad0432ba2761a382fc227be361f61711f0 --- /dev/null +++ b/PdFRT4oBgHgl3EQfIzcw/content/tmp_files/load_file.txt @@ -0,0 +1,803 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf,len=802 +page_content='Enhancement in physical properties of Pb-Based Perovskite Oxides (PbGeO3): Ab initio Calculation M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Agouri1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Waqdim1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Abbassi1,∗, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Ouali1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Taj1, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Manaut1,†, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Driouich1 1 Laboratory of Research in Physics and Engineering Sciences, Sultan Moulay Slimane University, Polydisciplinary Faculty, Beni Mellal, 23000, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' February 1, 2023 Abstract In the present paper, the electronic, structural, thermodynamic, and elastic properties of cubic PbGeO3 perovskite oxide are presented and computed using the WIEN2k code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The structural properties have been evaluated and they are in good agreement with the theoretical and exper- imental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' A phonon dispersion is made and it reveals that the cubic PbGeO3 perovskite is dynamically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' In addition, the electronic properties of PbGeO3 shows an opening gap energy, meaning a semiconductor behavior with an indirect band gap equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='67 eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Moreover, the obtained elastic constants of cubic PbGeO3 satisfy Born’s mechanical stability criteria, and this in- dicates that our compound behaves as a stable ductile material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The temperature-pressure effects on thermodynamic parameters are investigated using the Gibbs2 package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Finally, based on the obtained results about the cubic PbGeO3 perovskite properties, we assume that this compound will have potential applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Keywords: DFT, Elastic, Thermodynamic, Perovskite, WIEN2K, mBJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' ∗Corresponding author, E-mail: abbassi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='abder@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='com †Corresponding author, E-mail: b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='manaut@usms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='ma 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='13493v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='mtrl-sci] 31 Jan 2023 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Introduction Since the discovery of CaTiO3 [1], the perovskite oxides family has been a major subject of interest, and this is mainly due to its multi-functional character [2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' It has received great attention for its exploitation in many applications such as solar cells [4], spintronic and optoelectronic [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The exploitation of these materials mainly depends on their flexible structure, variable formula, and also their various properties [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Consequently, a lot of experimental and theoretical studies have proved that the perovskite oxides family has unique physical properties like photoelectric [9], magnetic [10], ferroelectric [11],.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='. etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Pb-based perovskite oxides have long been investigated for their rich and interesting properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' They are considered proper candidates in energy conversion such as in piezoelectric and ferroelec- tric devices [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Recently, several experimental and theoretical studies have been interested on Lead-based perovskite oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The unit cell compounds of PbXO3 (X = Ti, V ) were synthesized, and their crystal structures are determined using Neutron and X-ray diffraction [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' In ad- dition, these compounds have largely been used in ferroelectric and optical sensors [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The PbZrxTi1−xO3 (x = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='6, 1) were prepared and investigated, proving the exploitation of these compounds in optical applications [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The structural, magnetic, and electronic properties of tetragonal structures (Pbmm and P4/mmm) of PbMnO3 were calculated theoretically using the density functional theory (DFT) approach [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Besides, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Goodenough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=', showed in [19] the varied roles of Pb in transition-metal PbTMO3 (TM = V, Mn, Ni, Mn, Ti, Fe, Ru) perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Due to its excellent and unique physical properties, Cubic PbGeO3 perovskite oxide have received more attention in both experimental and theoretical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' In addition, the PbGeO3 perovskite crystallizes in cubic structure which is reported in experimental and theoretical works [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Us- ing X-ray photoelectron spectroscopy, they have calculated the binding energy of the PbGeO3 and Pb5Ge3O11 phases and showed their optical transmission characteristics [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Other researchers showed that PbGeO3 is considered an interesting choice for lithium batteries [21,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Theoretically, optoelectronic and thermoelectric properties of cubic PbGeO3 were evaluated within the DFT ap- proach [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' In this paper, we investigate the electronic, structural, thermodynamic and elastic properties of cu- bic PbGeO3 perovskite oxide using Full Potential Linearised Augmented Plane Wave (FP-LAPW) method within the DFT approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Our report is structured as follows: We start with computa- tional procedures, and then we analyze and discuss the obtained results about the studied physical 3 properties of PbGeO3 perovskite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Finally, a conclusion of the main results is given in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Computational details In this paper, we investigate the physical properties of cubic PbGeO3 perovskite oxide by using the FP-LAPW method within DFT approach [24] as implemented in the WIEN2k code [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Based on the Perdew-Burke-Ernzerhof approximation (PBE-GGA) [26] and modified Becke-Johnson (mBJ) exchange potential [27], we have studied the structural and electronic properties of the PbGeO3 perovskite oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The elastic parameters have been determined using ElaStic-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='1 package [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The separation energy between core and valence electrons is −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='0 Ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The number of plane waves is limited by RMT ×Kmax = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The lmax parameter is taken to be 10 and the Fourier expanded change density is Gmax = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The integration of first Brillouin zone is performed with (6 × 6 × 6) k-points grid in reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The crystal structure is designed using VESTA program [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The thermodynamic parameters of the cubic PbGeO3 are determined by using the quasi-harmonic Debye model [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The non-equilibrium Gibbs function G∗(P, V, T) is defined by the following equation : G∗(P, V, T) = E(V ) + PV + Hvibration[ΘD(V ), T], (1) where PV represents the constant hydrostatic pressure condition and E(V ) is the equilibrium energy per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The Hvibration[ΘD(V ), T] denotes the vibrational term, which can be written as: Hvibration (ΘD(V ), T) = mKBT �9ΘD 8T + 3 ln � 1 − e−ΘD/T � − D �ΘD T �� , (2) where m represents the number of atoms per formula and D � ΘD T � is the Debye integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The Debye temperature ΘD is expressed as : ΘD = ℏ KB � 6π2V 1/2m �1/3 f(σ) � Bs M , (3) where M stands for the molecular mass per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The adiabatic bulk modulus BT is approximately defined as the static compressibility : BT ≈ B(V ) = V d2E(V ) dV 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' (4) The G∗(P, V, T) function is minimized with respect to the volume V as: �dG∗(P, V, T) dV � P,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' (5) 4 By solving the equation (5), we find the thermal equation of states V (P, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The thermal expansion α, bulk modulus B and the heat capacity (at volume constant) CV are successively given by the following equations [30]: α = γCV BT V , (6) BT = V �d2G∗(P, V, T) d2V � P,T , (7) CV = 3mk � 4D �Θ T � − 3Θ/T eΘ/T − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' (8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Results and discussions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Structure and stability Perovskite oxide PbGeO3 was first optimized based on the experimental lattice parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' PbGeO3 has an ideal cubic phase with a space group Pm3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The atomic coordinates of the primitive cell of cubic PbGeO3 are defined as Pb : (0, 0, 0), Ge : (1/2, 1/2, 1/2) and O : (0, 1/2, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Figure 1 shows the variation of the total energy as a function of the unit cell volume in addition to the flexible structure of cubic PbGeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Figure 1: The crystal structure and the optimization plot of PbGeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The calculated values of lattice constant (a) and bulk modulus (B) of our compound are sum- marized in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' We notice that the obtained results are in good agreement with the theoretical and experimental works [20,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 46506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='23 46506,24 46506,25 Pb PL Pb P Energy (Ry) 46506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='26 Pb 46506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='27 Pb Pb 46506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='28 46506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='29 46506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='30 360 380 400 420 440 460 4805 Compound a(˚A) B(GPa) Methods PbGeO3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='8984 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='5279 Our work 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='9680 Exp [20] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='8320, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='8420, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='9002, 198, 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='8647, 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='7519, Theory [20,23] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='8404, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='8536, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='8150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='9745, 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='1913.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Table 1: Calculated lattice constant (a) and Bulk modulus (B) of PbGeO3 compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' In order to examine the dynamic stability of cubic PbGeO3, we have calculated the phonon dispersion using the supercell method within phonopy code [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Figure 2 presents the phonon dispersion of our material PbGeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' According to this figure, the phonon dispersion curve of our compound shows positive frequencies along the high symmetry directions, indicating the dynamic stability of the cubic PbGeO3 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Figure 2: Phonon dispersion curve of PbGeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Electronic properties Regarding the electronic properties of cubic PbGeO3, we have investigated the band structure, partial and total density of state using PBE + TB-mBJ exchange-correlation potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' We mention that, 25 20 15 Frequency 10 5 0 G M x R G6 the mBJ approach gives a large band gap energy, and it solves the problem of underestimation band gap energy [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Figure 3 shows the obtained electronic band structure of PbGeO3 along the high symmetry directions using the mBJ approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' We can see from this figure that PbGeO3 shows a semiconductor behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' In addition, we note that the valence band maximum (VBM) and the conduction band minimum (CBM) are placed, respectively, at X and Γ points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' This means that PbGeO3 has an indirect band gap equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='67 eV (Γ-X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Moreover, our result is consistent with other theoretical calculation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='Day et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Figure 3: Band structure of PbGeO3 using mBJ approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' For a better illustration of the contribution of different band energies in the band structure, we have calculated the partial and total density of states as presented in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' We note that the lower region of the valence band consists of all orbitals such as s − Pb, sp − Ge and p of Oxygen with hybridization between Pb and Oxygen in TDOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Near the Fermi level, the p−Oxygen is mixed with s − Pb which represents a strong hybridization of p and s of Oxygen and Lead, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The gap energy is clearly shown due to the contribution of the s − Ge and p orbitals of Oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' For the conduction band (CB), a large dispersion of s orbital of Ge is observed, and it forms a covalent chemical bond with the p orbital of oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Similar results are shown in the case of SrGeO3 [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Besides, we observe a hybrid state between the p orbitals of Pb and Oxygen around 5 eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 8 6 4- 2 - Energy (ev) 2 4 6- 8- R x M7 Figure 4: The partial and total density of the states of PbGeO3 using the mBJ approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Elastic properties Elastic constants play a significant role in material engineering as it provides important information about the ductility, brittleness, stillness, and also the mechanical stability of materials [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The elastic properties of cubic PbGeO3 are investigated using the ElaStic1-1 package [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' PbGeO3 is a cubic crystalline structure, which has three independent elastic constants (C11, C12 and C44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Table 2 shows the calculated elastic constants of our compound using the modified Becke Johnson (mBJ) approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' We note that C11 represents the longitudinal distortion (compression) and describes the hardness of the material, C12 is based on the transverse distortion (compression) and C44 represents the resistance to shear deformation, which is based on the shear modulus [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The mechanical stability is verified using Born’s criteria that are given by [35]: � � � � � � � � � C11 − C12 > 0, C44 > 0, C11 + 2C12 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' (9) From Table 2, the calculated elastic constants respect Born’s criteria, and this clearly means that cubic PbGeO3 is mechanically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' From table 2, one can see that the longitudinal distortion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='88 E s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='00 Ge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='10 DOs(States/eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content="05 00'0 Pb 66'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=',00 TDOS Pb 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='00 Ge 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='00 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='5 0 5 Enerov(ev)8 value is higher than C44 value, meaning a weak resistance to the shear modulus compared to the longitudinal distortion [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Elastic Coefficients Coefficient C11 C12 C44 PbGeO3 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='4 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='1 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='5 Table 2: The obtained elastic constants of the PbGeO3 compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Based on elastic constant Cαβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The mechanical constants such as shear modulus (G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' bulk modu- lus (B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Cauchy pressure (C”),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Pugh’s ratio (B/G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Poisson’s ratio (ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' anisotropy (A) are calculated through the VRH (Voigt-Reuss-Hill) approximation by using the following formulas [37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='38]: B = 2C12 + C11 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' G = GR + GV 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Y = 9BG 3B + G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' ν = 3B − 2G 2(G + 3B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' (10) where GR and GV are successively the shear modulus of Reuss and Voight approaches such that: GV = C11 + 3C44 − C12 5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' GR = 5C44(C11 − C12) 3(C11 + C12 + 4C44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' (11) The obtained results of mechanical parameters are listed in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The bulk and shear modulus can be used to measure the rigidity of materials [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The calculated G and B values are 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='03 GPa and 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='18 GPa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' We notice that the B value found from elastic constants is nearer to that obtained by fitting the Birch-Murnaghan equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' This comparison ensures that our computed elastic constants are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' In addition, Young’s modulus Y also deals with the hardness or stillness of materials [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The obtained value of Y is equal to 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='49 GPa, which is important as it is greater than 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Therefore, we assume that PbGeO3 behaves as a hard material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Anisotropy (A) is also an important parameter in industrial science for detecting micro-cracks in materials [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' From table 3 the calculated value of A is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='6, meaning that PbGeO3 shows an anisotropy aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The Poisson’s ratio (ν), Cauchy pressure (C”) and Pugh’s ratio (B/G) reveal the ductile or brittle aspect of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' It is well known that the critical value of Pugh’s ratio (B/G) which separates the ductile/brittle aspect is found to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Consequently, a material will show a ductile aspect for values of B/G higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='75, while it shows a brittle nature for values of B/G less than this critical value [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Another index of brittleness/ductility is the Cauchy pressure (C”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The positive and negative values of C” are, respectively, related to the ductility and brittleness nature [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The parameter of Poisson’s ratio (ν) is also a significant factor to distinguish the ductility/brittleness of materials with its critical value which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The material will be ductile (brittle) when the Poisson’s ratio is higher (less) than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='26 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Based on these roles and the obtained results, we conclude that PbGeO3 shows a ductility aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 9 Mechanical Constants Constant B(GPa) G(GPa) B/G Y (GPa) C”(GPa) ν A PbGeO3 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='18 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='37 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='49 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='67 Table 3: The obtained values of shear modulus G, bulk modulus B, Pugh’s ratio B/G, Young’s modulus Y , Poisson’s ratio ν, Cauchy pressure C ′′ and elastic anisotropic factor A of PbGeO3 compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Thermodynamic properties The study of thermodynamic properties depending on temperature and pressure provides detailed information about material applications and offers a point of view on their fabrication [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The thermodynamic parameters such as bulk modulus (B), volume (V ), thermal expansion (α), Deybe temperature (θD) and heat capacity (CV ) have been calculated using the Gibbs2 package within the quasi-harmonic approach [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The temperature-pressure effects on thermodynamic pa- rameters of cubic PbGeO3 perovskite oxide have been studied in the range of 0 to 1000 K for temperature and 0 to 30 GPa for pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Figure 5: Unit cell volume versus temperature of cubic PbGeO3 perovskite oxide at different pres- sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 430 0 GPa 10 GPa 420 20 GPa 30 GPa 410 400 V(bohr3) 390 380 370 360 350 340 0 100 200 300 400 500 600 700 800 900 1000 Temperature (K)10 The effect of both pressure and temperature on the unit cell volume of cubic PbGeO3 perovskite oxide is illustrated in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' We notice that the volume increases with increasing temperature (ex- pansion), and decreases with increasing pressure (compression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' We also remark that the calculated value of volume at 0 GPa and 0 K is in good agreement with what we found in the structural data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The variation of bulk modulus B of PbGeO3 versus temperature at certain pressures is shown in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' We notice that the effect of temperature and pressure on the bulk modulus B is opposite to their effect on the volume curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Indeed, the bulk modulus B increases with increasing pressure, and it decreases when we increase the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Therefore, the compressibility decreases with pressure at a certain temperature, while it increases with increasing temperature at a particular pressure [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Figure 6: Bulk modulus versus temperature of cubic PbGeO3 perovskite oxide at different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='.1 Thermal expansion The thermal expansion coefficient (α) of cubic PbGeO3 is calculated with respect to temperature at certain pressures as displayed in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' It is a significant parameter that provides information about the inter-atomic forces of materials, and it is also linked to the anharmonicity of the lattice interaction potential [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' From figure 7, it can be seen that the expansion coefficient α increases rapidly up to 300 K, then we observe a nearly linear increase at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' On the other side, increasing pressure at 300 0 GPa 10 GPa 280 20 GPa 30 GPa 260 Bulk Modulus (GPa) 240 220 200 180 160 140 120 0 100 200 400 500 300 600 700 800 900 1000 Temperature (K)11 a particular temperature leads to a decrease in the thermal expansion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' This result indi- cates that the PbGeO3 exhibits excellent volume invariance under high pressure [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Consequently, increasing pressure and temperature have opposite effects on thermal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Figure 7: Thermal expansion (α) as a function of temperature of cubic PbGeO3 perovskite oxide at different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='.2 Debye temperatures The Debye temperature is a special temperature of solids which exhibits the temperature at which the atomic vibrations of material reach their maximum of possible modes [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' It is a proper estimation of the rigidity of solids [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Figure 8 illustrates the Debye temperature versus temperature for PbGeO3 at given pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' It is obvious that the θD curve is approximately constant in the range of 0 to 100 K for all considered values of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' This result indicates that the crystal experiences a weak anharmonicity and a slight expansion in this temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Beyond 120 K, the Debye temperature θD is reduced gradually with increasing temperature, and this indicates a variation in the atomic vibration spectrum [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 6 0 GPa 10 GPa 20 GPa 30 GPa 5 0 0 100 200 300 400500 600 700 800 900 1000 Temperature (K)12 Figure 8: Debye temperature (θD) as a function of temperature of cubic PbGeO3 perovskite oxide at different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='.3 Heat capacity The heat capacity is not only a significant feature that provides the necessary information about the vibrational characteristics of the lattice but also an obligatory parameter for many applications [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The variation of the heat capacity CV of our compound as a function of temperature is shown in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' We notice that the heat capacity presents a similar behavior with different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' For a particular pressure, it can be seen that CV obeys T 3 at low temperatures, which accords with the simple Debye model [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Then, the CV increases slowly with increasing temperature, and it approaches the classical Dulong-Petit limit (3R × 5 atoms = 15R = 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='71 J/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content='mol) at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 0 GPa 10 GPa 600 20 GPa 30 GPa Debye temperature 550 500 450 400 0 100 200 300 400500 600 700 800 900 1000 Temperature (K)13 Figure 9: Heat capacity (CV ) with respect to temperature of cubic PbGeO3 perovskite oxide at different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Conclusion In this paper, we have studied the electronic, elastic, structural, and thermodynamic properties of cu- bic PbGeO3 based on the FP-LAPW method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' The mechanical and elastic parameters are evaluated, and they prove that the cubic PbGeO3 is mechanically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' In addition, the phonon dispersion shows positive frequencies, confirming that the PbGeO3 compound is dynamically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Next, we analyze the electronic properties which reveal that cubic PbGeO3 is a p-type semiconductor with an indirect band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Finally, the thermodynamic parameters such as volume, Debye temperature, bulk modulus, and heat capacity are predicted for the cubic PbGeO3 perovskite using quasi-harmonic Debye approximation with pressure and temperature in the range of 0 to 25 GPa and 0 to 1000 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' Overall, the discussed parameters exhibit the good efficiency of our material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQfIzcw/content/2301.13493v1.pdf'} +page_content=' 120 Dulong-Petit limit 100 80 60 40 0 GPa 20 10 GPa 20 GPa 30 GPa 0 100 200 300 600 700 800 0 400 500 900 1000 Temperature (K)14 References [1] E.' metadata={'source': 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Hancock,1★ A. J. Young,1 P. Chainakun2,3 +1HH Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK +2School of Physics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand +3Centre of Excellence in High Energy Physics and Astrophysics, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We fit a new vertically extended corona model to previously measured reverberation time lags +observed by XMM-Newton in two extremely variable Narrow Line Seyfert 1 Active Galactic +Nuclei (AGN), 1H 0707-495 and IRAS 13224-3809, in a variety of similarly observed flux +groups and explore the model in all observations over a 16 year period. The model employs +two X-ray sources located along the black hole rotational axis at height, ℎ1 and ℎ2 respectively. +These sources have their associated photon indices Γ1 and Γ2 which respond to fluctuations +in the disc with a maximum response duration of 𝑡max and a propagation delay between the +response of the two of 𝑡shift. We find that for 1H 0707-495, ℎ2 is significantly correlated with +Γ1 and anti-correlated with ionisation 𝜉. Whilst the 1H 0707-495 corona extends upwards, the +emission appears softer and the disc is less ionised. We find similarities in IRAS 13224-3809, +but significant anti-correlation between Γ2 and both 𝑡max and 𝑡shift. This suggests that when +the IRAS 13224-3809 corona becomes softer while extending vertically upwards, the overall +corona response occurs faster. This may also suggest that the inner disc also becomes more +active. In addition, Γ1 and Γ2 are extreme, relatively less variable, but more separate in IRAS +13224-3809 than in 1H 0707-495. This suggests that the IRAS 13224-3809 corona may be +more patchy in the sense that it has two more clear distinct spectral zones of Γ1 and Γ2 (possibly +relating to two distinct zones of coronal temperature) when compared to 1H 0707-495. +Key words: X-rays: individual: 1H 0707-495; X-rays: individual: IRAS13224-3809 +1 INTRODUCTION +The spectra of active galactic nuclei (AGN) have a component of +direct X-ray continuum emission, and that cold gas can reflect some +of this X-ray continuum (Pounds et al. 1990). The cold, optically +thick material seen through fluorescence and reflection is known to +occur in the presence of an accretion disc where X-rays produced by +inverse Compton scattering in a corona are reflected off the accretion +disc producing a modified spectrum emission with fluorescent Fe K +lines at 6.4 keV and other spectral features (George & Fabian 1991). +The first hints of reflection or reprocessing time-delay due to the +light travel time between the corona and disk were seen in XMM- +Newton observations of Ark564 (McHardy et al. 2007) which led +to the first robustly discovered delays in 1H0707-495 (Fabian et al. +2009) where the soft energy band (0.3−1 keV) lagged behind the hard +band (1 − 4 keV) by 30 s. Many reverberation lags have since been +discovered (e.g. Emmanoulopoulos et al. 2011; Kara et al. 2013b; +Zoghbi et al. 2013). +★ E-mail: steff.hancock@bristol.ac.uk +The soft negative time lag is the signature of relativistic reflection +that reverberates in response to continuum fluctuations and is inter- +preted as the light crossing time from the source to the reflecting +region, correlating positively with the black hole mass (De Marco +et al. 2013). Hard positive lags at lower frequencies are understood +to originate from fluctuations of the accretion rate propagating from +outer to inner radii, causing the hard X-rays produced at smaller radii +to respond after soft X-rays produced at larger radii (e.g. Kotov et al. +2001; Arévalo & Uttley 2006; Arévalo et al. 2008). Hard X-ray lags +(i.e. hard photon variability lagging soft photon variability) were ev- +ident in X-ray binaries before they were discovered in AGN (see e.g. +Miyamoto et al. 1988; Nowak et al. 1999). +The reproduction of soft reverberation lags from a compact corona +and the hard lags produced by propagating fluctuations through an +extended region whilst maintaining the features of the energy de- +pendence seen in the Fe K𝛼 line region is challenging and com- +putationally intensive. Motivated by these phenomena, Chainakun +& Young (2017), CY17 hereafter, developed an approximation of a +vertically extended source using two X-ray point sources located on +the rotation axis of the black hole. The model employs gravitational +© 2020 The Authors +arXiv:2301.04731v1 [astro-ph.HE] 11 Jan 2023 + +2 +S. Hancock et al. +units where the gravitational radius 𝑟𝑔 = 𝐺𝑀/𝑐2 and gravitational +time 𝑡𝑔 = 𝐺𝑀/𝑐3 (where 𝐺 is the gravitational constant, 𝑀 is the +black hole mass and 𝑐 is the speed of light). Instead of modelling the +propagating fluctuations, a phenomenological function of expected +source responses which react to these propagations is employed. The +two X-ray sources are allowed to vary as they respond to primary +intrinsic variations. The X-ray continuum variability depends on the +response of the source and the X-ray reflection depends also on the +disc response. Therefore the model has to predict the time lags from +both continuum X-ray sources and the associated disc responses. The +model was fitted to a 120 ks timing XMM-Newton observation of the +narrow-line Seyfert 1 galaxy PG 1244+026, examining the frequency +and energy where the lags were found. The model revealed hard and +soft X-ray sources at heights ∼ 6 𝑟𝑔 and ∼ 11 𝑟𝑔 respectively with +the upper source producing small amounts of reflection which sug- +gested a feasible geometry of a relativistic jet beaming away from the +disc. CY17 suggested that the continuum flux from the upper source +and the extra blackbody component that contribute significant flux +at energies < 1 keV are required to dilute the soft reverberation lags +and to reproduce the absence of soft lag in the lag-energy spectrum +of PG 1244+026. +The X-ray variability in 1H 0707-495 was found to be extreme +and may have both intrinsic and environmental absorption origins +(Parker et al. 2021). The geometry of 1H 0707-495 has been dis- +cussed by Szanecki et al. (2020) who developed a new extended +lamppost model which accounted for the spatial extent and rotation +of the X-ray source. The investigation of the location and size of the +corona indicated a compact corona at ∼ 2 𝑟𝑔 (highly centrally peaked +rather than extended) regardless of the effects of ionised absorption +from winds. They found no evidence that the size of the corona was +correlated to the luminosity as reported by Wilkins et al. (2014). +The time lags in IRAS 13224-3809 have been discussed by Alston +et al. (2020) who studied short-timescale variations and included +all relativistic effects allowing for ultra-fast outflows fitting multi- +ple epochs where the source height changed. They tackled inherent +degeneracies between the reverberation signal and black hole mass, +inner disc radius and height of the corona by tracking short-scale re- +verberation signatures where the source height changed. They found +that the height of the corona increased with increasing luminosity +and that black hole mass uncertainty estimates were comparable to +the leading optical reverberation method by Peterson et al. (2004). In +addition, Caballero-García et al. (2020) used all available data from +the XMM-Netwon archive to simultaneously fit various flux states of +the energy spectra and time lags using a new code which calculates +reflection spectra from the accretion disc in response to an X–ray +flare from a point source located above the black hole in accretion +disc lamp-post geometry. The model strongly favoured a maximally +spinning black hole and detected significant variations of the corona +height, increasing from 3 − 5 𝑟𝑔 at lower flux states and extending +to ∼ 10 − 20 𝑟𝑔 when the luminosity doubled. Recently, Chainakun +et al. (2022b) constrained the reverberation signatures that appeared +in the power spectral density of IRAS 13224-3809 and found that the +lamp-post source height increased from ∼ 3 𝑟𝑔 to ∼ 25 𝑟𝑔 with the +luminosity. +This study builds on the timing and spectral analysis of reverberat- +ing AGN reported by Hancock et al. (2022), HYC22 hereafter, using +a variety of similar spectral flux levels as found in the spectra of each +source. In addition we explore all suitable individual XMM-Newton +observations of 1H 0707-495 and IRAS 13224-3809 between 2000 +and 2016. Our aim is to develop the model initially created by CY17 +to explore these time lag signatures in an extended corona scenario. +These well studied AGN are selected due to their abundance of long +Table 1. XMM-Newton observations used in this sample. The information +Includes the name of the source, the Observation ID, year, exposure time and +effective exposure after cleaning. The group column refers to the low, medium +and high spectral flux groups. The low counts (lc) and high counts (hc) refer +to those to observations containing < 5 cts s−1 and > 5 cts s−1 respectively. +Source +Obs ID +Year +Exp [Eff] (ks) +Group +1H0707-495 +0110890201 +2000 +46[41] +Med (lc) +0148010301 +2002 +80[76] +Hi (hc) +0506200201 +2007 +41[38] +Lo (lc) +0506200301 +41[39] +Med (hc) +0506200401 +43[41] +Hi (hc) +0506200501 +47[41] +Hi (hc) +0511580101 +2008 +124[111] +Hi (hc) +0511580201 +124[93] +Hi (hc) +0511580301 +123[84] +Hi (hc) +0511580401 +122[81] +Hi (hc) +0653510301 +2010 +117[112] +Hi (hc) +0653510401 +128[118] +Hi (hc) +0653510501 +128[93] +Hi (hc) +0653510601 +129[105] +Hi (hc) +0554710801 +2011 +98[86] +Lo (lc) +IRAS13224-3809 +0110890101 +2002 +64[61] +Med (lc) +0673580101 +2011 +133[49] +Med (lc) +0673580201 +132[99] +Med (hc) +0673580301 +129[82] +Lo (lc) +0673580401 +135[113] +Med (hc) +0780560101 +2016 +141[141] +Med (hc) +0780561301 +141[127] +Med (hc) +0780561401 +141[126] +Med (hc) +0780561501 +141[126] +Med (hc) +0780561601 +141[137] +Med (hc) +0780561701 +141[123] +Med (hc) +0792180101 +141[123] +Med (hc) +0792180201 +141[129] +Med (hc) +0792180301 +141[129] +Lo (lc) +0792180401 +141[120] +Hi (hc) +0792180501 +138[122] +Med (lc) +0792180601 +136[122] +Hi (hc) +observations, however it should be noted that these are extreme nar- +row line Seyfert 1 galaxies. +2 OBSERVATIONS AND DATA REDUCTION +The data for all observations outlined in Table 1 were downloaded +from the XMM-Newton archive and processed using standard meth- +ods. The time lag estimates calculated between the soft (0.3−0.8 keV) +and hard (1 − 4 keV) energy bands reported by HYC22 have been +used throughout this study. +We initially assume 1H 0707-495 to have a black hole mass +log 𝑀 = 6.31𝑀⊙ (Bian & Zhao 2003) and redshift 𝑧 = 0.0411 +(Leighly 1999). The luminosity distance from the NED database is +187 Mpc. This AGN has been well documented to be dominated by +relativistically blurred reflection at either a low or a moderate incli- +nation angle (see e.g. Fabian et al. 2009; Zoghbi et al. 2010; Dauser +et al. 2012; Kara et al. 2013a; Caballero-García et al. 2018). We +employ the inclination angle 𝑖 = 53◦ as derived from the emissiv- +ity profile by Wilkins & Fabian (2011). For IRAS 13224-3809 we +assume a black hole mass of log 𝑀 = 6.30𝑀⊙ (Alston et al. 2018, +2020; Caballero-García et al. 2020). We also adopt appropriate val- +ues for redshift 𝑧 = 0.0406 and inclination 𝑖 = 64◦ as reported by +MNRAS 000, 1–13 (2020) + +Extended Corona Models of X-ray Reverberation in AGN +3 +Fabian et al. (2013). The luminosity distance from the NED database +is 310 Mpc. +3 THE EXTENDED CORONA MODEL (ECM) +The extended corona model (ECM) assumes a standard geometrically +thin, optically thick accretion disc (Shakura & Sunyaev 1973) which +extends from the innermost stable circular orbit (ISCO), or the radius +of marginal stability 𝑟ms, to 400 𝑟𝑔 around a central black hole. Note +that accreting supermassive black holes were mostly found to be +rapidly spinning (Reynolds 2021), therefore to limit the number of +free parameters and to avoid the model degeneracy, we fix the black +hole spin to be 𝑎 = 0.998. The accretion disc is illuminated by two +compact X-ray sources located on the symmetry axis at heights ℎ1 and +ℎ2 which are the lower and upper source heights respectively whose +amplitudes as a function of time are 𝑥1(𝑡) and 𝑥2(𝑡), respectively, +as in CY17. We maintain the two-source to investigate the extent of +the corona and a basic sketch of this scenario is shown in Figure 1. +Physically, the lower X-ray source may represent the base of a jet-like +structure or the lower region of a compact corona (Wilkins & Gallo +2015) and the upper source represents the farthest region from where +a flare or response from the disc is detected hence it is interpreted +as the upper extreme of the corona. A further plausible explanation +of this region could be due to a periodic vertical collimation of the +corona as a jet launching event subsides (Wilkins et al. 2015). These +explanations also suggest that photon emission is beamed vertically +away from the disc. +The photon trajectories were traced along Kerr geodesics as de- +scribed by Bardeen et al. (1972) and essentially outlined by CY17 +by first considering the flares of the two X-ray sources as two sepa- +rate delta functions and tracing the photons between the two sources, +the disc and the observer along Kerr geodesics. The full relativis- +tic effects outlined by Cunningham (1975) are included. The X-ray +reprocessing is modelled using REFLIONX (George & Fabian 1991; +Ross et al. 1999; Ross & Fabian 2005). In CY17, many of the free +parameters were reduced by making model assumptions based on the +physical environment of PG1244+026, for example, the inclination +angle, photon index and ionisation were fixed for simplicity at the +values suggested in previous literature (e.g. Kara et al. 2014). Here, +the inclination angle is fixed at 𝑖 = 53◦ for 1H 0707-495 (Wilkins +& Fabian 2011) and 𝑖 = 64◦ for IRAS 13224-3809 (Fabian et al. +2013), as described in Section 2, but both photon index and ionisa- +tion are allowed to be free. Note that a high inclination of 𝑖 > 60◦ +for IRAS 13224-3809 was also supported by the broadband spectral +fitting (Jiang et al. 2018) and simultaneous lag-frequency spectral +fitting (Alston et al. 2020). Fixing the inclination can help to avoid +degeneracies in the model. Then, the model contains an ionisation +parameter +𝜉(𝑟, 𝜙) = 4𝜋𝐹𝑡 (𝑟, 𝜙)/𝑛(𝑟) +(1) +where 𝐹𝑡 (𝑟, 𝜙) is the total flux due to both X-ray sources per unit +area of the disc at (𝑟, 𝜙), and 𝑛(𝑟) is the disc density, 𝑛(𝑟) ∝ 𝑟−𝑝. +The aim is to reproduce the high frequency soft (reverberation) lags +and the harder lags seen at lower frequencies which are associated +with propagating fluctuations in the disc. Kotov et al. (2001) and +Arévalo & Uttley (2006) describe the low frequency hard lags by +mass accretion fluctuations propagating inwards through the disc +where the central region contains a source of harder X-rays. That +is, the fluctuations modulate soft X-rays first resulting with the hard +lags, so the soft bands are dominant first before being overcome by +the hard band. Assuming these fluctuations cause the central X-ray +sources to respond at different times, the extended corona scenario +models this source response using a cut off power law +Ψ𝑖(𝑡) ∝ 𝑡−𝑞𝑖exp(−𝑡/𝑡max) +(2) +where the subscripts 𝑖 = 1 and 2 refer to the parameters of the lower +and upper sources, respectively. How the function decays with respect +to time is determined by 𝑞𝑖, where 𝑡 = 0 and 𝑡max is the beginning +and the end of the source response. Only the time difference between +the two responses is relevant. +In addition, the model makes use of the parameter 𝑡shift to delay the +response of the second source that reacts slower to primary variations +of the first source. In essence, these inward propagating fluctuations +can produce primary variations in the disc that will affect the flux of +the lower X-ray source at time 𝑡 = 1 and propagate upwards to the +upper X-ray source taking time 𝑡shift. While the disc response usually +obtained from the ray-tracing simulation is a function of energy and +time, the source response is independent of energy. However, the +source variability remains dependent of energy via 𝐹𝑖(𝐸) ∝ 𝐸−Γ𝑖, +where Γ𝑖 is the photon index of the X-ray continuum of the 𝑖th source. +The source variability in the energy band 𝐸 𝑗 is +𝑥𝑖(𝐸 𝑗, 𝑡) ∝ 𝐹𝑖(𝐸 𝑗)𝑥0(𝑡) ⊗ Ψ𝑖(𝑡), +(3) +so that the lower and upper sources respond in different ways to the +primary variations 𝑥0(𝑡), due to 𝐹𝑖(𝐸 𝑗) and Ψ𝑖(𝑡). We initially set 𝜉 +for a neutral to highly ionised environment where log 𝜉 = 0.0 − 3.0 +respectively and allow to vary during the fitting procedure. Other +initial assumptions are made by setting the Fe abundance to solar +and fixing the decay of the lower-source response 𝑞1 = 0.5. +We began by creating the disc model by integrating photon paths +from the observer to the disc assuming a maximum black hole spin +where 𝑎 = 0.998 with the disc extending from ISCO out to 400𝑟𝑔. +The next task was to integrate photon paths from a single X-ray source +to the disc, after which the spectra was calculated using REFLIONX +for a given source height and disc inclination already computed in the +previous steps. The final step was to calculate the reverberation signa- +tures by looking at the already computed source-to-disc and observer- +to-disc ray tracing runs. The disc response function, 𝜓𝑖(𝐸 𝑗, 𝑖), was +computed and the variability of the disc reflection can be calcu- +lated via a convolution term: 𝑥𝑖(𝐸 𝑗, 𝑡) ⊗ 𝜓𝑖(𝐸 𝑗, 𝑖). The observed +X-ray variability then can be written as the sum of the source and +the disc variability. The lag-frequency spectra were calculated from +the Fourier-phase differences between two energy bands, following +the standard techniques (e.g. Cackett et al. 2014; Emmanoulopoulos +et al. 2014; Chainakun & Young 2015). +Note that the light curve in each energy band always contains +both continuum and reflection components. The contamination of the +continuum flux in the reflection-dominated band, and the reflection +flux in the continuum-dominated band cause dilution effects which +can reduce the lag amplitude, meaning that the measured time lags +are shorter than the intrinsic time lags. The dilution effects modify the +shape of the lag-frequency spectrum without affecting the frequency +at which the lags occur. For discussions on dilution see, e.g., Uttley +et al. (2014); Kara et al. (2014); Chainakun & Young (2015). In order +to deal with dilution effects, the variability of the reflected photons of +the disc is normalized using the reflected response fraction (defined +as the reflected flux/continuum flux) of all energy bands 𝐸 𝑗. We +include a brightness parameter 𝑏 (as measured in the frame of the +observer) as a ratio of the brightness of the lower and upper source, +𝑏1 and 𝑏2 respectively by defining 𝑏 = 𝑏2/𝑏1 and fix 𝑏1 = 1 and +allow 𝑏 to vary between 1–3. The continuum flux of the lower source +will be less than the upper source due to its closer proximity to the +MNRAS 000, 1–13 (2020) + +4 +S. Hancock et al. +Figure 1. A simple sketch of the extended corona scenario showing the two +X-ray sources located on the rotation axis of an accreting black hole. The +arrows show the trajectories of the continuum and accretion disc reflected +photons. +black hole, therefore photons will be subject to light bending effects +towards the centre (Miniutti & Fabian 2004). +The model is capable of reproducing the prominent time lag fea- +tures seen in AGN when Γ1 ≠ Γ2 as seen in the top panel of Figure +2. In addition Γ1 > Γ2 for positive low freqency lags and Γ2 > Γ1 +for negaitive low frequency lags. As 𝑡max increases, so does the am- +plitude of the hard lags as seen in the bottom panel of Figure 2 and +aliasing effects (phase wrapping) are moved to lower frequencies and +positive lags will switch to negative lags. The low frequency hard +lags increase with energy and the softer reverberation lags dominate +at higher frequencies, consistent with the ‘two-mechanism’ features +of propagation and reflection and was also consistent with the tradi- +tional spectral features that are widely explained by reflection from +the inner disc. +We generated the initial model by running the code for a course +range of parameter values to fit the lag-frequency spectra of the +fully combined observations 1H0707-495 using an inclination angle +𝑖 = 53◦. We set the density profile to constant (𝑝 = 0) and the +response decay function 𝑞2 = 0.5 and iron abundance 𝑍Fe = 1.0. +The lower source height was fixed at 2 𝑟𝑔 and upper source height +was allowed to vary between 3−20 𝑟𝑔 thus obtaining the extent of the +corona. While fixing the lower source at 2 𝑟𝑔 is a pragmatic choice +that simplifies the fitting and limits the number of free parameters, +having one source this close to the horizon is comparable to the +limits by previous spectral or timing modelling, (e.g., Kara et al. +2013a; Caballero-García et al. 2018, 2020). In addition, Γ1 and Γ2 +were allowed to vary between 1.4 − 3.3 in line with REFLIONX, the +ionisation parameter specified at the ISCO could vary from 0 − 3 for +neutral and highly ionised exploration respectively. The brightness +parameter was also allowed to vary from 1 − 3. The remaining free +parameters were 𝑡max and 𝑡shift and we initially set these ranging from +50 − 600 𝑡𝑔 and 0 − 120 𝑡𝑔, respectively. The black hole mass for +each source was initially tested using sensible values from literature +(Bian & Zhao 2003; Alston et al. 2020) and allowed to vary during +the fitting procedure. +10-5 +10-4 +10-3 +10-2 +10-1 +100 +Frequency (1/tg) +35 +30 +25 +20 +15 +10 +5 +0 +5 +10 +Time Lag (tg) +Γ1 = 2.0, Γ2 = 1.7 +Γ1 = 2.1, Γ2 = 1.9 +Γ1 = 2.1, Γ2 = 2.0 +Γ1 = 2.1, Γ2 = 2.1 +Γ1 = 2.1, Γ2 = 2.3 +Γ1 = 2.1, Γ2 = 2.5 +Γ1 = 2.1, Γ2 = 2.6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +Frequency (1/tg) +10 +5 +0 +5 +10 +15 +Time Lag (tg) +tmax = 100tg +tmax = 300tg +tmax = 500tg +tmax = 700tg +Figure 2. The frequency dependent lags varying with Γ and 𝑡max. Both panels +were set at 𝑖 = 53◦, ℎ1 = 2, ℎ2 = 3, 𝐹𝑒 = 1.0, log 𝜉 = 1.0 and 𝑞2 = 0.5. The +top panel shows the frequency dependent lags varying with Γ1 and Γ2. The +positive low frequency fluctuation lags and the negative soft reverberation +lags are produced when Γ1 ≠ Γ2. The bottom panel shows lag behaviour with +different values of 𝑡max, where Γ1 = 2.0 and Γ2 = 1.5, 𝑏 = 2.0 and 𝑡shift = 10. +4 RESULTS +4.1 Model testing on 1H 0707-495 +The ECM fitting was developed using the Interactive Spectral Inter- +pretation System (ISIS) Version 1.6.2-27. Generally, much better fits +were obtained using single observations than combining the observa- +tions into groupings. Although spectral combinations are useful for +obtaining snapshots of the various epochs, the true variability is best +revealed by examining each observation in turn, however this is time +consuming and computationally expensive. Firstly, we independently +examined the posterior distribution for the mass using combined ob- +servation best fit. For this inspection we use the isis_emcee mod- +ule outlined at Remeis-Wiki (2018) by adopting the methods of the +MCMC hammer (Goodman & Weare 2010; Foreman-Mackey et al. +2013). We initiate 500 walkers and run the chain with 10,000 iter- +ations. The walkers converged tightly after 1,000 steps and settled +into an acceptance rate of ∼0.7 after ∼3,000 iterations to obtain a +MNRAS 000, 1–13 (2020) + +b +hExtended Corona Models of X-ray Reverberation in AGN +5 +6.1 +6.2 +6.3 +6.4 +0 +20 +40 +60 + lag_freq(1).log_mass + Probability +1H 0707-495 +6.2 +6.3 +6.4 +2.5×10−5 +3×10−5 + lag_freq(1).log_mass + lag_freq(1).norm +1H 0707-495 +6.1 +6.2 +6.3 +6.4 +0 +10 +20 +30 +40 + lag_freq(1).log_mass + Probability +IRAS 13224-3809 +6.2 +6.3 +6.4 +1.4×10−51.6×10−51.8×10−5 + lag_freq(1).log_mass + lag_freq(1).norm +IRAS 13224-3809 +Figure 3. Independent (emcee) tests of the posterior density for the combined +observations of 1H 0707-495 (top row) with mean mass 𝜇 = 6.22 (solid blue +line) and 𝜎 = 0.01 (black dashed lines). IRAS 13224-3809 is also shown +(bottom row) with mean mass 𝜇 = 6.34 and 𝜎 = 0.01. The contour plots of +the mass and normalisation are shown in the right column. All 𝑥-axes are +identical for comparison. +mass of log(𝑀/𝑀⊙) = 6.22 ± 0.01, where the errors are calculated +at the 90% confidence limit. The posterior density and contour plot is +shown in the top row of Figure 3. This value is comparable to that re- +ported in the literature (e.g. Zhou & Wang 2005; Zoghbi et al. 2010) +although uncertainties in the estimations have been acknowledged. +This initial model fit was reasonable where the 𝜒2 was 2.32 and the +upper source was located at 20.00+0.00 +−0.11 𝑟𝑔. +For this mass and geometry, the photon indices Γ1 and Γ2 were +well constrained at 2.6+0.01 +−0.07 and 2.1+0.72 +−0.01 respectively, where the +ionisation log 𝜉 ∼ 3.0. The upper source was exactly twice as bright +as the lowers source given by parameter 𝑏 ∼ 2. The source response +time from disc fluctuations 𝑡max = 250.00+67.82 +−0.04 +𝑡𝑔 and the time +taken for the fluctuations to propagate from the lower to upper source +𝑡shift = 20.00+11.05 +−11.92 𝑡𝑔. Although this initial fit to the combined data +is promising and can provide a good description of the data, we +acknowledge the reasonably large errors returned for 𝑡max, and the +loosely constrained errors found for 𝑡shift. In general, this is typical +of the fits found for all data. +Further complexity was revealed when attempting to standardise +the mass value for the remainder of the data to the value obtained +from model fitting via the University of Bristol high performance +computer BlueCrystal. A fixed mass of log(𝑀/𝑀⊙) = 6.22 gen- +erally returned much poorer fits and the fitting procedure remained +computationally intensive. +To develop this model further, the reverberation signatures were +read into a table model that can be used to fit the data in ISIS and +XSPEC. This step required the use of astropy and heasp, the latter +of which is a component of the standard HEASOFT installation +(NASA 2020). The table model was initially generated using Python +and the parameter space was given finer step sizes between parameter +values to obtain a finer grid. The output model was huge (>40 GB) +and the model loading time was too long for general usage, therefore +the parameter space was constrained further. The final table model +was 19 GB and the parameter space is outlined in Table 2. Although +Table 2. The extended corona model parameter space for both sources. Note +that 𝑓 denotes fixed parameter +Parameter +1H 0707-495 +IRAS 13224-3809 +log(𝑀/𝑀⊙) +5.5 − 8 +5.5 − 8 +𝑖 (◦) +53 𝑓 +64 𝑓 +𝑎 +0.998 𝑓 +0.998 𝑓 +ℎ1 (𝑟𝑔) +2 𝑓 +2 𝑓 +ℎ2 (𝑟𝑔) +3 − 20 +3 − 20 +Γ1 +1.5 − 3.3 +1.5 − 3.3 +Γ2 +1.4 − 3.2 +1.4 − 3.2 +𝑍Fe (solar) +1 𝑓 +1 𝑓 +log 𝜉 (ergs cm s−1) +0 − 3 +0 − 3 +𝑝 +0 𝑓 +0 𝑓 +𝑏 +1 − 3 +1 − 3 +𝑞2 +0.5 𝑓 +0.5 𝑓 +𝑡max (𝑡𝑔) +50 − 700 +50 − 1500 +𝑡shift (𝑡𝑔) +0 − 120 +0 − 150 +this is still very cumbersome, further constraints at this stage were +halted due to dynamic benefits of the model parameter space. +The first table model fit was conducted on the combined data for +1H 0707-495, achieving a very good model fit to the data where +𝜒2 = 0.501. This fitting was applied to all of the combination flux +groups for low, medium, high etc and the results are presented in +Table A1. The combined data and model fit along with the lag- +frequency spectrum file predicted by the model is presented in the top +panel of Figure 4. Of course, a full table model listing every possible +integer value of parameter space would be extremely difficult to +achieve given the intensive computational power required, so only the +closest lag-frequency, that is the predicted reverberation signatures +are shown unbinned at full resolution by the solid grey wavy lines. +4.2 Model testing on IRAS 13224-3809 +Once again we explore the initial model fit using isis_emcee with +500 walkers and 10,000 iterations and walkers converged tightly after +only ∼ 200 steps, settling into an acceptance rate of ∼0.7 after ∼300 +iterations. We obtained a mass of log(𝑀/𝑀⊙) = 6.34±0.01 for IRAS +13224-3809, where the errors are calculated at the 90% confidence +limit. Again we note the model capability of returning a mass value +closely comparable to log(𝑀/𝑀⊙) ∼ 6.3 as reported in, e.g., Alston +et al. (2020). The mass posterior density and contour plot, along with +the model fit is shown in bottom row of Figure 3. The ECM fit was +statistically good where 𝜒2 = 1.47 with the upper source located at +5.00+5.52 +−0.02 𝑟𝑔. For this mass and geometry, the photon indices Γ1 and +1 Note that the number of degrees of freedom depends on the binning being +used, and with lighter binning we would have more degrees of freedom. +We have chosen to bin the data more heavily to clearly show the time lag +versus frequency with higher signal-to-noise ratio, so there are fewer data +bins, which means that the number of degrees of freedom is very low. The +𝜒2 values reported here then are close to the reduced 𝜒2 values. This also +highlights a more general problem when we apply the extended corona model +while the quality of the time lags is limited. The data are however, still able +to strongly constrain our models, showing which regions of parameter space +provide a good description of the time lags and which can be ruled out. +MNRAS 000, 1–13 (2020) + +6 +S. Hancock et al. +10−4 +10−3 +−200 +0 +200 +400 +Frequency (Hz) +Time lag (s) +10−4 +10−3 +−500 +0 +500 +1000 +Frequency (Hz) +Time lag (s) +Figure 4. The combined lag-frequency Table model fits for 1H 0707-495 +(top panel) and IRAS 13224-3809 (bottom panel) showing the best model fit +(red) where the mass is variable. The closest reverberation signature is shown +by the grey wavy line. Note that the lags are estimated between the soft, +0.3 − 0.8 keV, and hard, 1 − 4 keV, bands. Furthermore, there is significant +cancellation of positive and negative lags due to phase wrapping and binning. +Γ2 were well constrained at 2.60+0.02 +−0.59 and 1.90 ± 0.01 respectively, +where the ionisation log 𝜉 ∼ 1.0. The upper source was about 3 times +as bright as the lowers source (𝑏 ∼ 3). The source response time from +disc fluctuations 𝑡max = 1300.00+0.00 +−304.36 𝑡𝑔. Also note that the ranges +of the parameters 𝑡max and 𝑡shift required adjusting up to 1500 𝑡𝑔 and +150 𝑡𝑔 respectively for this source as summarised in Table 2. The +model was able to provide a good statistical description of all data +when fitted via the table model. The best-fit result for the combined +observations is presented in the bottom panel of Figure 4. Note that +there is significant cancellation of positive and negative lags dues to +phase wrapping and binning. +4.3 Constraining the black hole mass +Before exploring the simultaneously fitted ECM model, we inspected +the variable mass model fits as a function of the negative (reverbera- +tion) time lags. For each source this relationship is shown in Figure 5, +suggesting that whilst the black hole mass and time lag relationship +is not evident for 1H 0707-495 (𝑝 > 0.05), it is moderately anti- +correlated in IRAS 13224-3809. Since the black hole mass for each +105 +106 +107 +M/M⊙ +10-1 +100 +101 +102 +103 +Time lag (s) +rs = 0.258 + p = 0.257 +1H 0707-495 +106 +107 +M/M⊙ +10-1 +100 +101 +102 +103 +Time lag (s) +rs = − 0.494 + p = 0.017 +IRAS 13224-3809 +Figure 5. The variable mass from the Table model fits as a function of the +reverberation lag for 1H 0707-495 (top) and IRAS 13224-3809 (bottom). +Other physical processes related to the coronal properties, rather than ge- +ometry, may involve in manifesting the lag-mass scaling relation under the +extended corona environment especially in IRAS 13224-3809 where the anti- +correlation between the lags and the mass is seen. Of course, there is only +one true value of the mass for each AGN. +source should be constant during these observations, the variable +mass found in IRAS 13224-3809 could be induced by other geomet- +ric effects in an extended corona environment that are not related to +the central mass. It could also be due to the fact that the height and +the mass are degenerate (Caballero-García et al. 2018, 2020), since +the mass and the source height affect the lags in a similar way. In any +case, our results suggest that it might be better to fix the black hole +mass when performing reverberation analysis. +To obtain the best mass value for all data, we use similar procedures +to Alston et al. (2020) by simultaneously fitting the data. We allowed +the mass to vary between values of log(𝑀/𝑀⊙) = 5.5 − 8.0 and +loaded the first data set and ran the table model fit, then added +the second data set where the identical parameters were tied for +each set. Essentially we leave the first mass as a free parameter and +subsequently tie the remaining data masses to the first data set. This +method of building one large parameter file was computationally +intensive. The results confirmed our expectations that not all data +sets would obtain individually excellent fits and, for 1H 0707-495, +the 𝜒2 statistics ranged from 0.19 − 2.98. This method, however, +helped to reduced the large errors for 𝑡max and 𝑡shift for the majority +of the data. The single mass value obtained for 1H 0707-495 was +log(𝑀/𝑀⊙) = 6.04 ± 0.01. The 1H 0707-495 mass constrained +MNRAS 000, 1–13 (2020) + +Extended Corona Models of X-ray Reverberation in AGN +7 +10−4 +10−3 +0.01 +0 +500 +Frequency (Hz) +Time lag (s) +log𝑀 = 6.7 +log𝑀 = 6.3 +log𝑀 = 6.0 +10−4 +10−3 +0.01 +0 +500 +Frequency (Hz) +Time lag (s) +ℎ2 = 20 𝑟𝑔 +ℎ2 = 10 𝑟𝑔 +ℎ2 = 5 𝑟𝑔 +Figure 6. Top panel: The variable mass behaviour for IRAS 13224-3809 +combined table model where ℎ2 = 13.0 𝑟𝑔 and log𝑀 = 6.0 (black), 6.3 (red) +and 6.7 (blue). Specific model parameters of the reverberation signature are +also provided. Bottom panel: The variable height behaviour is shown for the +same signature parameters and mass values as seen in the top panel, where +ℎ2 = 5 𝑟𝑔(black), 10 𝑟𝑔 (red) and 20 𝑟𝑔(blue). The other model parameters +were Γ1 = 2.6, Γ2 = 1.7, log 𝜉 = 3.0, 𝑏 = 2.0, 𝑡max = 300, 𝑡shift = 20 +(for both panels). Note that in the ECM model, the mass scales not only the +negative reverberation lags but also the positive lags. +here is slightly smaller than the mass of log(𝑀/𝑀⊙) = 6.37 as +reported in Zhou & Wang (2005). In fact, Caballero-García et al. +(2018) also adopted the mass from Zhou & Wang (2005) and showed +that if the mass is larger than this, the lamp-post model will not +be able to fit the reverberation time-lag data. Our obtained mass +for 1H 0707-495 is then still in the acceptable regime found by +Caballero-García et al. (2018). We repeated these methods to obtain +the single mass value for IRAS 13224-3809. All 17 data sets were +reasonably well constrained statistically where 𝜒2 ranged 0.17 – 2.75, +achieving a single mass value of log(𝑀/𝑀⊙) = 6.30 ± 0.01, which +is closely consistent with what obtained by Alston et al. (2020). +Figure 6 shows an overview of the behaviour of the time lags as a +function of frequency for different black hole mass values and upper +X-ray source locations. While the positive lags in previous literature +were usually modelled independently using a power-law function, the +ECM model can simultaneously produce both negative and positive +lags. +Table 3. The simultaneously fitted ECM Spearman’s rank correlations 𝑟𝑠 +and the associated 𝑝 value for each source and all data. Note that the first +row is showing the correlations for ℎ2 − 𝐿(2−10 keV) and the final row is the +Covering Fraction obtained from HYC22. The last column shows the results +when all data from both 1H 0707-495 and IRAS 13224-3809 are used to find +the correlation coefficients. +Par1 +Par2 +1H 0707-495 +IRAS 13224-3809 +1H and IRAS +𝑟𝑠 +𝑝 +𝑟𝑠 +𝑝 +𝑟𝑠 +𝑝 +ℎ2 +𝐿 +0.052 +0.853 +-0.127 +0.628 +-0.200 +0.272 +ℎ2 +Lag +0.312 +0.257 +0.556 +0.020 +0.418 +0.017 +ℎ2 +𝜉 +-0.631 +0.012 +-0.086 +0.742 +-0.423 +0.016 +ℎ2 +Γ1 +0.573 +0.026 +-0.199 +0.444 +0.131 +0.472 +ℎ2 +Γ2 +0.195 +0.486 +0.525 +0.030 +0.482 +0.005 +Γ1 +𝑡max +0.216 +0.440 +0.716 +0.001 +0.591 +0.001 +Γ1 +𝑡shift +0.327 +0.235 +0.303 +0.237 +0.381 +0.031 +Γ2 +𝑡max +-0.034 +0.904 +-0.725 +0.001 +-0.549 +0.001 +Γ2 +𝑡shift +-0.587 +0.021 +-0.648 +0.005 +-0.594 +0.00034 +𝑏 +𝑡max +-0.345 +0.125 +-0.661 +0.004 +-0.411 +0.019 +𝑏 +𝑡shift +-0.658 +0.008 +0.096 +0.714 +-0.125 +0.495 +𝑏 +Cvr +-0.603 +0.017 +0.214 +0.408 +-0.154 +0.402 +4.4 ECM correlations +We explore the model for correlations using the Spearman’s rank +method. For this we use all individual observation table model fits +returned from the single mass value phase for each AGN to maximise +the available data. We found moderate correlations between Γ1 and +𝑡max and between Γ2 and 𝑡max and 𝑡shift. On the other hand, large +errors are still a feature of 𝑡max therefore we acknowledge that these +correlations are not well constrained. Further investigation reveals +these correlations are often much stronger in IRAS 13224-3809 than +they are in 1H0707-495. The brightness parameters 𝑏 was strongly +anti-correlated with 𝑡max in IRAS 132224-3809, where no correlation +was evident in 1H 0707-495. Note that Caballero-García et al. (2018) +also suggested that the 1H 0707-495 mass should be fixed at values +below ∼ (2 − 3) × 106 𝑀⊙, otherwise the observed reverberation +lags cannot be explained. The time lag amplitude was moderately +correlated with the upper photon index Γ2 in all cases. The results +are presented in Table 3, which shows the parameters of interest and +the overview of the Spearman’s rank correlations𝑟𝑠 and their 𝑝 values +for each source in columns 3 and 4 and for any global correlations in +the final column. Note that the covering fraction is also tested from +the results reported in HYC22. We found no correlation between the +upper source height and the luminosity, however the time lags do +correlate moderately with the upper X-ray source height in 1H 0707- +495, and a stronger relationship is evident in IRAS 13224-3809. +Further consideration of the lags seen ≲ 250 s in the latter source +provides a much stronger correlation where 𝑟𝑠 = 0.68 with 𝑝-value = +0.005. The six strongest correlations found for these sources are +shown in Figure 7 and 8 respectively. The panels within these figures +also show where the limits of each parameter were reached as denoted +by the blue arrows. +Both AGN have a moderate relationship for Γ2 and 𝑡shift which +can be seen as the strongest global correlation where 𝑟𝑠 = −0.594 +and 𝑝-value = 0.00034. The lower source photon index Γ1 also +correlates strongly with 𝑡max in IRAS 13224-3809 where 𝑟𝑠 = 0.716 +and 𝑝-value = 0.001 with no counterpart correlation seen in 1H +MNRAS 000, 1–13 (2020) + +8 +S. Hancock et al. +0707-495. For the upper source Γ2 a moderate correlation is seen +with the brightness parameter 𝑏 in 1H 0707-495 and another strong +inverse correlation emerged with 𝑡max in IRAS 13224-3809 where +𝑟𝑠 = −0.725 and 𝑝-value = 0.001. Finally, a strong correlation found +in IRAS 13224-3809 and not in 1H 0707-495 was ΔΓ, (Γ1−Γ2), with +𝑡max where 𝑟𝑠 = 0.82 and 𝑝-value = 5.86 × 10−5. It is interesting +to note that unlike 1H 0707-495, IRAS 13224-3809 simultaneous +fitting does not always lead to Γ1 > Γ2. These results suggest that +the spectral properties of the corona may be specific to the source. +The Γ − 𝐿/𝐿Edd and Γ − ℎ2 relationships for these AGN are also +presented in Figure 9. It can be seen that lower and higher limits of Γ +and ℎ2 often reached their limits and these parameters may be wider +as indicated by the blue arrows in each panel. +In fact, the photon index Γ1 tends to increase sharply with in- +creasing luminosity above ∼ 2.5 × 1042 erg cm s−1 for both sources. +A similar trend in AGN data against the luminosity ratio 𝐿𝑥/𝐿Edd +has been reported by Yang et al. (2015) and interpreted as a two- +phase accretion flow model, although more data would be required +to investigate this scenario. For 1H 0707-495, the correlation is flat +at lower luminosity then a mild positive correlation kicks in when +the luminosity is 10−1.7 ≲ 𝐿/𝐿Edd ≲ 10−1.5. Contrarily, for IRAS +13224-3809, we see the progressively flat profile of Γ instead, and +their Γ1 and Γ2 values are extreme and more separated. The be- +haviour of Γ may suggest that contrasting mechanisms are driving +the variability in these AGN (within the model constraints). +5 DISCUSSION +The initial fitting of the ECM to the various flux groups achieved rea- +sonable fits and model descriptions, however these were not always +well constrained and error ranges often tended to the model maxi- +mum and minimum allowed values. The model can provide a good +statistical fit to the observed data when the mass is employed as a free +parameter and is capable of finding the upper X-ray source heights +between 3 – 20 𝑟𝑔. Fixing the mass at the single best value obtained +from the simultaneous fits also achieved similar results and whilst +the black hole mass of 1H 0707-495 dropped from 6.22 (obtained by +the independent emcee fit) to 6.04 ± 0.01 log(𝑀/𝑀⊙), both model +fits remain statistically reasonable. The black hole mass for IRAS +13224-3809 was estimated at 6.30 ± 0.01 log(𝑀/𝑀⊙) and remained +reasonably consistent with the independent measurement. These ob- +tained mass values contained very small errors and they fall within +the limits of the variable mass values from the model fits presented +in Table A1. +Caballero-García et al. (2020) fix the mass of IRAS 13224-3809 +at log(𝑀/𝑀⊙) = 6.30 and their model looks at the reverberation +from a single point source and predicts only the negative reverber- +ation lags, whilst the positive lags at low frequency are modelled +separately using the KYNXILREV phenomenological power law. Our +model is mass dependent and considers both soft and hard lag mech- +anisms simultaneously. Nevertheless, we note that a limiting factor +may be that we have fixed the lower source height at 2 𝑟𝑔; an as- +sumption required to ease the heavy computations required to create +the predicted time lags and hence we are estimating the extent of the +corona rather than its true size that may be obtained by employing ℎ1 +as a free parameter. This would be extremely intensive with the cur- +rent model and further development should maintain its complexity +whilst enhancing model computational performance. +Despite this, some meaningful relations among the ECM parame- +ters can still be inferred. For 1H 0707-495, we find significant corre- +lation (𝑝 < 0.05) between ℎ2 and Γ1, and significant anti-correlation +between ℎ2 and 𝜉. This means that when the 1H 0707-495 corona +extends upwards (larger ℎ2), the coronal emission seems to be softer +(larger Γ) while producing less overall ionisation on the disc (smaller +𝜉). This is expected since the vertically extended corona should pro- +duce less intense illumination pattern on the inner disc (e.g. Wilkins +& Fabian 2012). The tendency of softer corona with increasing its +vertical extent is also found in IRAS 13224-3809. This is in line +with Alston et al. (2020) and Chainakun et al. (2022b) where the +lamp-post geometry was used and the photon index of the continuum +was found to increase with the source height. In fact, Chainakun +et al. (2022b) also found that during the source height increases, the +disc itself generates more high-frequency variability, suggesting that +the inner disc becomes more active. Here, we find the significant +anticorrelation between Γ2 and both 𝑡max and 𝑡shift (𝑟𝑠 = −0.724 +and −0.648, respectively). Therefore, when the extending corona be- +comes softer, the source response as well as the signal propagation +occurs faster. This may agree with Chainakun et al. (2022b) that as +the source height increases the innermost region is more active so +that the corona requires shorter time in response. +Nevertheless, in the ECM environment, the time lags in IRAS +13224-3809 correlate stronger with ℎ2 than those found in 1H 0707- +495. Consideration of the lags seen under 250 s provides a much +stronger correlation where 𝑟𝑠 = 0.68 with 𝑝-value = 0.005, hence +there are hints that much longer time lags will still follow this cor- +relation albeit with a shallower slope. Furthermore, Chainakun et al. +(2019) pointed out that in order reproduce the time lags, their spher- +ical corona model required higher coronal temperatures for a lower +optical depth, 𝜏, supporting the 𝜏 and coronal temperature anti- +correlation argument discussed by Tortosa et al. (2018). Recently, +Chainakun et al. (2022a) presented a new method of predicting the +black hole mass of X-ray reverberating AGN using artificial neu- +ral networks and pointed out that the inconsistency of the lag-mass +relationship may be due to the lag amplitudes being more strongly +affected by other geometric effects that are not related to the mass +of the black hole. Also, there is a significantly larger number of +parameters in the ECM than in the lamp-post model. Perhaps, this +might explain the observed anti-correlation between the lags and the +mass seen in IRAS 13224-3809 under the extended corona environ- +ment when using individual observations (Fig. 5), that the lag-mass +relation is also modulated by other coronal parameters in the model. +In the lamp-post geometry, the X-ray source heights in IRAS +13224-3809 have been reported to correlate positively with the lumi- +nosity in the 2 – 10 keV energy range (Alston et al. 2020; Caballero- +García et al. 2020; Chainakun et al. 2022b). However, there was no +evidence of this relationship in this study. Whilst this is consistent +with the findings of Szanecki et al. (2020), it is in contrast to their +findings that the source is compact within a few gravitational radii +(when modelled using the relativistically smeared reflection from +the accretion disc). Variations of the inner regions and accretion flow +have been explained via transitions between the jet emitting disc +(JED) and standard accretion disc (SAD) framework (Marcel et al. +2022) and the framework of coupled hot accretion and jet model for +GBHBs and AGN (Yang et al. 2015), suggesting that a two-phase +accretion flow could be driving the observed variability. +MNRAS 000, 1–13 (2020) + +Extended Corona Models of X-ray Reverberation in AGN +9 +1.5 +2.0 +2.5 +3.0 +3.5 +Γ1 +0 +5 +10 +15 +20 +h2(rg) +rs = 0.573 + p = 0.026 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +ξECM (erg cm s−1) +0 +5 +10 +15 +20 +h2(rg) +rs = − 0.631 + p = 0.012 +0 +20 +40 +60 +80 +100 +120 +t-shift (tg) +1.0 +1.5 +2.0 +2.5 +3.0 +Γ2 +rs = − 0.587 + p = 0.021 +1.0 +1.5 +2.0 +2.5 +3.0 +b +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Γ2 +rs = 0.572 + p = 0.026 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Covering fraction +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +b +rs = − 0.603 + p = 0.017 +1.0 +1.5 +2.0 +2.5 +3.0 +b +0 +20 +40 +60 +80 +100 +120 +t-shift (tg) +rs = − 0.658 + p = 0.008 +Figure 7. The 1H 0707-495 ECM results showing moderate to strong correlations where the 𝑝-value <0.05. For clarity of the ℎ2 − 𝜉ECM correlations (upper +middle panel) have been heavily averaged into 4 groups and shown by the yellow points for data points located in quadrants identified by the grey dashed lines. +200 +0 +200 +400 +600 +800 +Time lag (s) +5 +10 +15 +20 +h2(rg) +rs = 0.556 + p = 0.020 +1.5 +2.0 +2.5 +3.0 +Γ2 +5 +10 +15 +20 +h2(rg) +rs = 0.525 + p = 0.030 +1.6 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +3.0 +3.2 +3.4 +Γ1 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +t-max (tg) +rs = 0.716 + p = 0.001 +1.5 +2.0 +2.5 +3.0 +Γ2 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +t-max (tg) +rs = − 0.725 + p = 0.001 +1.5 +2.0 +2.5 +3.0 +Γ2 +0 +20 +40 +60 +80 +100 +120 +140 +160 +t-shift (tg) +rs = − 0.648 + p = 0.005 +1.0 +1.5 +2.0 +2.5 +3.0 +b +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +t-max (tg) +rs = − 0.661 + p = 0.004 +Figure 8. The IRAS 13224-3809 ECM results showing moderate to strong correlations where the 𝑝-value <0.05. Each correlation has been divided into bin +sizes as shown by the vertical grey dashed lines and the resultant mean plot (and associated errors) has been shown by the large yellow data points for clarity of +each correlation. +MNRAS 000, 1–13 (2020) + +10 +S. Hancock et al. +For 1H 0707-495, the profile of Γ is flat at lower luminosity before +showing a mild positive correlation at 10−1.7 ≲ 𝐿/𝐿Edd ≲ 10−1.5 +(Figure 9). Although this is a small fraction of the Bolometric lumi- +nosity, similar findings have been reported by the two-phase accre- +tion Type II Luminous Hot Accretion Flow (LHAF) and disc-corona +regime, after which the evolutionary turnover point is reached when +the accretion rate changes. Furthermore, the evolutionary behaviour +of Γ1 and Γ2 is very similar when tracked against the Eddington +fraction. Although there is not a multitude of individual AGN cate- +gorised in the two-phase accretion scenario due to the requirement +for multiple long observations, a theoretical explanation of this be- +haviour was found in AGN and reported by Zdziarski et al. (2003) +who suggested that the emission of a cold medium irradiating hot +plasma could be due to the cooling via reprocessed emission of the +hot plasma causing the observed X-ray spectrum to soften, leading +to a higher value of Γ. +Contrarily, the Γ profile for IRAS 13224-3809 is relatively flat, +which is comparable to the Γ− 𝐿Bol/𝐿Edd behaviour of the jet-phase +regime of the JED-SAD framework where the Γ is determined by +the energy distribution of power law electrons in the jet and roughly +constant for sources with different luminosity’s, neither source fol- +lows the expected Γ ≈ 2.1. Furthermore, for IRAS 13224-3809, the +current model values of Γ1 are extreme although not uncommon (see +e.g., Wilkins et al. 2014) and Γ2 may drop well below 1.4 approach- +ing lower extreme values. There appears to be no relationship with +either source Γ and the accretion rate in IRAS 13224-3809. Similar +Γ behaviour was reported from RXTE observations of MCG-6-30-15 +where the spectral index increased with luminosity, reaching a final +upper value after which it remained roughly static as the luminosity +continued to increase (McHardy et al. 1999), possibly due to a harder +spectral component that does not change within the observation du- +ration as the continuum component steepens the spectra. +A steeper spectrum is expected for increasing luminosity and this +is evident in the spectral model for IRAS 13224-3809 as reported +in HYC22. The ECM scenario, however, shows very limited spec- +tral index variations for its flux variations. This has been previously +reported for NGC 5548 (Sobolewska & Papadakis 2009) and PG +0804+761 (Papadakis et al. 2003). Furthermore, Caballero-Garcia +et al. (2012) found no significant spectral variability in NGC 4151 +and NGC 2110, but reported the significant flux variations that in- +dicated the intrinsic variability of the central source in NGC 4388, +NGC 4945 and IC 4329. This raises the question of different spectral +states in AGN. However, 1H 0707-495 and IRAS 13224-3809 are +both classified Narrow-line Seyfert 1 galaxies where the 2–10 keV +flux variations are almost always associated with spectral variations +(Papadakis et al. 2002). The modelling applied thus far (spectral and +ECM models) suggests that different mechanisms may be contribut- +ing to the variability that may be explained by the differences in the +geometry of each source. +Note that variations in the optical depth 𝜏 of the corona cause spec- +tral changes in the Comptonised emission leading to a steeper spec- +trum and the temperature decreases with increasing 𝜏 (e.g. Haardt +et al. 1997), therefore the geometry of 1H 0707-495 should consist +of a smooth corona where the intrinsic flux varies in unison with Γ1 +and Γ2. IRAS 13224-3809 however, may contain a slightly different +geometry since the Γ1 and Γ2 values are more clearly separated, gen- +erally remaining relatively constant and occupied at the the extreme +values of the model especially when 5 𝑟𝑔 ≲ ℎ2 ≲ 10 𝑟𝑔 (Figure 9). +This suggests that IRAS 13224-3809 corona is more patchy (i.e. hav- +ing clear, distinct spectral zones of spectral Γ1 and Γ2 that may be +related to two distinct zones of corresponding 𝜏 and coronal temper- +ature), especially at locations where the corona ≲ 10 𝑟𝑔. +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +L(2 − 10keV)/LEdd (erg s−1) +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Γ +1H 0707-495 +Γ1 +Γ2 +1.5 +2.0 +2.5 +3.0 +3.5 +Γ +0 +5 +10 +15 +20 +h2(rg) +1H 0707-495 +Γ1 +Γ2 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +L(2 − 10keV)/LEdd (erg s−1) +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Γ +IRAS 13224-3809 +Γ1 +Γ2 +1.5 +2.0 +2.5 +3.0 +3.5 +Γ +0 +5 +10 +15 +20 +h2(rg) +IRAS 13224-3809 +Γ1 +Γ2 +Figure 9. The evolution of Γ as a function of the Eddington fraction and the +source height for 1H 0707-495 (top panels) and also for IRAS 13224-3809 +(bottom panels). The Γ1 and Γ2 values IRAS 13224-3809 are more separated +and occupied at more extreme values than those of 1H 0707-495, especially +when 5 𝑟𝑔 ≲ ℎ2 ≲ 10 𝑟𝑔. +MNRAS 000, 1–13 (2020) + +Extended Corona Models of X-ray Reverberation in AGN +11 +Global coronal correlations across AGN samples have remained +elusive and/or confusing (see e.g., Sarma et al. 2015; Hinkle & +Mushotzky 2021; Kamraj et al. 2022) the latter authors cautioning +the use of coronal parameters and the Eddington fraction relations +to infer properties of black hole systems. In contrast, this work finds +that strong relations are evident in the grouped data and when drilling +down to individual observations. These differences are possibly due +to the choice of spectral models and parameters explored. +Many of the 𝜒2 values fell below unity, suggesting that the model +may be over-fitted due to too many free parameters. This suggests +that deeper model assumptions leading to the freezing of at least one +more parameter may also be appropriate for future modelling. Some +parameter ranges may need to be slightly wider to accommodate +better statistics (whilst remaining feasible), especially for ℎ2, Γ1 and +Γ2. On the other hand, the large errors seen in 𝑡max and 𝑡shift will be +difficult to constrain with current observations and the large errors +associated with the low frequency fluctuation lags. More AGN need +to be explored by this model to test its capabilities against a wider +sample with higher quality time lags versus frequency allowing a +better statistical comparison of the data and models. +6 CONCLUSIONS +This work has explored the extended corona scenario using two X- +ray point sources. The lower source may represent the base of a +jet-like structure or the lower regions of a compact corona with the +upper source representing the extended region where the flares are +detected due to the periodic vertical collimation of a jet structure +or outer region of the corona. The model is capable of fitting the +lag-frequency of 1H 0707-495 and IRAS 13224-3809, where their +black hole mass can be constrained to log(𝑀/𝑀⊙) = 6.04 ± 0.01 +and 6.30 ± 0.01, respectively. +This work also supports the advantages of exploring corona cor- +relations using individual observations and also suggests that the +physics and geometry of the corona may diverge between different +sources. For the first time, we have gained an insight to the behaviour +of Γ for each of the X-ray sources located above the black hole axis +through the use of reverberation mapping. We find the tendency of +softer corona with increasing its vertical extent in both AGN. This +produces a much less-ionised disc which is evident in 1H 0707-495. +On the other hand, for IRAS 13224-3809 we also find the hint of +increasing Γ2 with decreasing 𝑡max and 𝑡shift, suggesting that shorter +propagating fluctuations and faster coronal response may be evident +when the corona extends vertically upwards. The intrinsic flux of +IRAS 13224-3809 show less variability with Γ and its corona may be +more patchy in a sense that contains clearer separate spectral zones +of distinct Γ1 and Γ2 that may also link to two clearer distinct zones +of coronal temperature than that of 1H 0707-495. +ACKNOWLEDGEMENTS +The calculations in this work were carried out using the high per- +formance computer BlueCrystal of the Advanced Computing Re- +search Centre, University of Bristol, UK. SH thanks the STFC for +funding and the Bristol, Cardiff & Swansea CDT Team for sup- +port. PC thanks funding support from (i) Suranaree University of +Technology (SUT), (ii) Thailand Science Research and Innovation +(TSRI), and (iii) National Science Research and Innovation Fund +(NSRF), project no. 160355. This research has made use of ISIS +functions (ISISscripts) provided by ECAP/Remeis observatory and +MIT (http://www.sternwarte.uni-erlangen.de/isis/). 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C., 2014, MNRAS, 443, 2746 +Wilkins D. R., Gallo L. C., Grupe D., Bonson K., Komossa S., Fabian A. C., +2015, MNRAS, 454, 4440 +Yang Q.-X., Xie F.-G., Yuan F., Zdziarski A. A., Gierliński M., Ho L. C., Yu +Z., 2015, MNRAS, 447, 1692 +Zdziarski A. A., Lubiński P., Gilfanov M., Revnivtsev M., 2003, MNRAS, +342, 355 +Zhou X.-L., Wang J.-M., 2005, ApJ, 618, L83 +Zoghbi A., Fabian A. C., Uttley P., Miniutti G., Gallo L. C., Reynolds C. S., +Miller J. M., Ponti G., 2010, MNRAS, 401, 2419 +Zoghbi A., Reynolds C., Cackett E. M., Miniutti G., Kara E., Fabian A. C., +2013, ApJ, 767, 121 +MNRAS 000, 1–13 (2020) + +Extended Corona Models of X-ray Reverberation in AGN +13 +APPENDIX A: TABLE MODEL ECM RESULTS +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–13 (2020) + +14 +S. Hancock et al. +Table A1. The Table Model results for 1H 0707-495 and IRAS 13224-3809 data to 90% confidence. Note that zero errors are a result of rounding to 3 sf and +also hitting upper or lower model parameter space limits. +Source +Obs Id +𝜒2 +log(𝑀/𝑀⊙) +ℎ2(𝑟𝑔) +Γ1 +Γ2 +log𝜉 (erg cm s−1) +𝑏 +𝑡max(𝑡𝑔) +𝑡shift(𝑡𝑔) +1H +Combined +0.50 +6.11+0.01 +−0.03 +3.49+0.35 +−0.36 +2.40+0.00 +−0.00 +1.94+0.01 +−0.01 +0.01+0.11 +−0.01 +3.00+0.00 +−0.04 +370.87+13.13 +−13.27 +17.06+0.20 +−0.20 +Hi-flux +0.07 +6.21+0.00 +−0.17 +10.04+0.71 +−0.70 +2.69+0.11 +−0.22 +1.93+0.43 +−0.03 +2.01+0.99 +−2.00 +3.00+0.18 +−0.28 +308.29+18.39 +−27.68 +15.91+4.96 +−2.49 +Med-flux +1.11 +5.31+0.19 +−0.03 +5.03+14.97 +−0.87 +2.57+0.24 +−0.05 +2.47+0.14 +−0.50 +0.04+2.95 +−0.05 +1.77+1.23 +−0.77 +667.13+32.87 +−567.13 +71.68+8.52 +−46.25 +Lo-flux +1.57 +6.21+0.26 +−0.39 +3.00+2.99 +−0.00 +3.10+0.00 +−0.03 +1.90+0.49 +−0.00 +2.99+0.00 +−0.99 +1.94+1.06 +−0.94 +615.53.+34.31 +−515.53 +59.99+46.70 +−23.68 +110890201 +1.74 +5.84+0.39 +−0.22 +19.30+0.67 +−16.30 +2.13+0.29 +−0.13 +1.70+0.80 +−0.00 +0.01+2.99 +−0.01 +1.87+0.48 +−0.87 +700.00+49.80 +−297.00 +43.70+56.30 +−33.70 +148010301 +0.92 +6.51+0.05 +−0.01 +16.00+4.00 +−11.20 +2.70+0.30 +−0.53 +1.70+0.80 +−0.00 +2.49+0.51 +−2.49 +3.00+0.00 +−0.48 +350.00+345.00 +−217.00 +18.50+16.20 +−8.47 +506200201 +0.12 +6.54+0.10 +−0.54 +3.58+7.67 +−0.58 +2.53+0.47 +−0.49 +1.70+0.80 +−0.00 +2.63+0.38 +−2.63 +1.95+1.05 +−0.95 +416.00+280.00 +−316.00 +50.00+35.10 +−14.30 +506200301 +0.00 +6.15+0.01 +−0.01 +18.20+1.81 +−0.56 +2.37+0.07 +−0.07 +1.72+0.09 +−0.02 +0.01+2.99 +−0.01 +2.76+0.09 +−0.09 +518.00+22.70 +−28.600 +34.10+0.70 +−0.70 +506200401 +0.81 +6.49+0.01 +−0.02 +13.60+0.24 +−0.24 +2.80+0.13 +−0.02 +1.70+0.02 +−0.00 +0.03+2.12 +−0.03 +2.87+0.13 +−0.15 +440.00+48.30 +−36.00 +20.50+0.67 +−1.10 +506200501 +2.75 +6.42+0.02 +−0.03 +17.50+2.55 +−0.73 +2.84+0.12 +−0.12 +1.70+0.13 +−0.00 +0.70+2.30 +−0.70 +2.63+0.21 +−0.20 +434.00+36.40 +−105.00 +21.10+1.38 +−2.13 +511580101 +0.81 +6.48+0.07 +−0.07 +4.07+1.50 +−1.07 +2.99+0.01 +−0.36 +2.70+0.00 +−1.00 +2.92+0.08 +−0.61 +3.00+0.00 +−1.63 +400.00+99.60 +−250.00 +120.00+0.00 +−11.70 +511580201 +0.38 +6.03+0.26 +−0.07 +4.14+15.90 +−1.14 +2.85+0.15 +−0.65 +1.70+0.80 +−0.00 +0.01+1.14 +−0.01 +1.00+1.29 +−0.00 +489.00+46.00 +−286.00 +110.00+10.00 +−50.00 +511580301 +0.00 +6.45+0.06 +−0.11 +16.50+3.05 +−5.52 +2.59+0.00 +−0.07 +1.73+0.14 +−0.03 +0.00+3.00 +−0.00 +2.37+0.24 +−0.39 +223.00+2.07 +−23.40 +1.20+0.14 +−2.09 +511580401 +1.18 +6.63+0.04 +−0.95 +20.00+0.02 +−1.72 +2.30+0.70 +−0.20 +1.70+0.79 +−0.00 +2.99+0.01 +−1.61 +2.17+0.52 +−0.44 +100.00+43.00 +−0.00 +25.02+1.39 +−3.52 +554710801 +0.00 +5.94+0.00 +−0.21 +17.00+3.00 ++0.32 +2.60+0.70 +−0.53 +2.10+0.02 +−0.40 +0.00+3.00 +−0.00 +3.00+0.00 +−0.32 +649.00+100.00 +−131.00 +29.2+7.12 +−9.21 +653510301 +0.01 +6.55+0.02 +−0.10 +13.30+3.34 +−0.00 +2.88+0.02 +−0.00 +2.26+0.18 +−0.09 +0.04+2.96 +−0.04 +3.00+0.00 +−0.05 +182.00+0.00 +−0.00 +19.20+4.38 +−0.00 +653510401 +0.00 +6.71+0.01 +−0.00 +17.30+2.70 +−3.67 +2.82+0.35 +−0.18 +1.85+0.30 +−0.15 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+−0.30 +1.55+1.45 +−1.55 +1.75+1.25 +−0.41 +186.99+524.52 +−86.99 +23.36+0.00 +−7.45 +110890101 +0.00 +6.99+0.01 +−0.96 +19.34+0.66 +−16.34 +3.00+0.10 +−1.00 +2.37+0.53 +−0.67 +2.69+0.31 +−2.69 +2.94+0.06 +−1.84 +108.76+1292.85 +−8.76 +47.94+44.89 +−22.53 +673580101 +0.39 +6.65+0.29 +−0.33 +17.99+2.01 +−14.99 +3.10+0.00 +−0.65 +1.70+0.90 +−0.00 +0.00+3.00 +−0.00 +2.22+0.78 +−1.22 +400.00+599.99 +−300 +48.16+91.81 +−33.39 +673580201 +0.21 +6.41+0.01 +−0.00 +16.00+1.43 +−4.65 +2.55+0.45 +−0.08 +1.73+0.42 +−0.03 +0.00+3.00 +−0.00 +2.82+0.18 +−0.49 +415.47+136.63 +−55.45 +23.06+1.99 +−5.42 +673580301 +0.07 +6.51+0.33 +−0.15 +10.36+9.64 +−1.15 +2.56+0.54 +−0.42 +2.12+0.53 +−0.42 +3.00+0.00 +−3.00 +2.98+0.02 +−1.54 +319.56+1106.79 +−95.52 +21.05+3.47 +−4.21 +673580401 +0.26 +6.67+0.00 +−0.00 +14.73+0.09 +−0.10 +2.59+0.02 +−0.02 +2.12+0.02 +−0.02 +1.90+0.38 +−0.35 +2.96+0.04 +−0.07 +333.93+13.87 +−14.22 +23.85+0.10 +−0.09 +780560101 +0.00 +6.89+0.01 +−0.57 +12.99+7.01 +−9.99 +2.59+0.51 +−0.35 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Models of X-ray Reverberation in AGN +15 +Table A2. The simultaneous fitting results for 1H 0707-495 and IRAS 13224-3809 where the black hole mass log(𝑀/𝑀⊙) = 6.04 ± 0.01 and 6.30 ± 0.01 +respectively (to 90% confidence). Note that zero errors are a result of rounding to 3 sf and also hitting upper or lower model parameter space limits. +Source +Obs Id +𝜒2 +ℎ2(𝑟𝑔) +Γ1 +Γ2 +log 𝜉 (erg cm s−1) +𝑏 +𝑡max(𝑡𝑔) +𝑡shift(𝑡𝑔) +1H 0707-495 +0110890201 +0.26 +11.10+2.02 +−0.59 +2.16+0.06 +−0.05 +1.67+0.04 +−0.03 +0.05+2.68 +−0.05 +2.18+0.14 +−0.14 +450.00+43.95 +−73.53 +30.53+0.99 +−3.71 +0148010301 +0.66 +7.38+0.30 +−0.27 +2.60+0.25 +−0.45 +1.46+0.87 +−0.06 +0.38+0.30 +−0.27 +1.00+.62 +−0.00 +450.00+170.87 +−81.06 +106.09+1.88 +−1.72 +0506200201 +0.34 +11.00+1.46 +−0.35 +2.27+0.03 +−0.00 +1.70+0.05 +−0.03 +2.00+0.33 +−2.00 +3.00+0.00 +−0.14 +488.13+30.99 +−33.09 +22.13+0.77 +−0.78 +0506200301 +0.49 +3.00+3.48 +−0.00 +2.00+0.35 +−0.04 +1.40+0.03 +−0.00 +2.41+0.17 +−0.17 +2.75+0.25 +−1.02 +600.05+20.62 +−150.53 +66.77+10.94 +−10.92 +0506200401 +0.76 +11.00+0.70 +−0.39 +2.55+0.02 +−0.03 +2.16+0.02 +−0.02 +0.95+1.74 +−0.95 +3.00+0.00 +−0.06 +540.99+18.15 +−33.66 +27.91+0.84 +−0.85 +0506200501 +2.98 +12.00+4.76 +−9.00 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+2.00+0.58 +−0.97 +799.85+0.15 +−373.13 +107.18+17.25 +−9.06 +MNRAS 000, 1–13 (2020) + diff --git a/T9E3T4oBgHgl3EQfzwt3/content/tmp_files/load_file.txt b/T9E3T4oBgHgl3EQfzwt3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..80670be98f705a442bb278351bf1f7ff652f1922 --- /dev/null +++ b/T9E3T4oBgHgl3EQfzwt3/content/tmp_files/load_file.txt @@ -0,0 +1,3060 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf,len=3059 +page_content='MNRAS 000, 1–13 (2020) Preprint 13 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 Extended Corona Models of X-ray Reverberation in the AGN 1H 0707-495 and IRAS 13224-3809 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Hancock,1★ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Young,1 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Chainakun2,3 1HH Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK 2School of Physics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand 3Centre of Excellence in High Energy Physics and Astrophysics, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We fit a new vertically extended corona model to previously measured reverberation time lags observed by XMM-Newton in two extremely variable Narrow Line Seyfert 1 Active Galactic Nuclei (AGN), 1H 0707-495 and IRAS 13224-3809, in a variety of similarly observed flux groups and explore the model in all observations over a 16 year period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The model employs two X-ray sources located along the black hole rotational axis at height, ℎ1 and ℎ2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' These sources have their associated photon indices Γ1 and Γ2 which respond to fluctuations in the disc with a maximum response duration of 𝑡max and a propagation delay between the response of the two of 𝑡shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We find that for 1H 0707-495, ℎ2 is significantly correlated with Γ1 and anti-correlated with ionisation 𝜉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Whilst the 1H 0707-495 corona extends upwards, the emission appears softer and the disc is less ionised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We find similarities in IRAS 13224-3809, but significant anti-correlation between Γ2 and both 𝑡max and 𝑡shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This suggests that when the IRAS 13224-3809 corona becomes softer while extending vertically upwards, the overall corona response occurs faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This may also suggest that the inner disc also becomes more active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In addition, Γ1 and Γ2 are extreme, relatively less variable, but more separate in IRAS 13224-3809 than in 1H 0707-495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This suggests that the IRAS 13224-3809 corona may be more patchy in the sense that it has two more clear distinct spectral zones of Γ1 and Γ2 (possibly relating to two distinct zones of coronal temperature) when compared to 1H 0707-495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Key words: X-rays: individual: 1H 0707-495;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' X-rays: individual: IRAS13224-3809 1 INTRODUCTION The spectra of active galactic nuclei (AGN) have a component of direct X-ray continuum emission, and that cold gas can reflect some of this X-ray continuum (Pounds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The cold, optically thick material seen through fluorescence and reflection is known to occur in the presence of an accretion disc where X-rays produced by inverse Compton scattering in a corona are reflected off the accretion disc producing a modified spectrum emission with fluorescent Fe K lines at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 keV and other spectral features (George & Fabian 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The first hints of reflection or reprocessing time-delay due to the light travel time between the corona and disk were seen in XMM- Newton observations of Ark564 (McHardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2007) which led to the first robustly discovered delays in 1H0707-495 (Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2009) where the soft energy band (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3−1 keV) lagged behind the hard band (1 − 4 keV) by 30 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Many reverberation lags have since been discovered (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Emmanoulopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Kara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2013b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Zoghbi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' ★ E-mail: steff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='hancock@bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='uk The soft negative time lag is the signature of relativistic reflection that reverberates in response to continuum fluctuations and is inter- preted as the light crossing time from the source to the reflecting region, correlating positively with the black hole mass (De Marco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Hard positive lags at lower frequencies are understood to originate from fluctuations of the accretion rate propagating from outer to inner radii, causing the hard X-rays produced at smaller radii to respond after soft X-rays produced at larger radii (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Kotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Arévalo & Uttley 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Arévalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Hard X-ray lags (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' hard photon variability lagging soft photon variability) were ev- ident in X-ray binaries before they were discovered in AGN (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Miyamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Nowak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The reproduction of soft reverberation lags from a compact corona and the hard lags produced by propagating fluctuations through an extended region whilst maintaining the features of the energy de- pendence seen in the Fe K𝛼 line region is challenging and com- putationally intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Motivated by these phenomena, Chainakun & Young (2017), CY17 hereafter, developed an approximation of a vertically extended source using two X-ray point sources located on the rotation axis of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The model employs gravitational © 2020 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='04731v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='HE] 11 Jan 2023 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Hancock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' units where the gravitational radius 𝑟𝑔 = 𝐺𝑀/𝑐2 and gravitational time 𝑡𝑔 = 𝐺𝑀/𝑐3 (where 𝐺 is the gravitational constant, 𝑀 is the black hole mass and 𝑐 is the speed of light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Instead of modelling the propagating fluctuations, a phenomenological function of expected source responses which react to these propagations is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The two X-ray sources are allowed to vary as they respond to primary intrinsic variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The X-ray continuum variability depends on the response of the source and the X-ray reflection depends also on the disc response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Therefore the model has to predict the time lags from both continuum X-ray sources and the associated disc responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The model was fitted to a 120 ks timing XMM-Newton observation of the narrow-line Seyfert 1 galaxy PG 1244+026, examining the frequency and energy where the lags were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The model revealed hard and soft X-ray sources at heights ∼ 6 𝑟𝑔 and ∼ 11 𝑟𝑔 respectively with the upper source producing small amounts of reflection which sug- gested a feasible geometry of a relativistic jet beaming away from the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' CY17 suggested that the continuum flux from the upper source and the extra blackbody component that contribute significant flux at energies < 1 keV are required to dilute the soft reverberation lags and to reproduce the absence of soft lag in the lag-energy spectrum of PG 1244+026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The X-ray variability in 1H 0707-495 was found to be extreme and may have both intrinsic and environmental absorption origins (Parker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The geometry of 1H 0707-495 has been dis- cussed by Szanecki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2020) who developed a new extended lamppost model which accounted for the spatial extent and rotation of the X-ray source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The investigation of the location and size of the corona indicated a compact corona at ∼ 2 𝑟𝑔 (highly centrally peaked rather than extended) regardless of the effects of ionised absorption from winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' They found no evidence that the size of the corona was correlated to the luminosity as reported by Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The time lags in IRAS 13224-3809 have been discussed by Alston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2020) who studied short-timescale variations and included all relativistic effects allowing for ultra-fast outflows fitting multi- ple epochs where the source height changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' They tackled inherent degeneracies between the reverberation signal and black hole mass, inner disc radius and height of the corona by tracking short-scale re- verberation signatures where the source height changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' They found that the height of the corona increased with increasing luminosity and that black hole mass uncertainty estimates were comparable to the leading optical reverberation method by Peterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In addition, Caballero-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2020) used all available data from the XMM-Netwon archive to simultaneously fit various flux states of the energy spectra and time lags using a new code which calculates reflection spectra from the accretion disc in response to an X–ray flare from a point source located above the black hole in accretion disc lamp-post geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The model strongly favoured a maximally spinning black hole and detected significant variations of the corona height, increasing from 3 − 5 𝑟𝑔 at lower flux states and extending to ∼ 10 − 20 𝑟𝑔 when the luminosity doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Recently, Chainakun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2022b) constrained the reverberation signatures that appeared in the power spectral density of IRAS 13224-3809 and found that the lamp-post source height increased from ∼ 3 𝑟𝑔 to ∼ 25 𝑟𝑔 with the luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This study builds on the timing and spectral analysis of reverberat- ing AGN reported by Hancock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2022), HYC22 hereafter, using a variety of similar spectral flux levels as found in the spectra of each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In addition we explore all suitable individual XMM-Newton observations of 1H 0707-495 and IRAS 13224-3809 between 2000 and 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Our aim is to develop the model initially created by CY17 to explore these time lag signatures in an extended corona scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' These well studied AGN are selected due to their abundance of long Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' XMM-Newton observations used in this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The information Includes the name of the source, the Observation ID, year, exposure time and effective exposure after cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The group column refers to the low, medium and high spectral flux groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The low counts (lc) and high counts (hc) refer to those to observations containing < 5 cts s−1 and > 5 cts s−1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='Source ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='Obs ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='Exp [Eff] (ks) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='Group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1H0707-495 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0110890201 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='46[41] ' metadata={'source': 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+page_content='observations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' however it should be noted that these are extreme nar- row line Seyfert 1 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2 OBSERVATIONS AND DATA REDUCTION The data for all observations outlined in Table 1 were downloaded from the XMM-Newton archive and processed using standard meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The time lag estimates calculated between the soft (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='8 keV) and hard (1 − 4 keV) energy bands reported by HYC22 have been used throughout this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We initially assume 1H 0707-495 to have a black hole mass log 𝑀 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='31𝑀⊙ (Bian & Zhao 2003) and redshift 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0411 (Leighly 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The luminosity distance from the NED database is 187 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This AGN has been well documented to be dominated by relativistically blurred reflection at either a low or a moderate incli- nation angle (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Zoghbi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Dauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Kara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2013a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Caballero-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We employ the inclination angle 𝑖 = 53◦ as derived from the emissiv- ity profile by Wilkins & Fabian (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For IRAS 13224-3809 we assume a black hole mass of log 𝑀 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='30𝑀⊙ (Alston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2018, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Caballero-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We also adopt appropriate val- ues for redshift 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0406 and inclination 𝑖 = 64◦ as reported by MNRAS 000, 1–13 (2020) Extended Corona Models of X-ray Reverberation in AGN 3 Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The luminosity distance from the NED database is 310 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 3 THE EXTENDED CORONA MODEL (ECM) The extended corona model (ECM) assumes a standard geometrically thin, optically thick accretion disc (Shakura & Sunyaev 1973) which extends from the innermost stable circular orbit (ISCO), or the radius of marginal stability 𝑟ms, to 400 𝑟𝑔 around a central black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that accreting supermassive black holes were mostly found to be rapidly spinning (Reynolds 2021), therefore to limit the number of free parameters and to avoid the model degeneracy, we fix the black hole spin to be 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The accretion disc is illuminated by two compact X-ray sources located on the symmetry axis at heights ℎ1 and ℎ2 which are the lower and upper source heights respectively whose amplitudes as a function of time are 𝑥1(𝑡) and 𝑥2(𝑡), respectively, as in CY17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We maintain the two-source to investigate the extent of the corona and a basic sketch of this scenario is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Physically, the lower X-ray source may represent the base of a jet-like structure or the lower region of a compact corona (Wilkins & Gallo 2015) and the upper source represents the farthest region from where a flare or response from the disc is detected hence it is interpreted as the upper extreme of the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' A further plausible explanation of this region could be due to a periodic vertical collimation of the corona as a jet launching event subsides (Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' These explanations also suggest that photon emission is beamed vertically away from the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The photon trajectories were traced along Kerr geodesics as de- scribed by Bardeen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (1972) and essentially outlined by CY17 by first considering the flares of the two X-ray sources as two sepa- rate delta functions and tracing the photons between the two sources, the disc and the observer along Kerr geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The full relativis- tic effects outlined by Cunningham (1975) are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The X-ray reprocessing is modelled using REFLIONX (George & Fabian 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Ross & Fabian 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In CY17, many of the free parameters were reduced by making model assumptions based on the physical environment of PG1244+026, for example, the inclination angle, photon index and ionisation were fixed for simplicity at the values suggested in previous literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Kara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Here, the inclination angle is fixed at 𝑖 = 53◦ for 1H 0707-495 (Wilkins & Fabian 2011) and 𝑖 = 64◦ for IRAS 13224-3809 (Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2013), as described in Section 2, but both photon index and ionisa- tion are allowed to be free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that a high inclination of 𝑖 > 60◦ for IRAS 13224-3809 was also supported by the broadband spectral fitting (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2018) and simultaneous lag-frequency spectral fitting (Alston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Fixing the inclination can help to avoid degeneracies in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Then, the model contains an ionisation parameter 𝜉(𝑟, 𝜙) = 4𝜋𝐹𝑡 (𝑟, 𝜙)/𝑛(𝑟) (1) where 𝐹𝑡 (𝑟, 𝜙) is the total flux due to both X-ray sources per unit area of the disc at (𝑟, 𝜙), and 𝑛(𝑟) is the disc density, 𝑛(𝑟) ∝ 𝑟−𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The aim is to reproduce the high frequency soft (reverberation) lags and the harder lags seen at lower frequencies which are associated with propagating fluctuations in the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Kotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2001) and Arévalo & Uttley (2006) describe the low frequency hard lags by mass accretion fluctuations propagating inwards through the disc where the central region contains a source of harder X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' That is, the fluctuations modulate soft X-rays first resulting with the hard lags, so the soft bands are dominant first before being overcome by the hard band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Assuming these fluctuations cause the central X-ray sources to respond at different times, the extended corona scenario models this source response using a cut off power law Ψ𝑖(𝑡) ∝ 𝑡−𝑞𝑖exp(−𝑡/𝑡max) (2) where the subscripts 𝑖 = 1 and 2 refer to the parameters of the lower and upper sources, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' How the function decays with respect to time is determined by 𝑞𝑖, where 𝑡 = 0 and 𝑡max is the beginning and the end of the source response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Only the time difference between the two responses is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In addition, the model makes use of the parameter 𝑡shift to delay the response of the second source that reacts slower to primary variations of the first source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In essence, these inward propagating fluctuations can produce primary variations in the disc that will affect the flux of the lower X-ray source at time 𝑡 = 1 and propagate upwards to the upper X-ray source taking time 𝑡shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' While the disc response usually obtained from the ray-tracing simulation is a function of energy and time, the source response is independent of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' However, the source variability remains dependent of energy via 𝐹𝑖(𝐸) ∝ 𝐸−Γ𝑖, where Γ𝑖 is the photon index of the X-ray continuum of the 𝑖th source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The source variability in the energy band 𝐸 𝑗 is 𝑥𝑖(𝐸 𝑗, 𝑡) ∝ 𝐹𝑖(𝐸 𝑗)𝑥0(𝑡) ⊗ Ψ𝑖(𝑡), (3) so that the lower and upper sources respond in different ways to the primary variations 𝑥0(𝑡), due to 𝐹𝑖(𝐸 𝑗) and Ψ𝑖(𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We initially set 𝜉 for a neutral to highly ionised environment where log 𝜉 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 respectively and allow to vary during the fitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Other initial assumptions are made by setting the Fe abundance to solar and fixing the decay of the lower-source response 𝑞1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We began by creating the disc model by integrating photon paths from the observer to the disc assuming a maximum black hole spin where 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='998 with the disc extending from ISCO out to 400𝑟𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The next task was to integrate photon paths from a single X-ray source to the disc, after which the spectra was calculated using REFLIONX for a given source height and disc inclination already computed in the previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The final step was to calculate the reverberation signa- tures by looking at the already computed source-to-disc and observer- to-disc ray tracing runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The disc response function, 𝜓𝑖(𝐸 𝑗, 𝑖), was computed and the variability of the disc reflection can be calcu- lated via a convolution term: 𝑥𝑖(𝐸 𝑗, 𝑡) ⊗ 𝜓𝑖(𝐸 𝑗, 𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The observed X-ray variability then can be written as the sum of the source and the disc variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The lag-frequency spectra were calculated from the Fourier-phase differences between two energy bands, following the standard techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Cackett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Emmanoulopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Chainakun & Young 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that the light curve in each energy band always contains both continuum and reflection components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The contamination of the continuum flux in the reflection-dominated band, and the reflection flux in the continuum-dominated band cause dilution effects which can reduce the lag amplitude, meaning that the measured time lags are shorter than the intrinsic time lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The dilution effects modify the shape of the lag-frequency spectrum without affecting the frequency at which the lags occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For discussions on dilution see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=', Uttley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Kara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Chainakun & Young (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In order to deal with dilution effects, the variability of the reflected photons of the disc is normalized using the reflected response fraction (defined as the reflected flux/continuum flux) of all energy bands 𝐸 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We include a brightness parameter 𝑏 (as measured in the frame of the observer) as a ratio of the brightness of the lower and upper source, 𝑏1 and 𝑏2 respectively by defining 𝑏 = 𝑏2/𝑏1 and fix 𝑏1 = 1 and allow 𝑏 to vary between 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The continuum flux of the lower source will be less than the upper source due to its closer proximity to the MNRAS 000, 1–13 (2020) 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Hancock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' A simple sketch of the extended corona scenario showing the two X-ray sources located on the rotation axis of an accreting black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The arrows show the trajectories of the continuum and accretion disc reflected photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' black hole, therefore photons will be subject to light bending effects towards the centre (Miniutti & Fabian 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The model is capable of reproducing the prominent time lag fea- tures seen in AGN when Γ1 ≠ Γ2 as seen in the top panel of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In addition Γ1 > Γ2 for positive low freqency lags and Γ2 > Γ1 for negaitive low frequency lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' As 𝑡max increases, so does the am- plitude of the hard lags as seen in the bottom panel of Figure 2 and aliasing effects (phase wrapping) are moved to lower frequencies and positive lags will switch to negative lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The low frequency hard lags increase with energy and the softer reverberation lags dominate at higher frequencies, consistent with the ‘two-mechanism’ features of propagation and reflection and was also consistent with the tradi- tional spectral features that are widely explained by reflection from the inner disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We generated the initial model by running the code for a course range of parameter values to fit the lag-frequency spectra of the fully combined observations 1H0707-495 using an inclination angle 𝑖 = 53◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We set the density profile to constant (𝑝 = 0) and the response decay function 𝑞2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 and iron abundance 𝑍Fe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The lower source height was fixed at 2 𝑟𝑔 and upper source height was allowed to vary between 3−20 𝑟𝑔 thus obtaining the extent of the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' While fixing the lower source at 2 𝑟𝑔 is a pragmatic choice that simplifies the fitting and limits the number of free parameters, having one source this close to the horizon is comparable to the limits by previous spectral or timing modelling, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=', Kara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2013a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Caballero-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2018, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In addition, Γ1 and Γ2 were allowed to vary between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 in line with REFLIONX, the ionisation parameter specified at the ISCO could vary from 0 − 3 for neutral and highly ionised exploration respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The brightness parameter was also allowed to vary from 1 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The remaining free parameters were 𝑡max and 𝑡shift and we initially set these ranging from 50 − 600 𝑡𝑔 and 0 − 120 𝑡𝑔, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The black hole mass for each source was initially tested using sensible values from literature (Bian & Zhao 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Alston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2020) and allowed to vary during the fitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 10-5 10-4 10-3 10-2 10-1 100 Frequency (1/tg) 35 30 25 20 15 10 5 0 5 10 Time Lag (tg) Γ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0, Γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='7 Γ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1, Γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='9 Γ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1, Γ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 Γ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1, Γ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1 Γ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1, Γ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 Γ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1, Γ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 Γ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1, Γ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='6 10-5 10-4 10-3 10-2 10-1 100 Frequency (1/tg) 10 5 0 5 10 15 Time Lag (tg) tmax = 100tg tmax = 300tg tmax = 500tg tmax = 700tg Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The frequency dependent lags varying with Γ and 𝑡max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Both panels were set at 𝑖 = 53◦, ℎ1 = 2, ℎ2 = 3, 𝐹𝑒 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0, log 𝜉 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 and 𝑞2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The top panel shows the frequency dependent lags varying with Γ1 and Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The positive low frequency fluctuation lags and the negative soft reverberation lags are produced when Γ1 ≠ Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The bottom panel shows lag behaviour with different values of 𝑡max, where Γ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 and Γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5, 𝑏 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 and 𝑡shift = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 4 RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1 Model testing on 1H 0707-495 The ECM fitting was developed using the Interactive Spectral Inter- pretation System (ISIS) Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Generally, much better fits were obtained using single observations than combining the observa- tions into groupings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Although spectral combinations are useful for obtaining snapshots of the various epochs, the true variability is best revealed by examining each observation in turn, however this is time consuming and computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Firstly, we independently examined the posterior distribution for the mass using combined ob- servation best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For this inspection we use the isis_emcee mod- ule outlined at Remeis-Wiki (2018) by adopting the methods of the MCMC hammer (Goodman & Weare 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We initiate 500 walkers and run the chain with 10,000 iter- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The walkers converged tightly after 1,000 steps and settled into an acceptance rate of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='7 after ∼3,000 iterations to obtain a MNRAS 000, 1–13 (2020) b hExtended Corona Models of X-ray Reverberation in AGN 5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 0 20 40 60 lag_freq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='log_mass Probability 1H 0707-495 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5×10−5 3×10−5 lag_freq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='log_mass lag_freq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='norm 1H 0707-495 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 0 10 20 30 40 lag_freq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='log_mass Probability IRAS 13224-3809 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4×10−51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='6×10−51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='8×10−5 lag_freq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='log_mass lag_freq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='norm IRAS 13224-3809 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Independent (emcee) tests of the posterior density for the combined observations of 1H 0707-495 (top row) with mean mass 𝜇 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='22 (solid blue line) and 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 (black dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' IRAS 13224-3809 is also shown (bottom row) with mean mass 𝜇 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='34 and 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The contour plots of the mass and normalisation are shown in the right column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' All 𝑥-axes are identical for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' mass of log(𝑀/𝑀⊙) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01, where the errors are calculated at the 90% confidence limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The posterior density and contour plot is shown in the top row of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This value is comparable to that re- ported in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Zhou & Wang 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Zoghbi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2010) although uncertainties in the estimations have been acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This initial model fit was reasonable where the 𝜒2 was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='32 and the upper source was located at 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='11 𝑟𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For this mass and geometry, the photon indices Γ1 and Γ2 were well constrained at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='07 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 respectively, where the ionisation log 𝜉 ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The upper source was exactly twice as bright as the lowers source given by parameter 𝑏 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The source response time from disc fluctuations 𝑡max = 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='00+67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='04 𝑡𝑔 and the time taken for the fluctuations to propagate from the lower to upper source 𝑡shift = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='00+11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='05 −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='92 𝑡𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Although this initial fit to the combined data is promising and can provide a good description of the data, we acknowledge the reasonably large errors returned for 𝑡max, and the loosely constrained errors found for 𝑡shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In general, this is typical of the fits found for all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Further complexity was revealed when attempting to standardise the mass value for the remainder of the data to the value obtained from model fitting via the University of Bristol high performance computer BlueCrystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' A fixed mass of log(𝑀/𝑀⊙) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='22 gen- erally returned much poorer fits and the fitting procedure remained computationally intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' To develop this model further, the reverberation signatures were read into a table model that can be used to fit the data in ISIS and XSPEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This step required the use of astropy and heasp, the latter of which is a component of the standard HEASOFT installation (NASA 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The table model was initially generated using Python and the parameter space was given finer step sizes between parameter values to obtain a finer grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The output model was huge (>40 GB) and the model loading time was too long for general usage, therefore the parameter space was constrained further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The final table model was 19 GB and the parameter space is outlined in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Although Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The extended corona model parameter space for both sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that 𝑓 denotes fixed parameter Parameter 1H 0707-495 IRAS 13224-3809 log(𝑀/𝑀⊙) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 − 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 − 8 𝑖 (◦) 53 𝑓 64 𝑓 𝑎 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='998 𝑓 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='998 𝑓 ℎ1 (𝑟𝑔) 2 𝑓 2 𝑓 ℎ2 (𝑟𝑔) 3 − 20 3 − 20 Γ1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 Γ2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2 𝑍Fe (solar) 1 𝑓 1 𝑓 log 𝜉 (ergs cm s−1) 0 − 3 0 − 3 𝑝 0 𝑓 0 𝑓 𝑏 1 − 3 1 − 3 𝑞2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 𝑓 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 𝑓 𝑡max (𝑡𝑔) 50 − 700 50 − 1500 𝑡shift (𝑡𝑔) 0 − 120 0 − 150 this is still very cumbersome, further constraints at this stage were halted due to dynamic benefits of the model parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The first table model fit was conducted on the combined data for 1H 0707-495, achieving a very good model fit to the data where 𝜒2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This fitting was applied to all of the combination flux groups for low, medium, high etc and the results are presented in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The combined data and model fit along with the lag- frequency spectrum file predicted by the model is presented in the top panel of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Of course, a full table model listing every possible integer value of parameter space would be extremely difficult to achieve given the intensive computational power required, so only the closest lag-frequency, that is the predicted reverberation signatures are shown unbinned at full resolution by the solid grey wavy lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2 Model testing on IRAS 13224-3809 Once again we explore the initial model fit using isis_emcee with 500 walkers and 10,000 iterations and walkers converged tightly after only ∼ 200 steps, settling into an acceptance rate of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='7 after ∼300 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We obtained a mass of log(𝑀/𝑀⊙) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='34±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 for IRAS 13224-3809, where the errors are calculated at the 90% confidence limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Again we note the model capability of returning a mass value closely comparable to log(𝑀/𝑀⊙) ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 as reported in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=', Alston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The mass posterior density and contour plot, along with the model fit is shown in bottom row of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The ECM fit was statistically good where 𝜒2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='47 with the upper source located at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='00+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='02 𝑟𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For this mass and geometry, the photon indices Γ1 and 1 Note that the number of degrees of freedom depends on the binning being used, and with lighter binning we would have more degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We have chosen to bin the data more heavily to clearly show the time lag versus frequency with higher signal-to-noise ratio, so there are fewer data bins, which means that the number of degrees of freedom is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The 𝜒2 values reported here then are close to the reduced 𝜒2 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This also highlights a more general problem when we apply the extended corona model while the quality of the time lags is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The data are however, still able to strongly constrain our models, showing which regions of parameter space provide a good description of the time lags and which can be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' MNRAS 000, 1–13 (2020) 6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Hancock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 10−4 10−3 −200 0 200 400 Frequency (Hz) Time lag (s) 10−4 10−3 −500 0 500 1000 Frequency (Hz) Time lag (s) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The combined lag-frequency Table model fits for 1H 0707-495 (top panel) and IRAS 13224-3809 (bottom panel) showing the best model fit (red) where the mass is variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The closest reverberation signature is shown by the grey wavy line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that the lags are estimated between the soft, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='8 keV, and hard, 1 − 4 keV, bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Furthermore, there is significant cancellation of positive and negative lags due to phase wrapping and binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Γ2 were well constrained at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='60+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='59 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 respectively, where the ionisation log 𝜉 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The upper source was about 3 times as bright as the lowers source (𝑏 ∼ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The source response time from disc fluctuations 𝑡max = 1300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='00 −304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='36 𝑡𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Also note that the ranges of the parameters 𝑡max and 𝑡shift required adjusting up to 1500 𝑡𝑔 and 150 𝑡𝑔 respectively for this source as summarised in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The model was able to provide a good statistical description of all data when fitted via the table model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The best-fit result for the combined observations is presented in the bottom panel of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that there is significant cancellation of positive and negative lags dues to phase wrapping and binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 Constraining the black hole mass Before exploring the simultaneously fitted ECM model, we inspected the variable mass model fits as a function of the negative (reverbera- tion) time lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For each source this relationship is shown in Figure 5, suggesting that whilst the black hole mass and time lag relationship is not evident for 1H 0707-495 (𝑝 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='05), it is moderately anti- correlated in IRAS 13224-3809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Since the black hole mass for each 105 106 107 M/M⊙ 10-1 100 101 102 103 Time lag (s) rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='258 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='257 1H 0707-495 106 107 M/M⊙ 10-1 100 101 102 103 Time lag (s) rs = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='494 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='017 IRAS 13224-3809 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The variable mass from the Table model fits as a function of the reverberation lag for 1H 0707-495 (top) and IRAS 13224-3809 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Other physical processes related to the coronal properties, rather than ge- ometry, may involve in manifesting the lag-mass scaling relation under the extended corona environment especially in IRAS 13224-3809 where the anti- correlation between the lags and the mass is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Of course, there is only one true value of the mass for each AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' source should be constant during these observations, the variable mass found in IRAS 13224-3809 could be induced by other geomet- ric effects in an extended corona environment that are not related to the central mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' It could also be due to the fact that the height and the mass are degenerate (Caballero-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2018, 2020), since the mass and the source height affect the lags in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In any case, our results suggest that it might be better to fix the black hole mass when performing reverberation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' To obtain the best mass value for all data, we use similar procedures to Alston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2020) by simultaneously fitting the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We allowed the mass to vary between values of log(𝑀/𝑀⊙) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 and loaded the first data set and ran the table model fit, then added the second data set where the identical parameters were tied for each set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Essentially we leave the first mass as a free parameter and subsequently tie the remaining data masses to the first data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This method of building one large parameter file was computationally intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The results confirmed our expectations that not all data sets would obtain individually excellent fits and, for 1H 0707-495, the 𝜒2 statistics ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='19 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This method, however, helped to reduced the large errors for 𝑡max and 𝑡shift for the majority of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The single mass value obtained for 1H 0707-495 was log(𝑀/𝑀⊙) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The 1H 0707-495 mass constrained MNRAS 000, 1–13 (2020) Extended Corona Models of X-ray Reverberation in AGN 7 10−4 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 0 500 Frequency (Hz) Time lag (s) log𝑀 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='7 log𝑀 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 log𝑀 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 10−4 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 0 500 Frequency (Hz) Time lag (s) ℎ2 = 20 𝑟𝑔 ℎ2 = 10 𝑟𝑔 ℎ2 = 5 𝑟𝑔 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Top panel: The variable mass behaviour for IRAS 13224-3809 combined table model where ℎ2 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 𝑟𝑔 and log𝑀 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 (black), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='3 (red) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='7 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Specific model parameters of the reverberation signature are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Bottom panel: The variable height behaviour is shown for the same signature parameters and mass values as seen in the top panel, where ℎ2 = 5 𝑟𝑔(black), 10 𝑟𝑔 (red) and 20 𝑟𝑔(blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The other model parameters were Γ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='6, Γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='7, log 𝜉 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0, 𝑏 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0, 𝑡max = 300, 𝑡shift = 20 (for both panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that in the ECM model, the mass scales not only the negative reverberation lags but also the positive lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' here is slightly smaller than the mass of log(𝑀/𝑀⊙) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='37 as reported in Zhou & Wang (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In fact, Caballero-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2018) also adopted the mass from Zhou & Wang (2005) and showed that if the mass is larger than this, the lamp-post model will not be able to fit the reverberation time-lag data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Our obtained mass for 1H 0707-495 is then still in the acceptable regime found by Caballero-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We repeated these methods to obtain the single mass value for IRAS 13224-3809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' All 17 data sets were reasonably well constrained statistically where 𝜒2 ranged 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='17 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='75, achieving a single mass value of log(𝑀/𝑀⊙) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01, which is closely consistent with what obtained by Alston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Figure 6 shows an overview of the behaviour of the time lags as a function of frequency for different black hole mass values and upper X-ray source locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' While the positive lags in previous literature were usually modelled independently using a power-law function, the ECM model can simultaneously produce both negative and positive lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The simultaneously fitted ECM Spearman’s rank correlations 𝑟𝑠 and the associated 𝑝 value for each source and all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that the first row is showing the correlations for ℎ2 − 𝐿(2−10 keV) and the final row is the Covering Fraction obtained from HYC22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The last column shows the results when all data from both 1H 0707-495 and IRAS 13224-3809 are used to find the correlation coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Par1 Par2 1H 0707-495 IRAS 13224-3809 1H and IRAS 𝑟𝑠 𝑝 𝑟𝑠 𝑝 𝑟𝑠 𝑝 ℎ2 𝐿 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} 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+page_content='019 𝑏 𝑡shift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='495 𝑏 Cvr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='408 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='154 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='402 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 ECM correlations We explore the model for correlations using the Spearman’s rank method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For this we use all individual observation table model fits returned from the single mass value phase for each AGN to maximise the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We found moderate correlations between Γ1 and 𝑡max and between Γ2 and 𝑡max and 𝑡shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' On the other hand, large errors are still a feature of 𝑡max therefore we acknowledge that these correlations are not well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Further investigation reveals these correlations are often much stronger in IRAS 13224-3809 than they are in 1H0707-495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The brightness parameters 𝑏 was strongly anti-correlated with 𝑡max in IRAS 132224-3809, where no correlation was evident in 1H 0707-495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that Caballero-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2018) also suggested that the 1H 0707-495 mass should be fixed at values below ∼ (2 − 3) × 106 𝑀⊙, otherwise the observed reverberation lags cannot be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The time lag amplitude was moderately correlated with the upper photon index Γ2 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The results are presented in Table 3, which shows the parameters of interest and the overview of the Spearman’s rank correlations𝑟𝑠 and their 𝑝 values for each source in columns 3 and 4 and for any global correlations in the final column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that the covering fraction is also tested from the results reported in HYC22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We found no correlation between the upper source height and the luminosity, however the time lags do correlate moderately with the upper X-ray source height in 1H 0707- 495, and a stronger relationship is evident in IRAS 13224-3809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Further consideration of the lags seen ≲ 250 s in the latter source provides a much stronger correlation where 𝑟𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='68 with 𝑝-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The six strongest correlations found for these sources are shown in Figure 7 and 8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The panels within these figures also show where the limits of each parameter were reached as denoted by the blue arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Both AGN have a moderate relationship for Γ2 and 𝑡shift which can be seen as the strongest global correlation where 𝑟𝑠 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='594 and 𝑝-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='00034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The lower source photon index Γ1 also correlates strongly with 𝑡max in IRAS 13224-3809 where 𝑟𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='716 and 𝑝-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='001 with no counterpart correlation seen in 1H MNRAS 000, 1–13 (2020) 8 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Hancock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 0707-495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For the upper source Γ2 a moderate correlation is seen with the brightness parameter 𝑏 in 1H 0707-495 and another strong inverse correlation emerged with 𝑡max in IRAS 13224-3809 where 𝑟𝑠 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='725 and 𝑝-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Finally, a strong correlation found in IRAS 13224-3809 and not in 1H 0707-495 was ΔΓ, (Γ1−Γ2), with 𝑡max where 𝑟𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='82 and 𝑝-value = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='86 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' It is interesting to note that unlike 1H 0707-495, IRAS 13224-3809 simultaneous fitting does not always lead to Γ1 > Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' These results suggest that the spectral properties of the corona may be specific to the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The Γ − 𝐿/𝐿Edd and Γ − ℎ2 relationships for these AGN are also presented in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' It can be seen that lower and higher limits of Γ and ℎ2 often reached their limits and these parameters may be wider as indicated by the blue arrows in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In fact, the photon index Γ1 tends to increase sharply with in- creasing luminosity above ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 × 1042 erg cm s−1 for both sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' A similar trend in AGN data against the luminosity ratio 𝐿𝑥/𝐿Edd has been reported by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2015) and interpreted as a two- phase accretion flow model, although more data would be required to investigate this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For 1H 0707-495, the correlation is flat at lower luminosity then a mild positive correlation kicks in when the luminosity is 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='7 ≲ 𝐿/𝐿Edd ≲ 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Contrarily, for IRAS 13224-3809, we see the progressively flat profile of Γ instead, and their Γ1 and Γ2 values are extreme and more separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The be- haviour of Γ may suggest that contrasting mechanisms are driving the variability in these AGN (within the model constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 5 DISCUSSION The initial fitting of the ECM to the various flux groups achieved rea- sonable fits and model descriptions, however these were not always well constrained and error ranges often tended to the model maxi- mum and minimum allowed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The model can provide a good statistical fit to the observed data when the mass is employed as a free parameter and is capable of finding the upper X-ray source heights between 3 – 20 𝑟𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Fixing the mass at the single best value obtained from the simultaneous fits also achieved similar results and whilst the black hole mass of 1H 0707-495 dropped from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='22 (obtained by the independent emcee fit) to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 log(𝑀/𝑀⊙), both model fits remain statistically reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The black hole mass for IRAS 13224-3809 was estimated at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 log(𝑀/𝑀⊙) and remained reasonably consistent with the independent measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' These ob- tained mass values contained very small errors and they fall within the limits of the variable mass values from the model fits presented in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Caballero-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2020) fix the mass of IRAS 13224-3809 at log(𝑀/𝑀⊙) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='30 and their model looks at the reverberation from a single point source and predicts only the negative reverber- ation lags, whilst the positive lags at low frequency are modelled separately using the KYNXILREV phenomenological power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Our model is mass dependent and considers both soft and hard lag mech- anisms simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Nevertheless, we note that a limiting factor may be that we have fixed the lower source height at 2 𝑟𝑔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' an as- sumption required to ease the heavy computations required to create the predicted time lags and hence we are estimating the extent of the corona rather than its true size that may be obtained by employing ℎ1 as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This would be extremely intensive with the cur- rent model and further development should maintain its complexity whilst enhancing model computational performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Despite this, some meaningful relations among the ECM parame- ters can still be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For 1H 0707-495, we find significant corre- lation (𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='05) between ℎ2 and Γ1, and significant anti-correlation between ℎ2 and 𝜉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This means that when the 1H 0707-495 corona extends upwards (larger ℎ2), the coronal emission seems to be softer (larger Γ) while producing less overall ionisation on the disc (smaller 𝜉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This is expected since the vertically extended corona should pro- duce less intense illumination pattern on the inner disc (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Wilkins & Fabian 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The tendency of softer corona with increasing its vertical extent is also found in IRAS 13224-3809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This is in line with Alston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2020) and Chainakun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2022b) where the lamp-post geometry was used and the photon index of the continuum was found to increase with the source height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In fact, Chainakun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2022b) also found that during the source height increases, the disc itself generates more high-frequency variability, suggesting that the inner disc becomes more active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Here, we find the significant anticorrelation between Γ2 and both 𝑡max and 𝑡shift (𝑟𝑠 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='724 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='648, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Therefore, when the extending corona be- comes softer, the source response as well as the signal propagation occurs faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This may agree with Chainakun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2022b) that as the source height increases the innermost region is more active so that the corona requires shorter time in response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Nevertheless, in the ECM environment, the time lags in IRAS 13224-3809 correlate stronger with ℎ2 than those found in 1H 0707- 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Consideration of the lags seen under 250 s provides a much stronger correlation where 𝑟𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='68 with 𝑝-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='005, hence there are hints that much longer time lags will still follow this cor- relation albeit with a shallower slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Furthermore, Chainakun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2019) pointed out that in order reproduce the time lags, their spher- ical corona model required higher coronal temperatures for a lower optical depth, 𝜏, supporting the 𝜏 and coronal temperature anti- correlation argument discussed by Tortosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Recently, Chainakun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2022a) presented a new method of predicting the black hole mass of X-ray reverberating AGN using artificial neu- ral networks and pointed out that the inconsistency of the lag-mass relationship may be due to the lag amplitudes being more strongly affected by other geometric effects that are not related to the mass of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Also, there is a significantly larger number of parameters in the ECM than in the lamp-post model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Perhaps, this might explain the observed anti-correlation between the lags and the mass seen in IRAS 13224-3809 under the extended corona environ- ment when using individual observations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 5), that the lag-mass relation is also modulated by other coronal parameters in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In the lamp-post geometry, the X-ray source heights in IRAS 13224-3809 have been reported to correlate positively with the lumi- nosity in the 2 – 10 keV energy range (Alston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Caballero- García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Chainakun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' However, there was no evidence of this relationship in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Whilst this is consistent with the findings of Szanecki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2020), it is in contrast to their findings that the source is compact within a few gravitational radii (when modelled using the relativistically smeared reflection from the accretion disc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Variations of the inner regions and accretion flow have been explained via transitions between the jet emitting disc (JED) and standard accretion disc (SAD) framework (Marcel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2022) and the framework of coupled hot accretion and jet model for GBHBs and AGN (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2015), suggesting that a two-phase accretion flow could be driving the observed variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' MNRAS 000, 1–13 (2020) Extended Corona Models of X-ray Reverberation in AGN 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 Γ1 0 5 10 15 20 h2(rg) rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='573 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 ξECM (erg cm s−1) 0 5 10 15 20 h2(rg) rs = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='631 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='012 0 20 40 60 80 100 120 t-shift (tg) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 Γ2 rs = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='587 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='021 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 Γ2 rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='572 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 Covering fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 b rs = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='603 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='017 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 b 0 20 40 60 80 100 120 t-shift (tg) rs = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='658 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='008 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The 1H 0707-495 ECM results showing moderate to strong correlations where the 𝑝-value <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For clarity of the ℎ2 − 𝜉ECM correlations (upper middle panel) have been heavily averaged into 4 groups and shown by the yellow points for data points located in quadrants identified by the grey dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 200 0 200 400 600 800 Time lag (s) 5 10 15 20 h2(rg) rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='556 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 Γ2 5 10 15 20 h2(rg) rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='525 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='030 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 Γ1 0 100 200 300 400 500 600 700 800 900 t-max (tg) rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='716 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 Γ2 0 100 200 300 400 500 600 700 800 900 t-max (tg) rs = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='725 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 Γ2 0 20 40 60 80 100 120 140 160 t-shift (tg) rs = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='648 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 b 0 100 200 300 400 500 600 700 800 900 t-max (tg) rs = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='661 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='004 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The IRAS 13224-3809 ECM results showing moderate to strong correlations where the 𝑝-value <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Each correlation has been divided into bin sizes as shown by the vertical grey dashed lines and the resultant mean plot (and associated errors) has been shown by the large yellow data points for clarity of each correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' MNRAS 000, 1–13 (2020) 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Hancock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For 1H 0707-495, the profile of Γ is flat at lower luminosity before showing a mild positive correlation at 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='7 ≲ 𝐿/𝐿Edd ≲ 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Although this is a small fraction of the Bolometric lumi- nosity, similar findings have been reported by the two-phase accre- tion Type II Luminous Hot Accretion Flow (LHAF) and disc-corona regime, after which the evolutionary turnover point is reached when the accretion rate changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Furthermore, the evolutionary behaviour of Γ1 and Γ2 is very similar when tracked against the Eddington fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Although there is not a multitude of individual AGN cate- gorised in the two-phase accretion scenario due to the requirement for multiple long observations, a theoretical explanation of this be- haviour was found in AGN and reported by Zdziarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2003) who suggested that the emission of a cold medium irradiating hot plasma could be due to the cooling via reprocessed emission of the hot plasma causing the observed X-ray spectrum to soften, leading to a higher value of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Contrarily, the Γ profile for IRAS 13224-3809 is relatively flat, which is comparable to the Γ− 𝐿Bol/𝐿Edd behaviour of the jet-phase regime of the JED-SAD framework where the Γ is determined by the energy distribution of power law electrons in the jet and roughly constant for sources with different luminosity’s, neither source fol- lows the expected Γ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Furthermore, for IRAS 13224-3809, the current model values of Γ1 are extreme although not uncommon (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=', Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2014) and Γ2 may drop well below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='4 approach- ing lower extreme values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' There appears to be no relationship with either source Γ and the accretion rate in IRAS 13224-3809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Similar Γ behaviour was reported from RXTE observations of MCG-6-30-15 where the spectral index increased with luminosity, reaching a final upper value after which it remained roughly static as the luminosity continued to increase (McHardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 1999), possibly due to a harder spectral component that does not change within the observation du- ration as the continuum component steepens the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' A steeper spectrum is expected for increasing luminosity and this is evident in the spectral model for IRAS 13224-3809 as reported in HYC22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The ECM scenario, however, shows very limited spec- tral index variations for its flux variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This has been previously reported for NGC 5548 (Sobolewska & Papadakis 2009) and PG 0804+761 (Papadakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Furthermore, Caballero-Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' (2012) found no significant spectral variability in NGC 4151 and NGC 2110, but reported the significant flux variations that in- dicated the intrinsic variability of the central source in NGC 4388, NGC 4945 and IC 4329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This raises the question of different spectral states in AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' However, 1H 0707-495 and IRAS 13224-3809 are both classified Narrow-line Seyfert 1 galaxies where the 2–10 keV flux variations are almost always associated with spectral variations (Papadakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The modelling applied thus far (spectral and ECM models) suggests that different mechanisms may be contribut- ing to the variability that may be explained by the differences in the geometry of each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that variations in the optical depth 𝜏 of the corona cause spec- tral changes in the Comptonised emission leading to a steeper spec- trum and the temperature decreases with increasing 𝜏 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Haardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 1997), therefore the geometry of 1H 0707-495 should consist of a smooth corona where the intrinsic flux varies in unison with Γ1 and Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' IRAS 13224-3809 however, may contain a slightly different geometry since the Γ1 and Γ2 values are more clearly separated, gen- erally remaining relatively constant and occupied at the the extreme values of the model especially when 5 𝑟𝑔 ≲ ℎ2 ≲ 10 𝑟𝑔 (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This suggests that IRAS 13224-3809 corona is more patchy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' hav- ing clear, distinct spectral zones of spectral Γ1 and Γ2 that may be related to two distinct zones of corresponding 𝜏 and coronal temper- ature), especially at locations where the corona ≲ 10 𝑟𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='06 L(2 − 10keV)/LEdd (erg s−1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 Γ 1H 0707-495 Γ1 Γ2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 Γ 0 5 10 15 20 h2(rg) 1H 0707-495 Γ1 Γ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='06 L(2 − 10keV)/LEdd (erg s−1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 Γ IRAS 13224-3809 Γ1 Γ2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='5 Γ 0 5 10 15 20 h2(rg) IRAS 13224-3809 Γ1 Γ2 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The evolution of Γ as a function of the Eddington fraction and the source height for 1H 0707-495 (top panels) and also for IRAS 13224-3809 (bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The Γ1 and Γ2 values IRAS 13224-3809 are more separated and occupied at more extreme values than those of 1H 0707-495, especially when 5 𝑟𝑔 ≲ ℎ2 ≲ 10 𝑟𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' MNRAS 000, 1–13 (2020) Extended Corona Models of X-ray Reverberation in AGN 11 Global coronal correlations across AGN samples have remained elusive and/or confusing (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=', Sarma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Hinkle & Mushotzky 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Kamraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 2022) the latter authors cautioning the use of coronal parameters and the Eddington fraction relations to infer properties of black hole systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' In contrast, this work finds that strong relations are evident in the grouped data and when drilling down to individual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' These differences are possibly due to the choice of spectral models and parameters explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Many of the 𝜒2 values fell below unity, suggesting that the model may be over-fitted due to too many free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This suggests that deeper model assumptions leading to the freezing of at least one more parameter may also be appropriate for future modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Some parameter ranges may need to be slightly wider to accommodate better statistics (whilst remaining feasible), especially for ℎ2, Γ1 and Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' On the other hand, the large errors seen in 𝑡max and 𝑡shift will be difficult to constrain with current observations and the large errors associated with the low frequency fluctuation lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' More AGN need to be explored by this model to test its capabilities against a wider sample with higher quality time lags versus frequency allowing a better statistical comparison of the data and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 6 CONCLUSIONS This work has explored the extended corona scenario using two X- ray point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The lower source may represent the base of a jet-like structure or the lower regions of a compact corona with the upper source representing the extended region where the flares are detected due to the periodic vertical collimation of a jet structure or outer region of the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The model is capable of fitting the lag-frequency of 1H 0707-495 and IRAS 13224-3809, where their black hole mass can be constrained to log(𝑀/𝑀⊙) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This work also supports the advantages of exploring corona cor- relations using individual observations and also suggests that the physics and geometry of the corona may diverge between different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' For the first time, we have gained an insight to the behaviour of Γ for each of the X-ray sources located above the black hole axis through the use of reverberation mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We find the tendency of softer corona with increasing its vertical extent in both AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This produces a much less-ionised disc which is evident in 1H 0707-495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' On the other hand, for IRAS 13224-3809 we also find the hint of increasing Γ2 with decreasing 𝑡max and 𝑡shift, suggesting that shorter propagating fluctuations and faster coronal response may be evident when the corona extends vertically upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The intrinsic flux of IRAS 13224-3809 show less variability with Γ and its corona may be more patchy in a sense that contains clearer separate spectral zones of distinct Γ1 and Γ2 that may also link to two clearer distinct zones of coronal temperature than that of 1H 0707-495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The calculations in this work were carried out using the high per- formance computer BlueCrystal of the Advanced Computing Re- search Centre, University of Bristol, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' SH thanks the STFC for funding and the Bristol, Cardiff & Swansea CDT Team for sup- port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' PC thanks funding support from (i) Suranaree University of Technology (SUT), (ii) Thailand Science Research and Innovation (TSRI), and (iii) National Science Research and Innovation Fund (NSRF), project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' 160355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' This research has made use of ISIS functions (ISISscripts) provided by ECAP/Remeis observatory and MIT (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='sternwarte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='uni-erlangen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='de/isis/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' We wish to thank the anonymous reviewer for comments and suggestions which has improved the quality of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' DATA AVAILABILITY The original data underlying this article were obtained from the XMM-Newton Observatory (http://nxsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='int).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The derived data generated in this research can be downloaded via http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='bris.' 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C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=', 2013, ApJ, 767, 121 MNRAS 000, 1–13 (2020) Extended Corona Models of X-ray Reverberation in AGN 13 APPENDIX A: TABLE MODEL ECM RESULTS This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' MNRAS 000, 1–13 (2020) 14 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Hancock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' The Table Model results for 1H 0707-495 and IRAS 13224-3809 data to 90% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Note that zero errors are a result of rounding to 3 sf and also hitting upper or lower model parameter space limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content=' Source Obs Id 𝜒2 log(𝑀/𝑀⊙) ℎ2(𝑟𝑔) Γ1 Γ2 log𝜉 (erg cm s−1) 𝑏 𝑡max(𝑡𝑔) 𝑡shift(𝑡𝑔) 1H Combined 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='50 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E3T4oBgHgl3EQfzwt3/content/2301.04731v1.pdf'} +page_content='01 −0.' metadata={'source': 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F-Yang-Mills connections are critical points of F-Yang Mills functional +on the space of connections of a principal fiber bundle, which is a generalization of +Yang-Mills connections, p-Yang-Mills connections and exponential Yang-Mills connec- +tions and so on. Here, F is a strictly increasing C2-function. In this paper, we extend +Simons theorem for an instability of Yang-Mills connections to F-Yang-Mills connec- +tions. We derive a sufficient condition that any non-flat, F-Yang-Mills connection +over convex hypersurfaces in a Euclidean space is instable. In the sphere case, this +condition is expressed by an inequality with respect to its dimension and a degree of +the differential of the function F. The proofs of the results are given by extending +Kobayashi-Ohnita-Takeuchi’s calculation to F-Yang-Mills connections. +Contents +1. +Introduction +1 +2. +Preliminaries +4 +3. +F-Yang-Mills functionals and F-Yang-Mills connections +8 +3.1. +Definition and the first variational formula +8 +3.2. +Instability and the second variational formula +9 +4. +A Simons type condition for instability of F-Yang-Mills connections +11 +4.1. +Analysis of the indices for F-harmonic forms (1) +11 +4.2. +Analysis of the indices for F-harmonic forms (2) +17 +4.3. +Instability of F-Yang-Mills connections over convex hypersurfaces in +Euclidean spaces +19 +References +22 +1. Introduction +A Yang-Mills connection is a critical point of the Yang-Mills functional defined on the +space of connections of any principal fiber bundle over a connected, closed Riemannian +manifolds. There are many developments in the theory of Yang-Mills connections. On +Date: January 11, 2023. +2020 Mathematics Subject Classification. Primary: 53C07, Secondary: 58E15. +Key words and phrases. F-Yang-Mills connection, instability, degree, F-harmonic form, index. +1 + +2 +KURANDO BABA AND KAZUTO SHINTANI +the other hand, several Yang-Mills type functionals were introduced and critical points +of such functionals have been studied, for example, p-Yang-Mills functional (Uhlenbeck +[19], Chen-Zhou [4]), exponential Yang-Mills functional (Matsuura-Urakawa [16], Wei +[20]) and the generalized Yang-Mills-Born-Infeld energy functional (Sibner-Sibner-Yang +[17], Dong-Wei [6], Gherghe [7]). +An F-Yang-Mills functional provides a unified description of the above functionals +(Jia-Zhou [10], Dong-Wei [6]). Here, F indicates a strictly increasing C2-function de- +fined on [0, c), 0 < c ≤ ∞. A critical point of the F-Yang-Mill functional is called +an F-Yang-Mills connection. The purpose of this paper is to study the stability of +F-Yang-Mills connections. More precisely, we would like to give a sufficient condition +that any non-flat, F-Yang-Mills connection is instable, which is an extension of the +following Simons theorem for the instability of Yang-Mills connections to F-Yang-Mills +connections. +Theorem 1.1 ([18]). For n > 4, any non-flat, Yang-Mills connection over the standard +sphere Sn is instable. +From this theorem the study of the weak stability for the usual Yang-Mills connec- +tions over Sn makes sense only for n ≤ 4. It is known that weakly stable Yang-Mills +connections over the 4-sphere are closely related to self-dual connections and anti-self- +dual connections. Indeed, Bourguignon-Lawson [3, Theorem B] proved that, in the case +when the structure group is a specific unitary group, any weakly stable Yang-Mills con- +nection over S4 is either self-dual or anti-self-dual. On the other hand, the construction +of these connections were given by Atiyah-Drinfeld-Hitchin-Manin [2]. We expect that +such studies can be explored for F-Yang-Mills connections. +This paper contributes to find a suitable extension of Theorem 1.1 for F-Yang-Mills +connections. In fact, we derive a Simons type condition for the instability of F-Yang- +Mills connections over convex hypersurfaces in a Euclidean space (see Theorem 4.11 +for the detail). The proof of this theorem is given by extending Kobayashi-Ohnita- +Takeuchi’s calculation [11] of the second variation of the usual Yang-Mills functional. +From Theorem 4.11, we have an extension of Theorem 1.1 as follows. +Theorem 1.2 (Corollary 4.12). Let dF ′ denote the degree of the differential F ′ defined +in Definition 4.8. Assume that the degree dF ′ is finite. Then, for n > 4dF ′ + 4, any +non-flat, F-Yang-Mills connection over Sn is instable. +Theorem 1.2 clarifies the importance of the finiteness of the degree dF ′ in order +to derive the Simons type condition for the instability of F-Yang-Mills connections. +For the usual Yang-Mills connections, this result coincides with Theorem 1.1 because +dF ′ = 0 holds. Furthermore, it can be verified that Theorem 1.2 also coincides with the +instability theorem for the p-Yang-Mills connections, which was given by Chen-Zhou +[4, Corollary 4.2]. We can find an alternative formula of the instability theorem for +F-Yang-Mills connections by Jia-Zhou [10, Corollary 16]. + +A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS +3 +On the other hand, in the case when F ′ has infinite degree, it is difficult to find +a sufficient condition for the instability of F-Yang-Mills connections under a general +setting. For example, the F-function corresponding to the exponential Yang-Mills func- +tional YM e or the generalized Yang-Mills-Born-Infeld energy functional YM ǫ=−1 with +minus sign has infinite degree. The stability of critical points of these functional were +studied by Matsuura-Urakawa for YM e and by Gherghe for YM ǫ=−1. For further de- +velopments we study the instability for the connections over Sn by means of the index +formula stated in Theorem 4.3. In fact, we derive a certain sufficient condition for +the instability of the connections by imposing the boundedness of its curvature (see, +Propositions 4.13 and 4.14 for the detail). +The organization of this paper is as follows: In Section 2, we review the basic no- +tions in the Yang-Mills theory, which are related to the present paper. We note that +Kobayashi-Ohnita-Takeuchi [11] studied the instability of Yang-Mills connections via +analysis of the indices for harmonic forms. Here, harmonic forms are defined as elements +in the zero eigenspace of Hodge-Laplacian. We recall Bochner-Weitzenb¨ock formula for +the Hodge-Laplacian, which is needed for our calculation. +In Section 3, we review +the basics for F-Yang-Mills connections. In Subsection 3.1, we recall the notion of F- +Yang-Mills connections and derive the F-Yang-Mills equation, i.e., the Euler-Lagrange +equation for the F-Yang-Mills functional. Motivated by the F-Yang-Mills equation, we +introduce the notion of F-harmonic forms (Definition 3.5). In Subsection 3.2, we recall +the definition of the instability of F-Yang-Mills connections and show the second vari- +ational formula for the F-Yang-Mills functional. Motivated by this formula, we define +the index of F-harmonic forms (Definition 3.8). In Section 4, we prove Theorem 1.2 via +analysis of the indices for F-harmonic forms. In Subsection 4.1, we extend the result +[11, (4.37) Proposition] for the index of harmonic forms to F-harmonic forms (Theorem +4.3). Following to Theorem 4.3, we need essentially to evaluate a quantity defined in +Definition 4.2. We also find that the key for proving Theorem 1.2 is to evaluate the +relation between F ′(∥ϕ∥2/2) and F ′′(∥ϕ∥2/2) for an F-harmonic 2-form ϕ. Motivated +by this consideration, in Subsection 4.2, we introduce the notion of the degree dF ′ of the +differential F ′ (Definition 4.8). In the case when dF ′ is finite, we derive an inequality for +the index of F-harmonic forms based on Theorem 4.3 (Theorem 4.10). In Subsection +4.3, we extend the result [11, (5.3) Theorem] for the instability of Yang-Mills connec- +tions to F-Yang-Mills connections by means of Theorem 4.10 (Theorem 4.11). As a +corollary of Theorem 4.11 we obtain Theorem 1.2 (Corollary 4.12). We prove Proposi- +tions 4.13 and 4.14. It is known that there are strong similarities between the theory of +Yang-Mills connections and that of harmonic maps. Finally, we discuss a counter part +of our results in the theory of harmonic maps. + +4 +KURANDO BABA AND KAZUTO SHINTANI +2. Preliminaries +Let (M, g) be an n-dimensional, connected, closed Riemannian manifold and D de- +note the Levi-Civita connection on M. Let G be a compact Lie group and g denote its +Lie algebra. We write the adjoint representation of G on g as Ad : G → GL(g). Let P +be a principal fiber bundle over M with structure group G. A g-valued 1-form A on P +is called a connection if A is of type Ad and A( ˜X) = X holds for all X ∈ g, where ˜X de- +notes the fundamental vector field on P associated with X. We denote by Ωk +Ad,hor(P, g) +the vector space of horizontal k-forms of type Ad on P with values in g. It is verified +that the curvature 2-form of a connection on P gives an element of Ω2 +Ad,hor(P, g). The +kernel of a connection on P defines an Ehresmann connection, that is, a right-invariant, +horizontal distribution on P. Then, it is known that this distribution is integrable if +and only if the curvature 2-form of A vanishes. A connection A is said to be flat, if its +curvature 2-form vanishes. For any two connections A, A′, the difference A − A′ is in +Ω1 +Ad,hor(P, g). Conversely, A+α gives another connection on P for all α ∈ Ω1 +Ad,hor(P, g). +Hence the set CP of connections on P becomes an affine space over the vector space +Ω1 +Ad,hor(P, g). +We make use of a different description of connections on P. Denote by gP = P ×Ad g +the adjoint bundle of P, that is, the associated vector bundle of P with the adjoint +representation Ad of G on g. It follows from [8, Theorem 5.13.4] that Ωk +Ad,hor(P, g) is +canonically isomorphic with the vector space of k-forms on M with values in gP, which +we write Ωk(gP) = Γ(ΛkT ∗M ⊗ gP). Any connection on P corresponds to a connection +on gP, i.e., a covariant derivative ∇ : Γ(gP) → Ω1(gP) on the bundle gP. It is shown +that the curvature R∇ of ∇ on gP is in Ω2(gP) (cf. [8, Corollary 5.13.5]). In what +follows, we identify CP with the set of connections on gP, which is an affine space over +the vector space Ω1(gP). +We give a fiber metric on gP which is compatible with connections on gP. Such a fiber +metric is induced from an Ad(G)-invariant inner product ⟨·, ·⟩ on g (cf. [8, Proposition +5.9.7]). In addition, ⟨·, ·⟩ also induces a pointwise inner product on the space Ωk(gP), +which is denoted by the same symbol ⟨·, ·⟩. We set ∥ϕ∥2 = ⟨ϕ, ϕ⟩ for ϕ ∈ Ωk(gP). Here, +we write ⟨ϕ, ψ⟩ (ϕ, ψ ∈ Ωk(gP)) by means of their components. We take an orthonormal +basis (e1, . . . , en) of the tangent space TxM at x ∈ M, and denote by (θ1, . . . , θn) its +dual basis. If we write +ϕ = 1 +k! +� +i1,...,ik +ϕei1,...,eikθi1 ∧ · · · ∧ θik , +ψ = 1 +k! +� +i1,...,ik +ψei1,...,eikθi1 ∧ · · · ∧ θik , +then we obtain +⟨ϕ, ψ⟩ = 1 +k! +� +i1,...,ik +⟨ϕei1,...,eik, ψei1,...,eik⟩ . + +A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS +5 +By integrating this pointwise inner product over M, we obtain an inner product on +Ωk(gP) as follows: +(ϕ, ψ) = +� +M +⟨ϕ, ψ⟩dv , +(2.1) +where dv denotes the Riemannian volume form on M. +For any connection ∇, the covariant exterior derivative d∇ : Ωk(gP) → Ωk+1(gP) is +given by +(d∇ϕ)X0,...,Xk = +k +� +i=0 +(−1)i(∇Xiϕ)X0,..., ˆ +Xi,...,Xk , +(2.2) +for ϕ ∈ Ωk(gP), where X0, . . . , Xk are tangent vectors of M. We denote by δ∇ the +formal adjoint operator of d∇, that is, δ∇ : Ωk(gP) → Ωk−1(gP) is defined by (d∇ψ, ϕ) = +(ψ, δ∇ϕ) for ϕ ∈ Ωk(gP) and ψ ∈ Ωk−1(gP). Following to [3, (2.13)], for any ϕ ∈ Ωk(gP), +δ∇ has the following expression: +(δ∇ϕ)X1,...,Xk−1 = − +n +� +j=1 +(∇ejϕ)ej,X1,...,Xk−1 . +For any connection ∇, the curvature 2-form R∇ satisfies d∇R∇ = 0, which is called the +Bianchi identity for ∇. In general, d∇ ◦ d∇ does not vanish. It is known that, if ∇ is +flat, then d∇ ◦ d∇ = 0 holds. This is an alternative interpretation of flat connections. +A Yang-Mills connection ∇ is defined as a critical point of the Yang-Mills functional +CP → R; ∇ �→ 1 +2 +� +M +∥R∇∥2dv . +It is shown that the Euler-Lagrange equation for this functional is given by δ∇R∇ = 0, +which is called the Yang-Mills equation. Hodge-Laplacian is defined by ∆∇ = d∇δ∇ + +δ∇d∇, which gives a differential operator on Ωk(gP). A gP-valued form ϕ is called a +harmonic form if ϕ satisfies ∆∇ϕ = 0. Then, it is verified that ϕ is harmonic if and +only if it satisfies d∇ϕ = 0 and δ∇ϕ = 0. This yields that the curvature form R∇ of a +Yang-Mills connection ∇ is a harmonic form. In Section 3, we will recall the notion of F- +Yang-Mills connections, which is an extension of Yang-Mills connections. Furthermore, +we will introduce the notion of F-harmonic forms as an extension of harmonic forms +(see Definition 3.5). +We show Bochner-Weitzenb¨ock formula for gP-valued forms, which describes the +relation between the Hodge-Laplacian and the rough Laplacian. This formula plays a +fundamental role in analysis of F-harmonic forms. In fact, we make use of this formula +to prove Proposition 4.7 in Section 4, which gives a method to calculate the differential +of the curvature R∇. We first recall the notion of the rough Laplacian, namely, it is +defined by +∇∗∇ϕ = − +n +� +j=1 +∇2 +ej,ejϕ, +ϕ ∈ Ωk(gP) , + +6 +KURANDO BABA AND KAZUTO SHINTANI +where ∇2 +X,Y = ∇X∇Y − ∇DXY . It is verified that ∇∗∇ is symmetric and non-negative. +This implies that a gP-valued form ϕ satisfies ∇∗∇ϕ = 0 if and only if ϕ is parallel +(∇ϕ = 0). We also recall Weitzenb¨ock curvature R∇ : Ωk(gP) → Ωk(gP) for k = 1, 2 as +follows. +In the case when k = 1, the operator R∇ : Ω1(gP) → Ω1(gP) is given by +R∇(α) = +� +i,j +[R∇ +ji, αj]θi , +for α = � +i αiθi ∈ Ω1(gP). If we set +[· ∧ ·] : Ω1(gP) × Ω1(gP) → Ω2(gP); [α ∧ β]X,Y = [αX, βY ] − [αY , βX] +then the following relation holds: +⟨R∇(α), α⟩ = ⟨[α ∧ α], R∇⟩ , +α ∈ Ω1(gP) . +(2.3) +We denote by R the Riemannian curvature on M, and by Ric : TxM → TxM (x ∈ M) +the Ricci curvature operator, that is, +Ric(X) = +n +� +i=1 +RX,eiei , +X ∈ TxM . +For α ∈ Ω1(gP), we define α ◦ Ric ∈ Ω1(gP) by (α ◦ Ric)X = αRic(X) for all X ∈ TxM. +Then, the following proposition shows the Bochner-Weitzenb¨ock formula for Ω1(gP). +Proposition 2.1 ([3, (3.2) Theorem]). For α ∈ Ω1(gE), we have +∆∇α = ∇∗∇α + α ◦ Ric + R∇(α) . +Next, let us consider the case when k = 2. We recall the Weitzenb¨ock curvature for +Ω2(gP), that is, R∇ : Ω2(gP) → Ω2(gP) is given by +R∇(ϕ)X,Y = +n +� +j=1 +� +[R∇ +ej,X, ϕej,Y ] − [R∇ +ej,Y , ϕej,X] +� +, +for ϕ ∈ Ω2(gP), where X, Y are tangent vector fields on M. We denote by X(M) the +space of tangent vector fields on M, and by Ω2(M) the space of 2-forms on M. For +ϕ ∈ Ω2(gP) and ω ∈ Ω2(M) ⊗ End(X(M)), we set +(ϕ ◦ ω)X,Y = 1 +2 +n +� +j=1 +ϕej,ωX,Y (ej) , +X, Y ∈ X(M) . +Here, we give a concrete example of ω, which appears in the Bochner-Weitzenb¨ock +formula for Ω2(gP). +Example 2.2. For any X, Y ∈ X(M), we set +(X ∧ Y )(Z) = ⟨X, Z⟩Y − ⟨Y, Z⟩X , +Z ∈ X(M) . + +A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS +7 +If I denotes the identity transformation on TxM, then Ric ∧ I ∈ Ω2(M) ⊗ End(X(M)) +is defined by +(Ric ∧ I)X,Y = Ric(X) ∧ Y + X ∧ Ric(Y ) , +X, Y ∈ X(M) . +We are ready to show the Bochner-Weitzenb¨ock formula for Ω2(gP). +Proposition 2.3 ([3, (3.10) Theorem]). For ϕ ∈ Ω2(gP), we have +∆∇ϕ = ∇∗∇ϕ + ϕ ◦ (Ric ∧ I + 2R) + R∇(ϕ) . +(2.4) +In order to evaluate the second term in (2.4), Kobayashi-Ohnita-Takeuchi [11, (4.36)] +introduced R(ϕ, ϕ) and Ric(ϕ, ϕ) for ϕ = (1/2) � +i,j ϕijθi ∧ θj ∈ Ω2(gP), which are +defined as follows: +R(ϕ, ϕ) = +� +i,j,k,l +Rijkl⟨ϕij, ϕkl⟩ , +Ric(ϕ, ϕ) = +� +i,j,k,l +Rikδjl⟨ϕij, ϕkl⟩ , +where Rijkl and Rik are the components of the Riemannian curvature R and the Ricci +curvature Ric on M, respectively, that is, R(ek, el)ej = � +i Ri +jklei = � +i Rijklei and +Rik = � +l Rlkli. By the definition, R(ϕ, ϕ) and Ric(ϕ, ϕ) are independent of the choice +of (e1, . . . , en). Here, we remark that, in the original definitions of R(ϕ, ϕ) and Ric(ϕ, ϕ), +the inner product (·, ·) as in (2.1) was used instead of ⟨·, ·⟩. +Then we have the following lemma. +Lemma 2.4. For any ϕ ∈ Ω2(gP), we have +⟨ϕ ◦ (Ric ∧ I + 2R), ϕ⟩ = Ric(ϕ, ϕ) − 1 +2R(ϕ, ϕ) . +Proof. A direct calculation shows +⟨ϕ ◦ Ric ∧ I, ϕ⟩ = 1 +2 +� +i,j +⟨(ϕ ◦ Ric ∧ I)ei,ej, ϕei,ej⟩ += 1 +2 +� +i,j,k +⟨ϕek,(Ric∧I)ei,ej (ek), ϕei,ej⟩ . +Then, by using ϕei,ej = −ϕej,ei, we get +⟨ϕ ◦ Ric ∧ I, ϕ⟩ = +� +i,j,k +Rik⟨ϕek,ej, ϕei,ej⟩ = Ric(ϕ, ϕ) . +In a similar manner, we can derive +⟨ϕ ◦ 2R, ϕ⟩ = −1 +2 +� +i,j,k,l +Rijkl⟨ϕij, ϕkl⟩ = −1 +2R(ϕ, ϕ) . +Thus, we have the assertion. +□ + +8 +KURANDO BABA AND KAZUTO SHINTANI +3. F-Yang-Mills functionals and F-Yang-Mills connections +3.1. Definition and the first variational formula. Let M be a connected, closed +Riemannian manifold and G be a compact connected Lie group. Let P = P(M, G) be +a principal fiber bundle over M with structure group G. We denote by gP the adjoint +bundle of P. Let 0 < c ≤ ∞ and F : [0, c) → R be a strictly increasing C2-function. +We set R≥0 = {a ∈ R | a ≥ 0}. +Definition 3.1. The F-Yang-Mills functional YM F : CP → R≥0 is defined by +YM F(∇) = +� +M +F(1 +2∥R∇∥2)dv . +A connection ∇ on gP is called a F-Yang-Mills connection if ∇ is a critical point of +YM F. Then, its curvature 2-form R∇ is also called the F-Yang-Mills field of ∇. +For example, if we take F(t) = t, then the F-Yang-Mills functional coincides with +the usual Yang-Mills functional. Other examples are found in Uhlenbeck ([19]), Sibner- +Sibner-Yang ([17]) and Matsuura-Urakawa ([16]). +Example 3.2. (1) Let p ≥ 2. If we put Fp(t) = (1/p)(2t)p/2, then the Fp-Yang-Mills +functional coincides with the p-Yang-Mills functional. A critical point of the p-Yang- +Mills functional is called a p-Yang-Mills connection (cf. [19]). +(2) Let ǫ = ±1. If we put Fǫ(t) = ǫ√1 + 2ǫt − ǫ, then the Fǫ-Yang-Mills functional +is called the generalized Yang-Mills-Born-Infeld energy functional with sign ǫ. We call +its critical point a critical connection of the functional (cf. [17]). +(3) If we put Fe(t) = et, then the Fe-Yang-Mills functional coincides with the expo- +nential Yang-Mills functional. A critical point of the exponential Yang-Mills functional +is called an exponential Yang-Mills connection (cf. [16]). +F-Yang-Mills connections are obtained by solving the Euler-Lagrange equation for +YM F. Here, we recall the first variational formula for the functional. +Proposition 3.3 ([6, Lemma 3.1], [10, (11)]). Let ∇t (|t| < ε) be a C∞-curve in CP +with ∇0 = ∇. If we put +α = d +dt +���� +t=0 +∇t ∈ Ω1(gP) , +then we have +d +dt +���� +t=0 +YM F(∇t) = +� +M +⟨δ∇(F ′(1 +2∥R∇∥2)R∇), α⟩dv . +Proof. Let ∇ ∈ CP and ∇t = ∇+At be a C∞-curve in CP through ∇, where At ∈ Ω1(gP) +with A0 = 0. Then the curvature of ∇t is given by +R∇t = R∇ + d∇At + 1 +2[At ∧ At] . + +A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS +9 +By a straightforward calculation, we have +d +dtYM F(∇t) = +� +M +d +dtF(1 +2∥R∇t∥2)dv += +� +M +F ′(1 +2∥R∇t∥2)⟨ d +dtR∇t, R∇t⟩dv += +� +M +F ′(1 +2∥R∇t∥2)⟨d∇ d +dtAt + [ d +dtAt ∧ At], R∇t⟩dv . +Let α = d +dt +���� +t=0 +∇t. The above equality becomes as follows +d +dt +���� +t=0 +YM F(∇t) = +� +M +F ′(1 +2∥R∇∥2)⟨R∇, d∇α⟩dv = +� +M +⟨δ∇(F ′(1 +2∥R∇∥2)R∇), α⟩dv . +Thus, we have complete the proof. +□ +From Proposition 3.3 we immediately get the Euler-Lagrange equation for YM F as +follows: +Corollary 3.4. ∇ is an F-Yang-Mills connection if and only if ∇ satisfies +δ∇(F ′(1 +2∥R∇∥2)R∇) = 0 . +(3.1) +We call (3.1) the F-Yang-Mills equation. Motivated by the F-Yang-Mills equation, +we introduce the notion of F-harmonic forms as follows. +Definition 3.5. A gP-valued form ϕ is said to be F-harmonic, if ϕ satisfies the following +two equations: +d∇ϕ = 0 , +δ∇(F ′(1 +2∥ϕ∥2)ϕ) = 0 . +(3.2) +We note that the curvature 2-form R∇ of an F-Yang-Mills connection ∇ is F- +harmonic. Indeed, R∇ satisfies (3.2) because of the Bianchi identity and the F-Yang +Mills equation for ∇. +3.2. Instability and the second variational formula. Let us consider the instabil- +ity for an F-Yang-Mills connection. We recall here the definition of this property. +Definition 3.6. An F-Yang-Mills connection ∇ is said to be weakly stable if the +following inequality holds for any α ∈ Ω1(gP): +d2 +dt2 +���� +t=0 +YM F(∇t) ≥ 0 +where +α = d +dt +���� +t=0 +∇t . +An F-Yang-Mills connection ∇ is said to be instable if ∇ is not weakly stable. +The following proposition gives the second variational formula for the F-Yang-Mills +functional. + +10 +KURANDO BABA AND KAZUTO SHINTANI +Proposition 3.7. Let ∇ be an F-Yang-Mills connection and ∇t (|t| < ε) be a C∞- +curve in CP with ∇0 = ∇. Then the second variation of the F-Yang-Mills functional is +given by the following: +d2 +dt2 +���� +t=0 +YM F(∇t) = +� +M +F ′′(1 +2∥R∇∥2)⟨d∇α, R∇⟩2dv ++ +� +M +F ′(1 +2∥R∇∥2) +� +⟨R∇(α), α⟩ + ∥d∇α∥2� +dv , +(3.3) +where α = d +dt +���� +t=0 +∇t. +Proof. A direct calculation yields +d +dtR∇t = d∇dAt +dt + 1 +2 +d +dt[At ∧ At] , +and +d2 +dt2R∇t = d∇( d2 +dt2At) + [ d2 +dt2At ∧ At] + [dAt +dt ∧ dAt +dt ] . +Hence we have +d +dt +���� +t=0 +R∇t = d∇α, +d2 +dt2 +���� +t=0 += d∇β + [α ∧ α]. +where α = d +dt +���� +t=0 +At and β = d2 +dt2 +���� +t=0 +At. We have +d2 +dt2 +���� +t=0 +YM F(∇t) += +� +M +F ′′(1 +2∥R∇∥2)⟨d∇α, R∇⟩2dv + +� +M +F ′(1 +2∥R∇∥2) +� +⟨[α ∧ α], R∇⟩ + ∥d∇α∥2� +dv ++ +� +M +F ′(1 +2∥R∇∥2)⟨d∇β, R∇⟩dv. +(3.4) +Then it can be verified that the third term of (3.4) vanishes. Indeed, since ∇ is an +F-Yang-Mills connection, we find +� +M +F ′(1 +2∥R∇∥2)⟨d∇β, R∇⟩dv = +� +M +⟨β, δ∇(F ′(1 +2∥R∇∥2)R∇)⟩dv = 0. +Therefore, we obtain +d2 +dt2 +���� +t=0 +YM F(∇t) += +� +M +F ′′(1 +2∥R∇∥2)⟨d∇α, R∇⟩2dv + +� +M +F ′(1 +2∥R∇∥2) +� +⟨[α ∧ α], R∇⟩ + ∥d∇α∥2� +dv += +� +M +F ′′(1 +2∥R∇∥2)⟨d∇α, R∇⟩2dv + +� +M +F ′(1 +2∥R∇∥2) +� +⟨R∇(α), α⟩ + ∥d∇α∥2� +dv. +Here, in the last equality we have used (2.3). Thus, we have complete the proof. +□ + +A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS +11 +An alternative expression of the second variational formula is found in [10, (20)]. The +difference between them is the integrand of the second term of (3.3). In the case when +YM F is the usual Yang-Mills functional (F(t) = t), F ′′(t) = 0 holds, so that the first +term of (3.3) vanishes. +Motivated by Proposition 3.7, we define the index for any F-harmonic 2-form as +follows: +Definition 3.8. The index of an F-harmonic form ϕ ∈ Ω2(gP) is defined by +Iϕ(α) = +� +M +F ′′(1 +2∥ϕ∥2)⟨d∇α, ϕ⟩2dv + +� +M +F ′(1 +2∥ϕ∥2) +� +⟨R∇(α), α⟩ + ∥d∇α∥2� +dv , +for any α ∈ Ω1(gP). +It follows from Proposition 3.7 that, for any F-Yang-Mills connection ∇, if ∇ is +weakly stable, then IR∇(α) ≥ 0 holds for all α ∈ Ω1(gP). In the next section, we will +derive a sufficient condition for the instability of F-Yang-Mills connections via analysis +of the indices for F-harmonic forms. +4. A Simons type condition for instability of F-Yang-Mills connections +4.1. Analysis of the indices for F-harmonic forms (1). Let M be an n-dimensional, +connected, closed Riemannian manifold and D denote the Levi-Civita connection on M. +Let P = P(M, G) be a principal fiber bundle over M with structure group G. Suppose +that the base space M is isometrically immersed in an N-dimensional Euclidean space +(RN, ⟨·, ·⟩) with n < N. Denote by h its second fundamental form. We shall make use +of the following convention on the ranges of indices: +1 ≤ A, B, C ≤ N, +1 ≤ i, j, k, l, m ≤ n, +n + 1 ≤ µ ≤ N . +Let (e1, . . . , en) be an orthonormal basis of TxM (x ∈ M) and (en+1, . . . , eN) be an +orthonormal basis of the normal space T ⊥ +x M of M in RN. Let (E1, . . . , EN) be the +canonical basis of RN. We denote by VA the tangent component of EA with respect +to the orthogonal decomposition RN = TxM ⊕ T ⊥ +x M. If we set vB +A = ⟨EA, eB⟩, then +the matrix (vB +A)1≤A,B≤N becomes orthogonal. The tangent vector field VA is given by +VA = � +i vi +Aei. Let hµ +ij denote the component of h(ei, ej) = � +µ hµ +ijeµ. Then we get the +following lemma. +Lemma 4.1. With the above settings, we obtain: +DeiVA = +� +j +� +µ +vµ +Ahµ +ijej . +Proof. We write DeiVA as DeiVA = � +j⟨DeiVA, ej⟩ej. In order to prove this lemma, it +is sufficient to verify ⟨DeiVA, ej⟩ = ⟨EA, h(ei, ej)⟩. Since M is isometrically immersed +in (RN, ⟨·, ·⟩), the Levi-Civita connection D on M is compatible with ⟨·, ·⟩. Hence we +have +ei⟨VA, ej⟩ = ⟨DeiVA, ej⟩ + ⟨VA, Deiej⟩ . +(4.1) + +12 +KURANDO BABA AND KAZUTO SHINTANI +On the other hand, since EA is parallel with respect to the canonical connection D0 on +RN, we have +ei⟨VA, ej⟩ = ei⟨EA, ej⟩ = ⟨EA, D0 +eiej⟩ = ⟨VA, Deiej⟩ + ⟨EA, h(ei, ej)⟩ . +(4.2) +Here, in the last equality, we have used the Gauss formula for the submanifold M in +RN. By comparing (4.1) to (4.2), we get ⟨DeiVA, ej⟩ = ⟨EA, h(ei, ej)⟩. Thus, we have +complete the proof. +□ +We evaluate the indices for F-harmonic 2-forms. More precisely, we calculate the +summation � +A Iϕ(ιVAϕ) for an F-harmonic 2-form ϕ, where ι denotes the interior +product. By Definition 3.8 we have +� +A +Iϕ(ιVAϕ) = +� +M +F ′′(1 +2∥ϕ∥2)⟨ +� +A +d∇(ιVAϕ), ϕ⟩2dv ++ +� +M +F ′(1 +2∥ϕ∥2) +� +⟨ +� +A +R∇(ιVAϕ), ιVAϕ⟩ + +� +A +∥d∇(ιVAϕ)∥2 +� +dv . +(4.3) +Following to [11, (4.37)], we define H(ϕ, ϕ) for any ϕ ∈ Ω2(gP) as follows : +H(ϕ, ϕ) = +� +i,j,k,l +� +µ +Hµhµ +ikδjl⟨ϕij, ϕkl⟩ , +where Hµ = � +m hµ +mm denotes the mean curvature of M in RN. +On the other hand, in the present case we introduce the following quantity. +Definition 4.2. We set +h1(ϕ, ϕ) = +� +µ +hµ +1(ϕ, ϕ)eµ , +hµ +1(ϕ, ϕ) = +� +i,j,k,l +hµ +ikδjl⟨ϕij, ϕkl⟩ . +Here, we note that H(ϕ, ϕ) and h1(ϕ, ϕ) are independent of the choice of (e1, . . . , en) +and (en+1, . . . , eN). In addition, for each µ, the component hµ +1(ϕ, ϕ) of h1(ϕ, ϕ) is also +independent of the choice of (e1, . . . , en). As shown later in Theorem 4.3, h1(ϕ, ϕ) is +needed to evaluate the first term in (4.3). +The purpose of this subsection is to prove the following theorem. +Theorem 4.3. With the above settings, we obtain: +� +A +Iϕ(ιVAϕ) = +� +M +F ′′(1 +2∥ϕ∥2)⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩dv ++ +� +M +F ′(1 +2∥ϕ∥2) {H(ϕ, ϕ) − 2Ric(ϕ, ϕ) + R(ϕ, ϕ)} dv . +(4.4) +This theorem is an extension of [11, (4.37) Proposition] to F-harmonic forms. +In order to prove Theorem 4.3, we first prepare some results. + +A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS +13 +Lemma 4.4. For any ϕ ∈ Ω2(gP), we have +� +A +⟨R∇(ιVAϕ), ιVAϕ⟩ = ⟨R∇(ϕ), ϕ⟩ . +(4.5) +Proof. We express ϕ and R∇ as follows: +ϕ = 1 +2 +� +i,j +ϕei,ejθi ∧ θj, +R∇ = 1 +2 +� +i,j +R∇ +ei,ejθi ∧ θj . +Then, by (2.3), we have +the l.h.s. of (4.5) = +� +A +⟨[ιVAϕ ∧ ιVAϕ], R∇⟩ += 1 +2 +� +A +� +i,j +2⟨[(ιVAϕ)ei, (ιVAϕ)ej], R∇ +ei,ej⟩ += +� +i,j,k +⟨[ϕek,ei, ϕek,ej], R∇ +ei,ej⟩ . +(4.6) +Here, in the last equality, we have used (ιVAϕ)ei = � +k vk +Aϕek,ei. On the other hand, by +the Ad(G)-invariance of ⟨·, ·⟩, we get the following two relations: +⟨[R∇ +ei,ek, ϕei,ej], ϕej,ek⟩ = −⟨ϕei,ej, [R∇ +ei,ek, ϕej,ek]⟩ , +⟨[R∇ +ei,ej, ϕei,ek], ϕej,ek⟩ = ⟨[ϕek,ei, ϕek,ej], R∇ +ei,ej⟩ . +By using these relations, we obtain +the r.h.s. of (4.5) = 1 +2 +� +i,j,k +� +⟨[R∇ +ei,ej, ϕei,ek], ϕej,ek⟩ − ⟨[R∇ +ei,ek, ϕei,ej], ϕej,ek⟩ +� += 1 +2 +� +i,j,k +� +⟨[R∇ +ei,ej, ϕei,ek], ϕej,ek⟩ + ⟨ϕei,ej, [R∇ +ei,ek, ϕej,ek]⟩ +� += +� +i,j,k +⟨[R∇ +ei,ej, ϕei,ek], ϕej,ek⟩ += +� +i,j,k +⟨[ϕek,ei, ϕek,ej], R∇ +ei,ej⟩ . +(4.7) +Comparing (4.6) to (4.7) we have the assertion. +□ +We define h2(ϕ, ϕ) and h′ +2(ϕ, ϕ) for ϕ ∈ Ω2(gE) as follows: +h2(ϕ, ϕ) = +� +i,j,k,l +� +µ +hµ +ikhµ +lj⟨ϕij, ϕkl⟩, +h′ +2(ϕ, ϕ) = +� +i,j,k,l,m +� +µ +hµ +mkhµ +miδjl⟨ϕij, ϕkl⟩ . +By the definition, h(ϕ, ϕ) and h′ +2(ϕ, ϕ) are independent of the choice of (e1, . . . , en) and +(en+1, . . . , eN). Then, we have the following lemma. +Lemma 4.5. Let ϕ be in Ω2(gP) satisfying d∇ϕ = 0. Then we have: + +14 +KURANDO BABA AND KAZUTO SHINTANI +(1) +� +A +∥d∇(ιVAϕ)∥2 = ∥∇ϕ∥2 + h2(ϕ, ϕ) + h′ +2(ϕ, ϕ). +(2) +� +A +⟨d∇(ιVAϕ), ϕ⟩2 = ∥ϕ∥2∥∇∥ϕ∥∥2 + ⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩. +Proof. Let ϕ ∈ Ω2(gP) with d∇ϕ = 0. We express ϕ and ∇ϕ as follows: +ϕ = 1 +2 +� +i,j +ϕijθi ∧ θj , +∇ϕ = 1 +2 +� +i,j,k +∇kϕijθk ⊗ (θi ∧ θj) . +(1) If we write +d∇(ιVAϕ) = 1 +2 +� +i,j +(d∇(ιVAϕ))ei,ejθi ∧ θj , +then, by means of (2.2), the component (d∇(ιVAϕ))ei,ej has the following expression: +(d∇(ιVAϕ))ei,ej = +� +k +� +µ +vµ +Ahµ +ikϕkj + +� +k +vk +A(∇eiϕ)ek,ej +− +�� +k +� +µ +vµ +Ahµ +jkϕki + +� +k +vk +A(∇ejϕ)ek,ei +� +. +(4.8) +Then, we have +� +A +∥d∇(ιVAϕ)∥2 = 1 +2 +� +A +� +i,j +⟨d∇(ιVAϕ)ei,ej, d∇(ιVAϕ)ei,ej⟩ += +� +i,j,k +{⟨∇iϕkj, ∇iϕkj⟩ − ⟨∇iϕkj, ∇jϕki⟩} ++ +� +i,j,k,l +� +µ +� +hµ +ikhµ +il⟨ϕkj, ϕlj⟩ − hµ +ikhµ +jl⟨ϕkj, ϕli⟩ +� +. +(4.9) +It can be shown that the second term of (4.9) coincides with h2(ϕ, ϕ) + h′ +2(ϕ, ϕ). On +the other hand, we make use of the condition d∇ϕ = 0 in order to verify that the first +term of (4.9) is equal to ∥∇ϕ∥2. Here, d∇ϕ = 0 yields +∇iϕkj + ∇jϕik + ∇kϕji = 0 . +(4.10) +By using this we obtain +� +i,j,k +⟨∇iϕkj, ∇jϕki⟩ = +� +i,j,k +⟨∇iϕkj, ∇iϕkj⟩ + +� +i,j,k +⟨∇iϕkj, ∇kϕji⟩ += +� +i,j,k +⟨∇iϕkj, ∇iϕkj⟩ + +� +i,j,k +⟨∇jϕik, ∇iϕkj⟩ , +that is, +� +i,j,k +⟨∇iϕkj, ∇jϕki⟩ = 1 +2 +� +i,j,k +⟨∇iϕkj, ∇iϕkj⟩ . +From the above arguments we obtain (1). + +A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS +15 +(2) By means of (4.8), we have +� +A +⟨d∇(ιVAϕ), ϕ⟩ += +� +i,j,k +i′,j′ +⟨ϕij, ∇iϕkj⟩⟨ϕi′j′, ∇i′ϕkj′⟩ + +� +i,j,k +i′,j′,k′ +� +µ +hµ +ikhµ +i′k′⟨ϕij, ϕkj⟩⟨ϕi′j′, ϕk′j′⟩ += +� +k +�� +i,j +⟨ϕij, ∇iϕkj⟩ +�2 ++ +� +i,j,k,l +i′,j′,k′,l′ +� +µ +hµ +ikhµ +i′k′δjlδj′l′⟨ϕij, ϕkl⟩⟨ϕi′j′, ϕk′l′⟩ . (4.11) +Then, we can verify that the second term of the right hand side of (4.11) coincides with +⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩. By using (4.10), we get +� +i,j +⟨ϕij, ∇iϕkj⟩ = 1 +2 +� +i,j +⟨ϕij, ∇kϕij⟩ = ⟨ϕ, ∇ekϕ⟩ = ∥ϕ∥(∇ek∥ϕ∥) . +Substituting this into the first term of the right hand side of (4.11), we have +� +k +�� +i,j +⟨ϕij, ∇iϕkj⟩ +�2 += ∥ϕ∥2 � +k +(∇ek∥ϕ∥)2 = ∥ϕ∥2∥∇∥ϕ∥∥2 . +From the above arguments we have complete the proof of this lemma. +□ +Here, we rewrite h2(ϕ, ϕ) and h′ +2(ϕ, ϕ) in terms of Ric(ϕ, ϕ), R(ϕ, ϕ) and H(ϕ, ϕ). +Lemma 4.6. For any ϕ ∈ Ω2(ϕ, ϕ), we get: +h2(ϕ, ϕ) = 1 +2R(ϕ, ϕ) , +h′ +2(ϕ, ϕ) = H(ϕ, ϕ) − Ric(ϕ, ϕ) . +(4.12) +Proof. It follows from the Gauss equation for M in RN ([13, Proposition 4.1, Chapter +VII]) that the following relation holds: +Rijkl = +� +µ +(hµ +ikhµ +jl − hµ +jkhµ +il) . +(4.13) +Then we obtain R(ϕ, ϕ) = � +i,j,k,l +� +µ(hµ +ikhµ +jl − hµ +jkhµ +il)⟨ϕij, ϕkl⟩ = 2h2(ϕ, ϕ). On the +other hand, (4.13) obeys Rik = � +µ (Hµhik − � +m hµ +imhµ +mk), from which we can derive +Ric(ϕ, ϕ) = H(ϕ, ϕ) − h′ +2(ϕ, ϕ). Thus, we have proved this lemma. +□ +Substituting (4.12) into Lemma 4.5, (1) we have +� +A +∥d∇(ιVAϕ)∥2 = ∥∇ϕ∥2 + 1 +2R(ϕ, ϕ) + H(ϕ, ϕ) − Ric(ϕ, ϕ) . +(4.14) +We also get +H(ϕ, ϕ) − 2Ric(ϕ, ϕ) + R(ϕ, ϕ) = −H(ϕ, ϕ) + 2(h2(ϕ, ϕ) + h′ +2(ϕ, ϕ)) . +(4.15) + +16 +KURANDO BABA AND KAZUTO SHINTANI +The Bochner-Weitzenb¨ock formula gives a way to calculate the differential of an F- +harmonic form. Indeed, we make use of this formula to prove the following theorem, +which is a generalization of [9, Lemma 8]. +Proposition 4.7. For any F-harmonic form ϕ ∈ Ω2(gP), we have: +� +M +F ′′(1 +2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2dv + +� +M +F ′(1 +2∥ϕ∥2) +� +⟨R∇(ϕ), ϕ⟩ + ∥∇ϕ∥2� +dv += − +� +M +F ′(1 +2∥ϕ∥2)⟨ϕ ◦ (Ric ∧ I + 2R), ϕ⟩dv . +(4.16) +Proof. Let x ∈ M and (e1, . . . , en) be an orthonormal basis of TxM. +We extend +(e1, . . . , en) to a local orthonormal frame field so that (De1)(x) = 0, . . . , (Den)(x) = 0. +Then, at the point x, we have +∆F(1 +2∥ϕ∥2) = − +� +∇ei∇eiF(1 +2∥ϕ∥2) += −F ′′(1 +2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2 + F ′(1 +2∥ϕ∥2) · 1 +2∆∥ϕ∥2 . +(4.17) +From Proposition 2.3 and d∇ϕ = 0, we can derive +1 +2∆∥ϕ∥2 = ⟨d∇δ∇ϕ, ϕ⟩ − ⟨ϕ ◦ (Ric ∧ I + 2R), ϕ⟩ − ⟨R∇(ϕ), ϕ⟩ − ∥∇ϕ∥2 . +Substituting this into the right hand side of the second term of (4.17), we obtain +∆F(1 +2∥ϕ∥2) += −F ′′(1 +2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2 − F ′(1 +2∥∇ϕ∥2) +� +⟨R∇(ϕ), ϕ⟩ + ∥∇ϕ∥2� +− F ′(1 +2∥ϕ∥2)⟨ϕ ◦ (Ric ∧ I + 2R), ϕ⟩ + F ′(1 +2∥∇ϕ∥2)⟨d∇δ∇ϕ, ϕ⟩ . +By integrating both sides over M, the left hand side vanishes because of Green’s +theorem ([12, Appendix 6]), and the right hand side is equal to +− +� +M +F ′′(1 +2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2dv − +� +M +F ′(1 +2∥∇ϕ∥2) +� +⟨R∇(ϕ), ϕ⟩ + ∥∇ϕ∥2� +dv +− +� +M +F ′(1 +2∥ϕ∥2)⟨ϕ ◦ (Ric ∧ I + 2R), ϕ⟩dv . +Here, we have used the second equality in (3.5). From the above arguments we have +the assertion. +□ +We are ready to prove Theorem 4.3. + +A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS +17 +Proof of Theorem 4.3. Let ϕ be an F-harmonic 2-form. By using Lemmas 4.4, 4.5, (2) +and (4.14), the summation (4.3) is rewritten as follows: +� +A +Iϕ(ιVAϕ) = +� +M +F ′′(1 +2∥ϕ∥2)⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩dv ++ +� +M +F ′′(1 +2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2dv + +� +M +F ′(1 +2∥ϕ∥2) +� +⟨R∇(ϕ), ϕ⟩ + ∥∇ϕ∥2� +dv ++ +� +M +F ′(1 +2∥ϕ∥2) +� +H(ϕ, ϕ) − Ric(ϕ, ϕ) + 1 +2R(ϕ, ϕ) +� +dv . +(4.18) +By rewriting the right hand side of (4.16) by means of Lemma 2.4, we obtain +� +M +F ′′(1 +2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2dv + +� +M +F ′(1 +2∥ϕ∥2) +� +⟨R∇(ϕ), ϕ⟩ + ∥∇ϕ∥2� +dv += +� +M +F ′(1 +2∥ϕ∥2) +� +−Ric(ϕ, ϕ) + 1 +2R(ϕ, ϕ) +� +dv . +Substituting this into (4.18), we can derive (4.4). Thus, we have complete the proof. +□ +4.2. Analysis of the indices for F-harmonic forms (2). We will perform further +calculations of the summation � +A Iϕ(ιVAϕ) in terms of Theorem 4.3. In our calcula- +tions, the key is to evaluate the relation between F ′(∥ϕ∥2/2) and F ′′(∥ϕ∥2/2) in (4.4). +So, we define the degree of the differential F ′ as follows. +Definition 4.8. Let F be a strictly increasing C2-function defined on [0, c), 0 < c ≤ ∞. +The degree of F ′ is defined by +dF ′ = sup +00 +tF ′′ +p (t) +F ′p(t) = sup +t>0 +(p − 2)t(2t) +p−4 +2 +(2t) +p−2 +2 += p − 2 +2 +. +(2) For F = Fǫ (ǫ = ±1), which is defined on [0, ∞) if ǫ = 1; on [0, 1/2) if ǫ = −1, +from +tF ′′ +ǫ (t) +F ′ǫ(t) = −1 +2 + +1 +2(1 + 2ǫt) , +we get dF ′ +ǫ=1 = 0 and dF ′ +ǫ=−1 = ∞. +(3) For F = Fe, we have F ′ +e(t) = F ′′ +e (t) = et. Hence we have dF ′e = ∞. + +18 +KURANDO BABA AND KAZUTO SHINTANI +In what follows, we assume that the degree dF ′ is finite. Let ϕ ∈ Ω2(gP) be a non- +zero, F-harmonic form. The norm ∥ϕ∥ of ϕ gives a smooth function on M. We define +a closed subset M0 in M as follows: +M0 = {x ∈ M | ∥ϕ∥(x) = 0} . +Then it is verified that M0 has measure zero in M by means of the connectedness of M +and ϕ ̸≡ 0. Since dF ′ is finite, we have +F ′′(1 +2∥ϕ∥2) ≤ +2 +∥ϕ∥2F ′(1 +2∥ϕ∥2) · dF ′ +on M − M0 . +Hence (4.4) yields +� +A +Iϕ(ιVAϕ) = +� +M−M0 +F ′′(1 +2∥ϕ∥2)⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩dv ++ +� +M−M0 +F ′(1 +2∥ϕ∥2){H(ϕ, ϕ) − 2Ric(ϕ, ϕ) + R(ϕ, ϕ)}dv +≤ +� +M−M0 +2 +∥ϕ∥2F ′(1 +2∥ϕ∥2) · dF ′⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩dv ++ +� +M−M0 +F ′(1 +2∥ϕ∥2){H(ϕ, ϕ) − 2Ric(ϕ, ϕ) + R(ϕ, ϕ)}dv . (4.19) +So, if we put +B(ϕ, ϕ) = dF ′⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩ + ∥ϕ∥2 +2 +{H(ϕ, ϕ) − 2Ric(ϕ, ϕ) + R(ϕ, ϕ)} += dF ′⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩ + ∥ϕ∥2 +2 +{−H(ϕ, ϕ) + 2(h2(ϕ, ϕ) + h′ +2(ϕ, ϕ))} ,(4.20) +then, by (4.19), we obtain +� +A +I(ιVAϕ) ≤ +� +M−M0 +2 +∥ϕ∥2F ′(1 +2∥ϕ∥2)B(ϕ, ϕ)dv . +(4.21) +From the above argument, we conclude: +Theorem 4.10. Let M be a connected, closed Riemannian manifold isometrically +immersed in RN. Assume that the degree dF ′ is finite. Then, for any non-zero, F- +harmonic 2-form ϕ, we have the inequality (4.21). Furthermore, if B(ϕ, ϕ) < 0 holds, +then we have � +A Iϕ(ιVAϕ) < 0. +Here, we remark that B(ϕ, ϕ) is independent of the choice of orthonormal bases +(e1, . . . , en) of TxM and (en+1, . . . , eN) of T ⊥ +x M. In particular, the inequality B(ϕ, ϕ) < +0 is invariant under the orthonormal basis changes. + +A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS +19 +4.3. Instability of F-Yang-Mills connections over convex hypersurfaces in +Euclidean spaces. Let ϕ be an F-harmonic 2-form. Let M be a connected, compact, +convex hypersurface in an (n + 1)-Euclidean space Rn+1 and λ1, . . . , λn be its principal +curvatures. Without loss of generalities, we may assume that λi is positive for each i. +It follows from hn+1 +ij += λiδij that H(ϕ, ϕ) is expressed as follows: +H(ϕ, ϕ) = +� +i,j +�� +m +λm +� +λi∥ϕij∥2 . +Furthermore, we have ⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩ = � +i,j,i′,j′ λiλi′∥ϕij∥2∥ϕi′j′∥2, h2(ϕ, ϕ) = +� +i,j λiλj∥ϕij∥2 and h′ +2(ϕ, ϕ) = � +i,j λ2 +i ∥ϕij∥2. Substituting these into (4.20), we get: +B(ϕ, ϕ) = +� +i,j,i′,j′ +Biji′∥ϕij∥2∥ϕi′j′∥2 , +Biji′ = dF ′λiλi′ + 1 +4 +� +− +�� +m +λm +� +λi + 2λiλj + 2λ2 +i +� +. +If Bijk is negative for each i, j, k, then we obtain B(ϕ, ϕ) < 0. Then, Theorem 4.10 +yields +� +A +Iϕ(ιVAϕ) < 0 . +(4.22) +On the other hand, the inequality Bijk < 0 is rewritten as +λi +� +m̸=i,j +λm > λi (λi + λj + 4dF ′λk) , +that is, +� +m̸=i,j +λm > λi + λj + 4dF ′λk . +(4.23) +From this argument, (4.23) gives a sufficient condition that any non-flat, F-Yang-Mills +connection over M is instable. In order to prove this, we assume for contradiction that +there exists a non-flat, weakly stable F-Yang-Mills connection ∇ over M. Applying +ϕ = R∇ ∈ Ω2(gP) to (4.22), we have +� +A +IR∇(ιVAR∇) < 0 . +On the other hand, it follows from the weak instability of ∇ that IR∇(ιVAR∇) ≥ 0 holds +for each A. This obeys +� +A +IR∇(ιVAR∇) ≥ 0 , +which is a contradiction. Therefore we have derived the following theorem. + +20 +KURANDO BABA AND KAZUTO SHINTANI +Theorem 4.11. Let λ1, . . . , λn be the principal curvatures of a connected, compact, +convex hypersurface M in Rn+1. +Assume that the degree dF ′ is finite. +Then, any +non-flat, F-Yang-Mills connection over M is instable if the following condition holds: +� +m̸=i,j +λm > λi + λj + 4dF ′λk +(1 ≤ i, j, k ≤ n) . +Let us consider the case when M is the standard n-sphere Sn ⊂ Rn+1. If we denote +by r the radius of Sn, then the principal curvatures λi are equal to 1/r. Hence we have +the following result as a corollary of Theorem 4.11. +Corollary 4.12. If the inequality +n > 4dF ′ + 4 +(4.24) +holds, then any non-flat, F-Yang-Mills connection over Sn is instable. +We give an application of Corollary 4.12 for F-Yang-Mills connections as in Example +3.2. As shown in Example 4.9, for F = Fp (p ≥ 2), Fǫ=1, the degree dF ′ is finite. +(1) In the case of F = Fp, (4.24) reduces to n > 2p. Hence, if n > 2p, then any +non-flat, p-Yang-Mills connection over Sn is instable. This result coincides with the +results of Simons ([18]) for p = 2 and Chen-Zhou ([4, Corollary 4.2]) for p ≥ 2. +(2) In the case of F = Fǫ=1, we have obtained dF ′ +ǫ=1 = 0. Thus, if n > 4, then any +non-flat, critical connection of the generalized Yang-Mills-Born-Infeld energy functional +with positive sign is instable. +By means of Theorem 4.3, we give an observation for the instability of an F-Yang- +Mills connection in the case when F ′ has infinite degree. Here, we recall that Theorem +4.3 does not require no assumptions about the finiteness of dF ′. Now, let us consider the +instability of critical connections of YM ǫ=−1 and exponential Yang-Mills connections, +which are examples of F-Yang-Mills connections with dF ′ = ∞. We first consider the +case when F = Fǫ=−1. Based on the domain of definition for Fǫ=−1, we assume that the +Fǫ=−1-harmonic form ϕ = R∇ ∈ Ω2(gP) satisfies ∥ϕ∥ < 1. From hn+1 +ij += (1/r)δij we get +H(ϕ, ϕ) = 2n +r2 ∥ϕ∥2, +⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩ = 4 +r2∥ϕ∥4, +h2(ϕ, ϕ) = h′ +2(ϕ, ϕ) = 2 +r2∥ϕ∥2 . +By Theorem 4.3 and (4.15), we have +� +A +(IϕιVAϕ) += +� +Sn +4 +r2F ′′ +ǫ=−1(1 +2∥ϕ∥2)∥ϕ∥4dv + +� +Sn F ′ +ǫ=−1(1 +2∥ϕ∥2) +� +−2(n − 4) +r2 +� +∥ϕ∥2dv . +By using +F ′ +ǫ=−1(1 +2∥ϕ∥2) = +1 +� +1 − ∥ϕ∥2 , +F ′′ +ǫ=−1(1 +2∥ϕ∥2) = +1 +(1 − ∥ϕ∥2) +� +1 − ∥ϕ∥2 , + +A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS +21 +we obtain +� +A +Iϕ(ιVAϕ) = 2 +r2 +� +Sn +∥ϕ∥2 +� +1 − ∥ϕ∥2 +� +2 +1 − ∥ϕ∥2 − (n − 2) +� +dv . +(4.25) +From this argument, if the integrand of the right hand side of (4.25) is negative on Sn, +then � +A Iϕ(ιVAϕ) < 0 holds. Thus, we derive the following proposition. +Proposition 4.13. Let ∇ be a non-flat, critical connection over the standard n-sphere +Sn for the generalized Yang-Mills-Born-Infeld energy functional YM ǫ=−1 with negative +sign. If n > 4 and the curvature 2-form R∇ satisfies +∥R∇∥ < +� +n − 4 +n − 2 , +then ∇ is instable. +We give an analogous result for exponential Yang-Mills connections. Let ∇ be an +exponential Yang-Mills connection over Sn and ϕ be in Ω2(gP). A similar calculation +shows +� +A +(IϕιVAϕ) = 2 +r2 +� +Sn exp(1 +2∥ϕ∥2)∥ϕ∥2 � +2∥ϕ∥2 − (n − 4) +� +dv . +From this we conclude: +Proposition 4.14. Let ∇ be a non-flat, exponential Yang-Mills connection over the +standard n-sphere Sn. If n > 4 and the curvature 2-form R∇ satisfies +∥R∇∥ < +� +n − 4 +2 +, +then ∇ is instable. +There are strong similarities between the theory of Yang-Mills connections and that +of harmonic maps, which are critical points of a certain energy functional defied on the +space of smooth map between Riemannian manifolds. Finally, we discuss a counter part +of our results in the theory of harmonic maps as follows: Ara [1] introduced the notion +of F-harmonic maps as a generalization of harmonic maps, p-harmonic maps and so on. +He ([1, Theorem 7.1]) also derived the instability theorem of F-harmonic maps from +a closed Riemannian manifold into the n-dimensional standard sphere Sn, which is an +extension of the results by Leung [15] for harmonic maps and by Cheung-Leung [5] for +p-harmonic maps. By means of Ara’s result, the finiteness of the degree dF ′ in the sense +of Definition 4.8 yields the following statement as a counter part of Corollary 4.12: If +the inequality +n > 2dF ′ + 2 +holds, then any non-constant F-harmonic map from a connected, closed Riemannian +manifold into Sn is instable. +This inequality is a natural extension of Leung’s one + +22 +KURANDO BABA AND KAZUTO SHINTANI +[15, Corollary 1]. We can also find a counter part of Proposition 4.14 in the theory of +exponentially harmonic maps due to Koh [14, Theorem, p. 212]. +References +[1] M. Ara, “Geometry of F-Harmonic Maps”, Kodai. Math. J., 22 (1999), 243–263. +[2] M.F. Atiyah, +V.G. Drinfeld, +N.J. Hitchin, +Yu.I. Manin, +“Construction of instantons”, +Phys. Lett. A, 65, (1978), 185–187. +[3] J. P. Bourguignon, H. B. Lawson, Jr. “Stability and Isolation Phenomena for Yang-Mills Fields”, +Commun. Math. Phys., 79, (1981), 189–230. +[4] Q. Chen, Z.-R. Zhou, “On Gap Properties and Instabilities of p-Yang-Mills Fields”, Cand. J. +Math., 59, (2007), 1245–1259. +[5] L. F. Cheung, P. F. Leung, “Some results on stable p-harmonic maps”, Glasgow Math. J., 36 +(1994), 77–80. +[6] Y. Dong, S. W. Wei, “On Vanishing Theorems for Vector Bundle Valued p-Forms and their +Applications”, Commun. Math. Phys., 304, (2011), 329–368. +[7] C. Gherghe, “On a Yang-Mills Type Functional”, SIGMA, 15 (2019), 022, pp. 8. +[8] M. Hamilton, “Mathematical Gauge Theory”, Springer Cham, 2017. +[9] G.-Y. Jia, Z.-R. Zhou, “Gaps of F-Yang-Mills fields on submanifolds”, Tsukuba J. Math, 36, +(2012), 121–134. +[10] G.-Y. Jia, Z.-R. Zhou, “Stabilities of F-Yang-Mills Fields on Submanifolds”, Archivum Mathe- +maticum, 49, (2013), 125–139. +[11] S. Kobayashi, Y. Ohnita and M. Takeuchi, “On instability of Yang-Mills connections”, Math. Z., +193, (1986), 165–189. +[12] S. Kobayashi, K. Nomizu, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, +Inc. 1963. +[13] S. Kobayashi, K. Nomizu, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, +Inc. 1969. +[14] S. E. Koh, “A nonexistence theorem for stable exponentially harmonic maps”, Bull. Korean +Math. Soc., 32 (1995), 211–214. +[15] P. F. Leung, “On the stability of harmonic maps, Harmonic Maps”, Lecture Notes in Mathematics, +949, Springer Verlag, 1982, 122–129. +[16] F. Matsuura, H. Urakawa, “On exponential Yang-Mills connections”, Journal of Geometry and +Physics, 17 (1995), 73–89. +[17] L. Sibner, R. Sibner and Y. S. Yang, “Generalized Bernstein property and gravitational strings +in Born-Infeld theory”, Nonlinearity, 20, (2007), 1193–1213. +[18] J. Simons, “Gauge Fields”, Tokyo Symposium on Minimal Submanifolds and Geodesics, 1977. +[19] K. Uhlenbeck, “Connections with Lp bounds on curvature”, Comm. Math. Phys., 83, (1982), +31–42. +[20] S. W. Wei, “On exponential Yang-Mills fields and p-Yang-Mills fields”, arXiv:2205.03016v1 +[math.DG] 6 May 2022. + +A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS +23 +Department of Mathematics, Faculty of Science and Technology, Tokyo University +of Science, Noda, Chiba, 278-8510, Japan +Email address: kurando.baba@rs.tus.ac.jp +Department of Mathematics, Graduate School of Science and Technology, Tokyo +University of Science, Noda, Chiba, 278-8510, Japan +Email address: 6121505@ed.tus.ac.jp + diff --git a/TNE3T4oBgHgl3EQfEAkP/content/tmp_files/load_file.txt b/TNE3T4oBgHgl3EQfEAkP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d0bb1b098444d7f4868a1af7929ad9b5af60e363 --- /dev/null +++ b/TNE3T4oBgHgl3EQfEAkP/content/tmp_files/load_file.txt @@ -0,0 +1,827 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf,len=826 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='04291v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='DG] 11 Jan 2023 A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS KURANDO BABA AND KAZUTO SHINTANI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' F-Yang-Mills connections are critical points of F-Yang Mills functional on the space of connections of a principal fiber bundle, which is a generalization of Yang-Mills connections, p-Yang-Mills connections and exponential Yang-Mills connec- tions and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Here, F is a strictly increasing C2-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In this paper, we extend Simons theorem for an instability of Yang-Mills connections to F-Yang-Mills connec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We derive a sufficient condition that any non-flat, F-Yang-Mills connection over convex hypersurfaces in a Euclidean space is instable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In the sphere case, this condition is expressed by an inequality with respect to its dimension and a degree of the differential of the function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The proofs of the results are given by extending Kobayashi-Ohnita-Takeuchi’s calculation to F-Yang-Mills connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Preliminaries 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' F-Yang-Mills functionals and F-Yang-Mills connections 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Definition and the first variational formula 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Instability and the second variational formula 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A Simons type condition for instability of F-Yang-Mills connections 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Analysis of the indices for F-harmonic forms (1) 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Analysis of the indices for F-harmonic forms (2) 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Instability of F-Yang-Mills connections over convex hypersurfaces in Euclidean spaces 19 References 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Introduction A Yang-Mills connection is a critical point of the Yang-Mills functional defined on the space of connections of any principal fiber bundle over a connected, closed Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' There are many developments in the theory of Yang-Mills connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' On Date: January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Primary: 53C07, Secondary: 58E15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' F-Yang-Mills connection, instability, degree, F-harmonic form, index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' 1 2 KURANDO BABA AND KAZUTO SHINTANI the other hand, several Yang-Mills type functionals were introduced and critical points of such functionals have been studied, for example, p-Yang-Mills functional (Uhlenbeck [19], Chen-Zhou [4]), exponential Yang-Mills functional (Matsuura-Urakawa [16], Wei [20]) and the generalized Yang-Mills-Born-Infeld energy functional (Sibner-Sibner-Yang [17], Dong-Wei [6], Gherghe [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' An F-Yang-Mills functional provides a unified description of the above functionals (Jia-Zhou [10], Dong-Wei [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Here, F indicates a strictly increasing C2-function de- fined on [0, c), 0 < c ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A critical point of the F-Yang-Mill functional is called an F-Yang-Mills connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The purpose of this paper is to study the stability of F-Yang-Mills connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' More precisely, we would like to give a sufficient condition that any non-flat, F-Yang-Mills connection is instable, which is an extension of the following Simons theorem for the instability of Yang-Mills connections to F-Yang-Mills connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1 ([18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For n > 4, any non-flat, Yang-Mills connection over the standard sphere Sn is instable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' From this theorem the study of the weak stability for the usual Yang-Mills connec- tions over Sn makes sense only for n ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' It is known that weakly stable Yang-Mills connections over the 4-sphere are closely related to self-dual connections and anti-self- dual connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Indeed, Bourguignon-Lawson [3, Theorem B] proved that, in the case when the structure group is a specific unitary group, any weakly stable Yang-Mills con- nection over S4 is either self-dual or anti-self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' On the other hand, the construction of these connections were given by Atiyah-Drinfeld-Hitchin-Manin [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We expect that such studies can be explored for F-Yang-Mills connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' This paper contributes to find a suitable extension of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1 for F-Yang-Mills connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In fact, we derive a Simons type condition for the instability of F-Yang- Mills connections over convex hypersurfaces in a Euclidean space (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='11 for the detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The proof of this theorem is given by extending Kobayashi-Ohnita- Takeuchi’s calculation [11] of the second variation of the usual Yang-Mills functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='11, we have an extension of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2 (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let dF ′ denote the degree of the differential F ′ defined in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Assume that the degree dF ′ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then, for n > 4dF ′ + 4, any non-flat, F-Yang-Mills connection over Sn is instable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2 clarifies the importance of the finiteness of the degree dF ′ in order to derive the Simons type condition for the instability of F-Yang-Mills connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For the usual Yang-Mills connections, this result coincides with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1 because dF ′ = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Furthermore, it can be verified that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2 also coincides with the instability theorem for the p-Yang-Mills connections, which was given by Chen-Zhou [4, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We can find an alternative formula of the instability theorem for F-Yang-Mills connections by Jia-Zhou [10, Corollary 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS 3 On the other hand, in the case when F ′ has infinite degree, it is difficult to find a sufficient condition for the instability of F-Yang-Mills connections under a general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For example, the F-function corresponding to the exponential Yang-Mills func- tional YM e or the generalized Yang-Mills-Born-Infeld energy functional YM ǫ=−1 with minus sign has infinite degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The stability of critical points of these functional were studied by Matsuura-Urakawa for YM e and by Gherghe for YM ǫ=−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For further de- velopments we study the instability for the connections over Sn by means of the index formula stated in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In fact, we derive a certain sufficient condition for the instability of the connections by imposing the boundedness of its curvature (see, Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='13 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='14 for the detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The organization of this paper is as follows: In Section 2, we review the basic no- tions in the Yang-Mills theory, which are related to the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We note that Kobayashi-Ohnita-Takeuchi [11] studied the instability of Yang-Mills connections via analysis of the indices for harmonic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Here, harmonic forms are defined as elements in the zero eigenspace of Hodge-Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We recall Bochner-Weitzenb¨ock formula for the Hodge-Laplacian, which is needed for our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In Section 3, we review the basics for F-Yang-Mills connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1, we recall the notion of F- Yang-Mills connections and derive the F-Yang-Mills equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=', the Euler-Lagrange equation for the F-Yang-Mills functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Motivated by the F-Yang-Mills equation, we introduce the notion of F-harmonic forms (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2, we recall the definition of the instability of F-Yang-Mills connections and show the second vari- ational formula for the F-Yang-Mills functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Motivated by this formula, we define the index of F-harmonic forms (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In Section 4, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2 via analysis of the indices for F-harmonic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1, we extend the result [11, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='37) Proposition] for the index of harmonic forms to F-harmonic forms (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Following to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3, we need essentially to evaluate a quantity defined in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We also find that the key for proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2 is to evaluate the relation between F ′(∥ϕ∥2/2) and F ′′(∥ϕ∥2/2) for an F-harmonic 2-form ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Motivated by this consideration, in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2, we introduce the notion of the degree dF ′ of the differential F ′ (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In the case when dF ′ is finite, we derive an inequality for the index of F-harmonic forms based on Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3, we extend the result [11, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3) Theorem] for the instability of Yang-Mills connec- tions to F-Yang-Mills connections by means of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='10 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' As a corollary of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='11 we obtain Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2 (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We prove Proposi- tions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='13 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' It is known that there are strong similarities between the theory of Yang-Mills connections and that of harmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Finally, we discuss a counter part of our results in the theory of harmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' 4 KURANDO BABA AND KAZUTO SHINTANI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Preliminaries Let (M, g) be an n-dimensional, connected, closed Riemannian manifold and D de- note the Levi-Civita connection on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let G be a compact Lie group and g denote its Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We write the adjoint representation of G on g as Ad : G → GL(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let P be a principal fiber bundle over M with structure group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A g-valued 1-form A on P is called a connection if A is of type Ad and A( ˜X) = X holds for all X ∈ g, where ˜X de- notes the fundamental vector field on P associated with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We denote by Ωk Ad,hor(P, g) the vector space of horizontal k-forms of type Ad on P with values in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' It is verified that the curvature 2-form of a connection on P gives an element of Ω2 Ad,hor(P, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The kernel of a connection on P defines an Ehresmann connection, that is, a right-invariant, horizontal distribution on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then, it is known that this distribution is integrable if and only if the curvature 2-form of A vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A connection A is said to be flat, if its curvature 2-form vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For any two connections A, A′, the difference A − A′ is in Ω1 Ad,hor(P, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Conversely, A+α gives another connection on P for all α ∈ Ω1 Ad,hor(P, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Hence the set CP of connections on P becomes an affine space over the vector space Ω1 Ad,hor(P, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We make use of a different description of connections on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Denote by gP = P ×Ad g the adjoint bundle of P, that is, the associated vector bundle of P with the adjoint representation Ad of G on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' It follows from [8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4] that Ωk Ad,hor(P, g) is canonically isomorphic with the vector space of k-forms on M with values in gP, which we write Ωk(gP) = Γ(ΛkT ∗M ⊗ gP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Any connection on P corresponds to a connection on gP, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=', a covariant derivative ∇ : Γ(gP) → Ω1(gP) on the bundle gP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' It is shown that the curvature R∇ of ∇ on gP is in Ω2(gP) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' [8, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In what follows, we identify CP with the set of connections on gP, which is an affine space over the vector space Ω1(gP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We give a fiber metric on gP which is compatible with connections on gP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Such a fiber metric is induced from an Ad(G)-invariant inner product ⟨·, ·⟩ on g (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' [8, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In addition, ⟨·, ·⟩ also induces a pointwise inner product on the space Ωk(gP), which is denoted by the same symbol ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We set ∥ϕ∥2 = ⟨ϕ, ϕ⟩ for ϕ ∈ Ωk(gP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Here, we write ⟨ϕ, ψ⟩ (ϕ, ψ ∈ Ωk(gP)) by means of their components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We take an orthonormal basis (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , en) of the tangent space TxM at x ∈ M, and denote by (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , θn) its dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' If we write ϕ = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=',ik ϕei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=',eikθi1 ∧ · · · ∧ θik , ψ = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=',ik ψei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=',eikθi1 ∧ · · · ∧ θik , then we obtain ⟨ϕ, ψ⟩ = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=',ik ⟨ϕei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=',eik, ψei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=',eik⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS 5 By integrating this pointwise inner product over M, we obtain an inner product on Ωk(gP) as follows: (ϕ, ψ) = � M ⟨ϕ, ψ⟩dv , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1) where dv denotes the Riemannian volume form on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For any connection ∇, the covariant exterior derivative d∇ : Ωk(gP) → Ωk+1(gP) is given by (d∇ϕ)X0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=',Xk = k � i=0 (−1)i(∇Xiϕ)X0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=', ˆ Xi,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=',Xk , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2) for ϕ ∈ Ωk(gP), where X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , Xk are tangent vectors of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We denote by δ∇ the formal adjoint operator of d∇, that is, δ∇ : Ωk(gP) → Ωk−1(gP) is defined by (d∇ψ, ϕ) = (ψ, δ∇ϕ) for ϕ ∈ Ωk(gP) and ψ ∈ Ωk−1(gP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Following to [3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='13)], for any ϕ ∈ Ωk(gP), δ∇ has the following expression: (δ∇ϕ)X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=',Xk−1 = − n � j=1 (∇ejϕ)ej,X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=',Xk−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For any connection ∇, the curvature 2-form R∇ satisfies d∇R∇ = 0, which is called the Bianchi identity for ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In general, d∇ ◦ d∇ does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' It is known that, if ∇ is flat, then d∇ ◦ d∇ = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' This is an alternative interpretation of flat connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A Yang-Mills connection ∇ is defined as a critical point of the Yang-Mills functional CP → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' ∇ �→ 1 2 � M ∥R∇∥2dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' It is shown that the Euler-Lagrange equation for this functional is given by δ∇R∇ = 0, which is called the Yang-Mills equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Hodge-Laplacian is defined by ∆∇ = d∇δ∇ + δ∇d∇, which gives a differential operator on Ωk(gP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A gP-valued form ϕ is called a harmonic form if ϕ satisfies ∆∇ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then, it is verified that ϕ is harmonic if and only if it satisfies d∇ϕ = 0 and δ∇ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' This yields that the curvature form R∇ of a Yang-Mills connection ∇ is a harmonic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In Section 3, we will recall the notion of F- Yang-Mills connections, which is an extension of Yang-Mills connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Furthermore, we will introduce the notion of F-harmonic forms as an extension of harmonic forms (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We show Bochner-Weitzenb¨ock formula for gP-valued forms, which describes the relation between the Hodge-Laplacian and the rough Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' This formula plays a fundamental role in analysis of F-harmonic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In fact, we make use of this formula to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='7 in Section 4, which gives a method to calculate the differential of the curvature R∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We first recall the notion of the rough Laplacian, namely, it is defined by ∇∗∇ϕ = − n � j=1 ∇2 ej,ejϕ, ϕ ∈ Ωk(gP) , 6 KURANDO BABA AND KAZUTO SHINTANI where ∇2 X,Y = ∇X∇Y − ∇DXY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' It is verified that ∇∗∇ is symmetric and non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' This implies that a gP-valued form ϕ satisfies ∇∗∇ϕ = 0 if and only if ϕ is parallel (∇ϕ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We also recall Weitzenb¨ock curvature R∇ : Ωk(gP) → Ωk(gP) for k = 1, 2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In the case when k = 1, the operator R∇ : Ω1(gP) → Ω1(gP) is given by R∇(α) = � i,j [R∇ ji, αj]θi , for α = � i αiθi ∈ Ω1(gP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' If we set [· ∧ ·] : Ω1(gP) × Ω1(gP) → Ω2(gP);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' [α ∧ β]X,Y = [αX, βY ] − [αY , βX] then the following relation holds: ⟨R∇(α), α⟩ = ⟨[α ∧ α], R∇⟩ , α ∈ Ω1(gP) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3) We denote by R the Riemannian curvature on M, and by Ric : TxM → TxM (x ∈ M) the Ricci curvature operator, that is, Ric(X) = n � i=1 RX,eiei , X ∈ TxM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For α ∈ Ω1(gP), we define α ◦ Ric ∈ Ω1(gP) by (α ◦ Ric)X = αRic(X) for all X ∈ TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then, the following proposition shows the Bochner-Weitzenb¨ock formula for Ω1(gP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1 ([3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2) Theorem]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For α ∈ Ω1(gE), we have ∆∇α = ∇∗∇α + α ◦ Ric + R∇(α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Next, let us consider the case when k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We recall the Weitzenb¨ock curvature for Ω2(gP), that is, R∇ : Ω2(gP) → Ω2(gP) is given by R∇(ϕ)X,Y = n � j=1 � [R∇ ej,X, ϕej,Y ] − [R∇ ej,Y , ϕej,X] � , for ϕ ∈ Ω2(gP), where X, Y are tangent vector fields on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We denote by X(M) the space of tangent vector fields on M, and by Ω2(M) the space of 2-forms on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For ϕ ∈ Ω2(gP) and ω ∈ Ω2(M) ⊗ End(X(M)), we set (ϕ ◦ ω)X,Y = 1 2 n � j=1 ϕej,ωX,Y (ej) , X, Y ∈ X(M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Here, we give a concrete example of ω, which appears in the Bochner-Weitzenb¨ock formula for Ω2(gP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For any X, Y ∈ X(M), we set (X ∧ Y )(Z) = ⟨X, Z⟩Y − ⟨Y, Z⟩X , Z ∈ X(M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS 7 If I denotes the identity transformation on TxM, then Ric ∧ I ∈ Ω2(M) ⊗ End(X(M)) is defined by (Ric ∧ I)X,Y = Ric(X) ∧ Y + X ∧ Ric(Y ) , X, Y ∈ X(M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We are ready to show the Bochner-Weitzenb¨ock formula for Ω2(gP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3 ([3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='10) Theorem]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For ϕ ∈ Ω2(gP), we have ∆∇ϕ = ∇∗∇ϕ + ϕ ◦ (Ric ∧ I + 2R) + R∇(ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4) In order to evaluate the second term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4), Kobayashi-Ohnita-Takeuchi [11, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='36)] introduced R(ϕ, ϕ) and Ric(ϕ, ϕ) for ϕ = (1/2) � i,j ϕijθi ∧ θj ∈ Ω2(gP), which are defined as follows: R(ϕ, ϕ) = � i,j,k,l Rijkl⟨ϕij, ϕkl⟩ , Ric(ϕ, ϕ) = � i,j,k,l Rikδjl⟨ϕij, ϕkl⟩ , where Rijkl and Rik are the components of the Riemannian curvature R and the Ricci curvature Ric on M, respectively, that is, R(ek, el)ej = � i Ri jklei = � i Rijklei and Rik = � l Rlkli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' By the definition, R(ϕ, ϕ) and Ric(ϕ, ϕ) are independent of the choice of (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Here, we remark that, in the original definitions of R(ϕ, ϕ) and Ric(ϕ, ϕ), the inner product (·, ·) as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1) was used instead of ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For any ϕ ∈ Ω2(gP), we have ⟨ϕ ◦ (Ric ∧ I + 2R), ϕ⟩ = Ric(ϕ, ϕ) − 1 2R(ϕ, ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A direct calculation shows ⟨ϕ ◦ Ric ∧ I, ϕ⟩ = 1 2 � i,j ⟨(ϕ ◦ Ric ∧ I)ei,ej, ϕei,ej⟩ = 1 2 � i,j,k ⟨ϕek,(Ric∧I)ei,ej (ek), ϕei,ej⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then, by using ϕei,ej = −ϕej,ei, we get ⟨ϕ ◦ Ric ∧ I, ϕ⟩ = � i,j,k Rik⟨ϕek,ej, ϕei,ej⟩ = Ric(ϕ, ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In a similar manner, we can derive ⟨ϕ ◦ 2R, ϕ⟩ = −1 2 � i,j,k,l Rijkl⟨ϕij, ϕkl⟩ = −1 2R(ϕ, ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Thus, we have the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' □ 8 KURANDO BABA AND KAZUTO SHINTANI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' F-Yang-Mills functionals and F-Yang-Mills connections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Definition and the first variational formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let M be a connected, closed Riemannian manifold and G be a compact connected Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let P = P(M, G) be a principal fiber bundle over M with structure group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We denote by gP the adjoint bundle of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let 0 < c ≤ ∞ and F : [0, c) → R be a strictly increasing C2-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We set R≥0 = {a ∈ R | a ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The F-Yang-Mills functional YM F : CP → R≥0 is defined by YM F(∇) = � M F(1 2∥R∇∥2)dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A connection ∇ on gP is called a F-Yang-Mills connection if ∇ is a critical point of YM F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then, its curvature 2-form R∇ is also called the F-Yang-Mills field of ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For example, if we take F(t) = t, then the F-Yang-Mills functional coincides with the usual Yang-Mills functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Other examples are found in Uhlenbeck ([19]), Sibner- Sibner-Yang ([17]) and Matsuura-Urakawa ([16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (1) Let p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' If we put Fp(t) = (1/p)(2t)p/2, then the Fp-Yang-Mills functional coincides with the p-Yang-Mills functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A critical point of the p-Yang- Mills functional is called a p-Yang-Mills connection (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (2) Let ǫ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' If we put Fǫ(t) = ǫ√1 + 2ǫt − ǫ, then the Fǫ-Yang-Mills functional is called the generalized Yang-Mills-Born-Infeld energy functional with sign ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We call its critical point a critical connection of the functional (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (3) If we put Fe(t) = et, then the Fe-Yang-Mills functional coincides with the expo- nential Yang-Mills functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A critical point of the exponential Yang-Mills functional is called an exponential Yang-Mills connection (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' F-Yang-Mills connections are obtained by solving the Euler-Lagrange equation for YM F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Here, we recall the first variational formula for the functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3 ([6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1], [10, (11)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let ∇t (|t| < ε) be a C∞-curve in CP with ∇0 = ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' If we put α = d dt ���� t=0 ∇t ∈ Ω1(gP) , then we have d dt ���� t=0 YM F(∇t) = � M ⟨δ∇(F ′(1 2∥R∇∥2)R∇), α⟩dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let ∇ ∈ CP and ∇t = ∇+At be a C∞-curve in CP through ∇, where At ∈ Ω1(gP) with A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then the curvature of ∇t is given by R∇t = R∇ + d∇At + 1 2[At ∧ At] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS 9 By a straightforward calculation, we have d dtYM F(∇t) = � M d dtF(1 2∥R∇t∥2)dv = � M F ′(1 2∥R∇t∥2)⟨ d dtR∇t, R∇t⟩dv = � M F ′(1 2∥R∇t∥2)⟨d∇ d dtAt + [ d dtAt ∧ At], R∇t⟩dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let α = d dt ���� t=0 ∇t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The above equality becomes as follows d dt ���� t=0 YM F(∇t) = � M F ′(1 2∥R∇∥2)⟨R∇, d∇α⟩dv = � M ⟨δ∇(F ′(1 2∥R∇∥2)R∇), α⟩dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Thus, we have complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' □ From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3 we immediately get the Euler-Lagrange equation for YM F as follows: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' ∇ is an F-Yang-Mills connection if and only if ∇ satisfies δ∇(F ′(1 2∥R∇∥2)R∇) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1) We call (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1) the F-Yang-Mills equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Motivated by the F-Yang-Mills equation, we introduce the notion of F-harmonic forms as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A gP-valued form ϕ is said to be F-harmonic, if ϕ satisfies the following two equations: d∇ϕ = 0 , δ∇(F ′(1 2∥ϕ∥2)ϕ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2) We note that the curvature 2-form R∇ of an F-Yang-Mills connection ∇ is F- harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Indeed, R∇ satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2) because of the Bianchi identity and the F-Yang Mills equation for ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Instability and the second variational formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let us consider the instabil- ity for an F-Yang-Mills connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We recall here the definition of this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' An F-Yang-Mills connection ∇ is said to be weakly stable if the following inequality holds for any α ∈ Ω1(gP): d2 dt2 ���� t=0 YM F(∇t) ≥ 0 where α = d dt ���� t=0 ∇t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' An F-Yang-Mills connection ∇ is said to be instable if ∇ is not weakly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The following proposition gives the second variational formula for the F-Yang-Mills functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' 10 KURANDO BABA AND KAZUTO SHINTANI Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let ∇ be an F-Yang-Mills connection and ∇t (|t| < ε) be a C∞- curve in CP with ∇0 = ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then the second variation of the F-Yang-Mills functional is given by the following: d2 dt2 ���� t=0 YM F(∇t) = � M F ′′(1 2∥R∇∥2)⟨d∇α, R∇⟩2dv + � M F ′(1 2∥R∇∥2) � ⟨R∇(α), α⟩ + ∥d∇α∥2� dv , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3) where α = d dt ���� t=0 ∇t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A direct calculation yields d dtR∇t = d∇dAt dt + 1 2 d dt[At ∧ At] , and d2 dt2R∇t = d∇( d2 dt2At) + [ d2 dt2At ∧ At] + [dAt dt ∧ dAt dt ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Hence we have d dt ���� t=0 R∇t = d∇α, d2 dt2 ���� t=0 = d∇β + [α ∧ α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' where α = d dt ���� t=0 At and β = d2 dt2 ���� t=0 At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We have d2 dt2 ���� t=0 YM F(∇t) = � M F ′′(1 2∥R∇∥2)⟨d∇α, R∇⟩2dv + � M F ′(1 2∥R∇∥2) � ⟨[α ∧ α], R∇⟩ + ∥d∇α∥2� dv + � M F ′(1 2∥R∇∥2)⟨d∇β, R∇⟩dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4) Then it can be verified that the third term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Indeed, since ∇ is an F-Yang-Mills connection, we find � M F ′(1 2∥R∇∥2)⟨d∇β, R∇⟩dv = � M ⟨β, δ∇(F ′(1 2∥R∇∥2)R∇)⟩dv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Therefore, we obtain d2 dt2 ���� t=0 YM F(∇t) = � M F ′′(1 2∥R∇∥2)⟨d∇α, R∇⟩2dv + � M F ′(1 2∥R∇∥2) � ⟨[α ∧ α], R∇⟩ + ∥d∇α∥2� dv = � M F ′′(1 2∥R∇∥2)⟨d∇α, R∇⟩2dv + � M F ′(1 2∥R∇∥2) � ⟨R∇(α), α⟩ + ∥d∇α∥2� dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Here, in the last equality we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Thus, we have complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' □ A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS 11 An alternative expression of the second variational formula is found in [10, (20)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The difference between them is the integrand of the second term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In the case when YM F is the usual Yang-Mills functional (F(t) = t), F ′′(t) = 0 holds, so that the first term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Motivated by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='7, we define the index for any F-harmonic 2-form as follows: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The index of an F-harmonic form ϕ ∈ Ω2(gP) is defined by Iϕ(α) = � M F ′′(1 2∥ϕ∥2)⟨d∇α, ϕ⟩2dv + � M F ′(1 2∥ϕ∥2) � ⟨R∇(α), α⟩ + ∥d∇α∥2� dv , for any α ∈ Ω1(gP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' It follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='7 that, for any F-Yang-Mills connection ∇, if ∇ is weakly stable, then IR∇(α) ≥ 0 holds for all α ∈ Ω1(gP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In the next section, we will derive a sufficient condition for the instability of F-Yang-Mills connections via analysis of the indices for F-harmonic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A Simons type condition for instability of F-Yang-Mills connections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Analysis of the indices for F-harmonic forms (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let M be an n-dimensional, connected, closed Riemannian manifold and D denote the Levi-Civita connection on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let P = P(M, G) be a principal fiber bundle over M with structure group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Suppose that the base space M is isometrically immersed in an N-dimensional Euclidean space (RN, ⟨·, ·⟩) with n < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Denote by h its second fundamental form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We shall make use of the following convention on the ranges of indices: 1 ≤ A, B, C ≤ N, 1 ≤ i, j, k, l, m ≤ n, n + 1 ≤ µ ≤ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , en) be an orthonormal basis of TxM (x ∈ M) and (en+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , eN) be an orthonormal basis of the normal space T ⊥ x M of M in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , EN) be the canonical basis of RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We denote by VA the tangent component of EA with respect to the orthogonal decomposition RN = TxM ⊕ T ⊥ x M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' If we set vB A = ⟨EA, eB⟩, then the matrix (vB A)1≤A,B≤N becomes orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The tangent vector field VA is given by VA = � i vi Aei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let hµ ij denote the component of h(ei, ej) = � µ hµ ijeµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then we get the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' With the above settings, we obtain: DeiVA = � j � µ vµ Ahµ ijej .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We write DeiVA as DeiVA = � j⟨DeiVA, ej⟩ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In order to prove this lemma, it is sufficient to verify ⟨DeiVA, ej⟩ = ⟨EA, h(ei, ej)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Since M is isometrically immersed in (RN, ⟨·, ·⟩), the Levi-Civita connection D on M is compatible with ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Hence we have ei⟨VA, ej⟩ = ⟨DeiVA, ej⟩ + ⟨VA, Deiej⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1) 12 KURANDO BABA AND KAZUTO SHINTANI On the other hand, since EA is parallel with respect to the canonical connection D0 on RN, we have ei⟨VA, ej⟩ = ei⟨EA, ej⟩ = ⟨EA, D0 eiej⟩ = ⟨VA, Deiej⟩ + ⟨EA, h(ei, ej)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2) Here, in the last equality, we have used the Gauss formula for the submanifold M in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' By comparing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1) to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2), we get ⟨DeiVA, ej⟩ = ⟨EA, h(ei, ej)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Thus, we have complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' □ We evaluate the indices for F-harmonic 2-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' More precisely, we calculate the summation � A Iϕ(ιVAϕ) for an F-harmonic 2-form ϕ, where ι denotes the interior product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='8 we have � A Iϕ(ιVAϕ) = � M F ′′(1 2∥ϕ∥2)⟨ � A d∇(ιVAϕ), ϕ⟩2dv + � M F ′(1 2∥ϕ∥2) � ⟨ � A R∇(ιVAϕ), ιVAϕ⟩ + � A ∥d∇(ιVAϕ)∥2 � dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3) Following to [11, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='37)], we define H(ϕ, ϕ) for any ϕ ∈ Ω2(gP) as follows : H(ϕ, ϕ) = � i,j,k,l � µ Hµhµ ikδjl⟨ϕij, ϕkl⟩ , where Hµ = � m hµ mm denotes the mean curvature of M in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' On the other hand, in the present case we introduce the following quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We set h1(ϕ, ϕ) = � µ hµ 1(ϕ, ϕ)eµ , hµ 1(ϕ, ϕ) = � i,j,k,l hµ ikδjl⟨ϕij, ϕkl⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Here, we note that H(ϕ, ϕ) and h1(ϕ, ϕ) are independent of the choice of (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , en) and (en+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , eN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In addition, for each µ, the component hµ 1(ϕ, ϕ) of h1(ϕ, ϕ) is also independent of the choice of (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' As shown later in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3, h1(ϕ, ϕ) is needed to evaluate the first term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The purpose of this subsection is to prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' With the above settings, we obtain: � A Iϕ(ιVAϕ) = � M F ′′(1 2∥ϕ∥2)⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩dv + � M F ′(1 2∥ϕ∥2) {H(ϕ, ϕ) − 2Ric(ϕ, ϕ) + R(ϕ, ϕ)} dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4) This theorem is an extension of [11, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='37) Proposition] to F-harmonic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In order to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3, we first prepare some results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS 13 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For any ϕ ∈ Ω2(gP), we have � A ⟨R∇(ιVAϕ), ιVAϕ⟩ = ⟨R∇(ϕ), ϕ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We express ϕ and R∇ as follows: ϕ = 1 2 � i,j ϕei,ejθi ∧ θj, R∇ = 1 2 � i,j R∇ ei,ejθi ∧ θj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3), we have the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='5) = � A ⟨[ιVAϕ ∧ ιVAϕ], R∇⟩ = 1 2 � A � i,j 2⟨[(ιVAϕ)ei, (ιVAϕ)ej], R∇ ei,ej⟩ = � i,j,k ⟨[ϕek,ei, ϕek,ej], R∇ ei,ej⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='6) Here, in the last equality, we have used (ιVAϕ)ei = � k vk Aϕek,ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' On the other hand, by the Ad(G)-invariance of ⟨·, ·⟩, we get the following two relations: ⟨[R∇ ei,ek, ϕei,ej], ϕej,ek⟩ = −⟨ϕei,ej, [R∇ ei,ek, ϕej,ek]⟩ , ⟨[R∇ ei,ej, ϕei,ek], ϕej,ek⟩ = ⟨[ϕek,ei, ϕek,ej], R∇ ei,ej⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' By using these relations, we obtain the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='5) = 1 2 � i,j,k � ⟨[R∇ ei,ej, ϕei,ek], ϕej,ek⟩ − ⟨[R∇ ei,ek, ϕei,ej], ϕej,ek⟩ � = 1 2 � i,j,k � ⟨[R∇ ei,ej, ϕei,ek], ϕej,ek⟩ + ⟨ϕei,ej, [R∇ ei,ek, ϕej,ek]⟩ � = � i,j,k ⟨[R∇ ei,ej, ϕei,ek], ϕej,ek⟩ = � i,j,k ⟨[ϕek,ei, ϕek,ej], R∇ ei,ej⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='7) Comparing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='6) to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='7) we have the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' □ We define h2(ϕ, ϕ) and h′ 2(ϕ, ϕ) for ϕ ∈ Ω2(gE) as follows: h2(ϕ, ϕ) = � i,j,k,l � µ hµ ikhµ lj⟨ϕij, ϕkl⟩, h′ 2(ϕ, ϕ) = � i,j,k,l,m � µ hµ mkhµ miδjl⟨ϕij, ϕkl⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' By the definition, h(ϕ, ϕ) and h′ 2(ϕ, ϕ) are independent of the choice of (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , en) and (en+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , eN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let ϕ be in Ω2(gP) satisfying d∇ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then we have: 14 KURANDO BABA AND KAZUTO SHINTANI (1) � A ∥d∇(ιVAϕ)∥2 = ∥∇ϕ∥2 + h2(ϕ, ϕ) + h′ 2(ϕ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (2) � A ⟨d∇(ιVAϕ), ϕ⟩2 = ∥ϕ∥2∥∇∥ϕ∥∥2 + ⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let ϕ ∈ Ω2(gP) with d∇ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We express ϕ and ∇ϕ as follows: ϕ = 1 2 � i,j ϕijθi ∧ θj , ∇ϕ = 1 2 � i,j,k ∇kϕijθk ⊗ (θi ∧ θj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (1) If we write d∇(ιVAϕ) = 1 2 � i,j (d∇(ιVAϕ))ei,ejθi ∧ θj , then, by means of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2), the component (d∇(ιVAϕ))ei,ej has the following expression: (d∇(ιVAϕ))ei,ej = � k � µ vµ Ahµ ikϕkj + � k vk A(∇eiϕ)ek,ej − �� k � µ vµ Ahµ jkϕki + � k vk A(∇ejϕ)ek,ei � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='8) Then, we have � A ∥d∇(ιVAϕ)∥2 = 1 2 � A � i,j ⟨d∇(ιVAϕ)ei,ej, d∇(ιVAϕ)ei,ej⟩ = � i,j,k {⟨∇iϕkj, ∇iϕkj⟩ − ⟨∇iϕkj, ∇jϕki⟩} + � i,j,k,l � µ � hµ ikhµ il⟨ϕkj, ϕlj⟩ − hµ ikhµ jl⟨ϕkj, ϕli⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='9) It can be shown that the second term of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='9) coincides with h2(ϕ, ϕ) + h′ 2(ϕ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' On the other hand, we make use of the condition d∇ϕ = 0 in order to verify that the first term of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='9) is equal to ∥∇ϕ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Here, d∇ϕ = 0 yields ∇iϕkj + ∇jϕik + ∇kϕji = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='10) By using this we obtain � i,j,k ⟨∇iϕkj, ∇jϕki⟩ = � i,j,k ⟨∇iϕkj, ∇iϕkj⟩ + � i,j,k ⟨∇iϕkj, ∇kϕji⟩ = � i,j,k ⟨∇iϕkj, ∇iϕkj⟩ + � i,j,k ⟨∇jϕik, ∇iϕkj⟩ , that is, � i,j,k ⟨∇iϕkj, ∇jϕki⟩ = 1 2 � i,j,k ⟨∇iϕkj, ∇iϕkj⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' From the above arguments we obtain (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS 15 (2) By means of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='8), we have � A ⟨d∇(ιVAϕ), ϕ⟩ = � i,j,k i′,j′ ⟨ϕij, ∇iϕkj⟩⟨ϕi′j′, ∇i′ϕkj′⟩ + � i,j,k i′,j′,k′ � µ hµ ikhµ i′k′⟨ϕij, ϕkj⟩⟨ϕi′j′, ϕk′j′⟩ = � k �� i,j ⟨ϕij, ∇iϕkj⟩ �2 + � i,j,k,l i′,j′,k′,l′ � µ hµ ikhµ i′k′δjlδj′l′⟨ϕij, ϕkl⟩⟨ϕi′j′, ϕk′l′⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='11) Then, we can verify that the second term of the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='11) coincides with ⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' By using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='10), we get � i,j ⟨ϕij, ∇iϕkj⟩ = 1 2 � i,j ⟨ϕij, ∇kϕij⟩ = ⟨ϕ, ∇ekϕ⟩ = ∥ϕ∥(∇ek∥ϕ∥) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Substituting this into the first term of the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='11), we have � k �� i,j ⟨ϕij, ∇iϕkj⟩ �2 = ∥ϕ∥2 � k (∇ek∥ϕ∥)2 = ∥ϕ∥2∥∇∥ϕ∥∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' From the above arguments we have complete the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' □ Here, we rewrite h2(ϕ, ϕ) and h′ 2(ϕ, ϕ) in terms of Ric(ϕ, ϕ), R(ϕ, ϕ) and H(ϕ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For any ϕ ∈ Ω2(ϕ, ϕ), we get: h2(ϕ, ϕ) = 1 2R(ϕ, ϕ) , h′ 2(ϕ, ϕ) = H(ϕ, ϕ) − Ric(ϕ, ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' It follows from the Gauss equation for M in RN ([13, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='1, Chapter VII]) that the following relation holds: Rijkl = � µ (hµ ikhµ jl − hµ jkhµ il) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='13) Then we obtain R(ϕ, ϕ) = � i,j,k,l � µ(hµ ikhµ jl − hµ jkhµ il)⟨ϕij, ϕkl⟩ = 2h2(ϕ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' On the other hand, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='13) obeys Rik = � µ (Hµhik − � m hµ imhµ mk), from which we can derive Ric(ϕ, ϕ) = H(ϕ, ϕ) − h′ 2(ϕ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Thus, we have proved this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' □ Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='12) into Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='5, (1) we have � A ∥d∇(ιVAϕ)∥2 = ∥∇ϕ∥2 + 1 2R(ϕ, ϕ) + H(ϕ, ϕ) − Ric(ϕ, ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='14) We also get H(ϕ, ϕ) − 2Ric(ϕ, ϕ) + R(ϕ, ϕ) = −H(ϕ, ϕ) + 2(h2(ϕ, ϕ) + h′ 2(ϕ, ϕ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='15) 16 KURANDO BABA AND KAZUTO SHINTANI The Bochner-Weitzenb¨ock formula gives a way to calculate the differential of an F- harmonic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Indeed, we make use of this formula to prove the following theorem, which is a generalization of [9, Lemma 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' For any F-harmonic form ϕ ∈ Ω2(gP), we have: � M F ′′(1 2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2dv + � M F ′(1 2∥ϕ∥2) � ⟨R∇(ϕ), ϕ⟩ + ∥∇ϕ∥2� dv = − � M F ′(1 2∥ϕ∥2)⟨ϕ ◦ (Ric ∧ I + 2R), ϕ⟩dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let x ∈ M and (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , en) be an orthonormal basis of TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We extend (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , en) to a local orthonormal frame field so that (De1)(x) = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' , (Den)(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Then, at the point x, we have ∆F(1 2∥ϕ∥2) = − � ∇ei∇eiF(1 2∥ϕ∥2) = −F ′′(1 2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2 + F ′(1 2∥ϕ∥2) · 1 2∆∥ϕ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='17) From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3 and d∇ϕ = 0, we can derive 1 2∆∥ϕ∥2 = ⟨d∇δ∇ϕ, ϕ⟩ − ⟨ϕ ◦ (Ric ∧ I + 2R), ϕ⟩ − ⟨R∇(ϕ), ϕ⟩ − ∥∇ϕ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Substituting this into the right hand side of the second term of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='17), we obtain ∆F(1 2∥ϕ∥2) = −F ′′(1 2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2 − F ′(1 2∥∇ϕ∥2) � ⟨R∇(ϕ), ϕ⟩ + ∥∇ϕ∥2� − F ′(1 2∥ϕ∥2)⟨ϕ ◦ (Ric ∧ I + 2R), ϕ⟩ + F ′(1 2∥∇ϕ∥2)⟨d∇δ∇ϕ, ϕ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' By integrating both sides over M, the left hand side vanishes because of Green’s theorem ([12, Appendix 6]), and the right hand side is equal to − � M F ′′(1 2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2dv − � M F ′(1 2∥∇ϕ∥2) � ⟨R∇(ϕ), ϕ⟩ + ∥∇ϕ∥2� dv − � M F ′(1 2∥ϕ∥2)⟨ϕ ◦ (Ric ∧ I + 2R), ϕ⟩dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Here, we have used the second equality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' From the above arguments we have the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' □ We are ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' A SIMONS TYPE CONDITION FOR INSTABILITY OF F-YANG-MILLS CONNECTIONS 17 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let ϕ be an F-harmonic 2-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' By using Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='5, (2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='14), the summation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3) is rewritten as follows: � A Iϕ(ιVAϕ) = � M F ′′(1 2∥ϕ∥2)⟨h1(ϕ, ϕ), h1(ϕ, ϕ)⟩dv + � M F ′′(1 2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2dv + � M F ′(1 2∥ϕ∥2) � ⟨R∇(ϕ), ϕ⟩ + ∥∇ϕ∥2� dv + � M F ′(1 2∥ϕ∥2) � H(ϕ, ϕ) − Ric(ϕ, ϕ) + 1 2R(ϕ, ϕ) � dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='18) By rewriting the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='16) by means of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4, we obtain � M F ′′(1 2∥ϕ∥2)∥ϕ∥2∥∇∥ϕ∥∥2dv + � M F ′(1 2∥ϕ∥2) � ⟨R∇(ϕ), ϕ⟩ + ∥∇ϕ∥2� dv = � M F ′(1 2∥ϕ∥2) � −Ric(ϕ, ϕ) + 1 2R(ϕ, ϕ) � dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Substituting this into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='18), we can derive (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Thus, we have complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Analysis of the indices for F-harmonic forms (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' We will perform further calculations of the summation � A Iϕ(ιVAϕ) in terms of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' In our calcula- tions, the key is to evaluate the relation between F ′(∥ϕ∥2/2) and F ′′(∥ϕ∥2/2) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' So, we define the degree of the differential F ′ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' Let F be a strictly increasing C2-function defined on [0, c), 0 < c ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE3T4oBgHgl3EQfEAkP/content/2301.04291v1.pdf'} +page_content=' The degree of F ′ is defined by dF ′ = sup 0 0. For all c, c1, and c2, such that δ(c, c1) = c2, |c2| = |c| − 1. +Intrinsic functions are curried and produce intrinsic or intrinsic functions of one +arity less through δ. For example, for + ∈ C, we have δ(δ(+, 1), 2) = 3, |+| = 2, +|δ(+, 1)| = 1, and |δ(δ(+, 1), 2)| = 0. Next, randomness in our semantics is +deterministic via a trace of random draws in the style of Kozen [22]. +Definition 3 (Traces). The set S of traces is the set such that, for all s ∈ S, +s is a sequence of intrinsics from C with arity 0. +In the following, we use the notation [c1, c2, . . . , cn] for sequences and ∥ for +sequence concatenation. For example, [c1, c2] ∥ [c2, c4] = [c1, c2, c3, c4]. We also +use subscripts to select elements in a sequence, e.g., [c1, c2, c3, c4]2 = c2. In +practice, traces are often sequences of real numbers, e.g., [1.1, 3.2, 8.4] ∈ S. +Fig. 3 presents the semantics as a relation ρ ⊢ t s⇓w +l v over P × T × S × R × +L × V . L is the set of sequences over X, i.e., sequences of names. For example, +[x, y, z] ∈ L, where x, y, z ∈ X. We use l ∈ L to track the sequence of let- +bindings during evaluation. For example, evaluating let x = 1 in let y = 2 +in x + y results in l = [x, y]. In Section 4, we use the sequence of encountered +let-bindings to define alignment. For simplicity, from now on we assume that +bound variables are always unique (i.e., variable shadowing is impossible). + +10 +D. Lundén et al. +It is helpful to think of ρ, t, and s as the input to ⇓, and l, w and v as the out- +put. In the environment ρ, t, with trace s, evaluates to v, encounters the sequence +of let bindings l, and accumulates the weight w. The trace s is the sequence of +all random draws, and each random draw in (Assume) consumes precisely one +element of s. The rule (Let) tracks the sequence of bindings by adding x at the +correct position in l. The number w is the likelihood of the execution—the prob- +ability density of all draws in the program, registered at (Assume), combined +with direct likelihood modifications, registered at (Weight). The remaining as- +pects of the semantics are standard (see, e.g., Kahn [20]). To give an example of +the semantics, we have +∅ ⊢ tgeo +[true,true,true,false]⇓0.5·1.5·0.5·1.5·0.5·1.5·0.5 +[geometric,x,x,x,x] +4 +(2) +for the particular execution of tgeo making three recursive calls. Next, we for- +malize and apply the alignment analysis to (1). +4 +Alignment Analysis +This section presents the main contribution of this paper: automatic alignment +in PPLs. Section 4.1 introduces A-normal form and gives a precise definition of +alignment. Section 4.2 formalizes and proves the correctness of the alignment +analysis. Lastly, Section 4.3 discusses a dynamic version of alignment. +4.1 +A-Normal Form and Alignment +To reason about all subterms t′ of a program t and to enable the analysis in +Section 4.2, we need to uniquely label all subterms. A straightforward approach +is to use variable names within the program itself as labels (remember that +we assume bound variables are always unique). This leads us to the standard +A-normal form (ANF) representation of programs [11]. +Definition 4 (A-normal form). +tANF ::= x | let x = t′ +ANF in tANF +t′ +ANF ::= x | c | λx. tANF | x y +| if x then tANF else tANF | assume x | weight x +(3) +We use TANF to denote the set of all terms tANF. Unlike t ∈ T , tANF ∈ TANF +enforces that a variable bound by a let labels each subterm in the program. +Furthermore, we can automatically transform any program in T to a semantically +equivalent TANF program, and TANF ⊂ T . Therefore, we assume in the remainder +of the paper that all terms are in ANF. +Given the importance of alignment in universal PPLs, it is somewhat surpris- +ing that there are no previous attempts to give a formal definition of its meaning. +Here, we give a first such formal definition, but before defining alignment, we +require a way to restrict, or filter, sequences. + +Automatic Alignment in Higher-Order PPLs +11 +Definition 5 (Restriction of sequences). For all l ∈ L and Y ⊆ X, l|Y (the +restriction of l to Y ) is the subsequence of l with all elements not in Y removed. +For example, [x, y, z, y, x]|{x,z} = [x, z, x]. We now formally define alignment. +Definition 6 (Alignment). For t ∈ TANF, the set At ⊆ X is the largest set +such that, for arbitrary ∅ ⊢ t s1⇓w1 +l1 v1 and ∅ ⊢ t s2⇓w2 +l2 v2, l1|At = l2|At. +The aligned expressions—the expressions in a program bound by a let to a +variable name in At—are those that occur in the same order in every program +execution, regardless of random draws. Note that we seek the largest set, as +At = ∅ is always a trivial solution. Assume we have a program such that l = +[x, y, x, z, x] or l = [x, y, x, z, x, y] are the only possibilities. Then, At = {x, z}. +Next, assume that we transform the programs in Fig. 2a and Fig. 1a to ANF. The +expression labeled by x in Fig. 2a is then clearly not aligned, as random draws +determine how many times it executes (l could be, e.g., [x, x] or [x, x, x, x]). On +the other hand, the expression n (line 13) in Fig. 1a is aligned, as its number +and order of evaluations do not depend on any random draws. +Definition 6 is context insensitive: for each name x in the program, the ex- +pression bound by x is either aligned or unaligned. One could also consider a +context-sensitive definition of alignment in which x can be aligned in some con- +texts and unaligned in others. A context could, for example, be the sequence of +function applications (i.e., the call stack) leading up to an expression. Consider- +ing different contexts for x is complicated and difficult to take full advantage of. +We justify the choice of context-insensitive alignment with the real-world models +in Section 7, neither of which requires a context-sensitive alignment. +With alignment defined, we now move on to the static alignment analysis. +4.2 +Alignment Analysis +The basis for the alignment analysis is 0-CFA [33,41]—a static analysis frame- +work for higher-order functional programs. The prefix 0 indicates that 0-CFA is +context insensitive. There is also a set of analyses k-CFA [29] that adds increas- +ing amounts (with k ∈ N) of context sensitivity to 0-CFA. We could use such +analyses with a context-sensitive version of Definition 6. However, the potential +benefit of k-CFA is also offset by the worst-case exponential time complexity, +already at k = 1. In contrast, the time complexity of 0-CFA is polynomial (cu- +bic in the worst-case). The alignment analysis for the models in Section 7 runs +instantaneously, justifying that the time complexity is not a problem in practice. +The extensions to 0-CFA required to analyze alignment are non-trivial to +design, but the resulting formalization is surprisingly simple. The challenge is +instead to prove that the extensions correctly capture the alignment property +from Definition 6. We extend 0-CFA to analyze stochastic values and alignment +in programs t ∈ TANF. As with most static analyses, our analysis is sound +but conservative (i.e., sound but incomplete)—the analysis may mark aligned +expressions of programs as unaligned, but not vice versa. That the analysis is +conservative does not degrade the alignment analysis results for any model in + +12 +D. Lundén et al. +1 let n1 = ¬ in let n2 = ¬ in +2 let one = 1 in +3 let half = 0.5 in let c = true in +4 let f1 = λx1. let t1 = weight one in x1 in +5 let f2 = λx2. let t2 = weight one in t2 in +6 let f3 = λx3. let t3 = weight one in t3 in +7 let f4 = λx4. let t4 = weight one in t4 in +8 let bern = Bernoulli in +9 let d1 = bern half in +10 let a1 = assume d1 +11 let v1 = f1 one in +12 let v2 = n1 a1 in +13 let v3 = n2 c in +14 let f5 = +15 +if a1 then let t5 = f4 one in f2 +16 +else f3 +17 in +18 let v4 = f5 one in +19 let i1 = +20 +if c then let t6 = f1 one in t6 +21 +else one +22 in i1 +Fig. 4: A program texample ∈ TANF illustrating the analysis. +Section 7, which justifies the approach. We divide the formal analysis into two +algorithms. Algorithm 1 generates constraints for t that a valid analysis solution +must satisfy. This section describes Algorithm 1 and the generated constraints. +Appendix B.1 provides the second algorithm, Algorithm 4, that computes a +solution satisfying the generated constraints. We provide examples of applying +Algorithm 4 here, but defer the complete description to Appendix B.1. +For soundness of the analysis, we require ⟨λx. t, ρ⟩ ̸∈ C (recall that C is +the set of intrinsics). That is, closures are not in C. By Definition 3, this im- +plies that closures are not in the sample space of probability distributions in D +and that evaluating intrinsics never produces closures (this would unnecessarily +complicate the analysis without any benefit). +In addition to standard 0-CFA constraints, Algorithm 1 generates new con- +straints for stochastic values and unalignment. We use the contrived but illus- +trative program in Fig. 4 as an example. Note that, while omitted from Fig. 4 +for ease of presentation, the analysis also supports recursion introduced through +let rec. Stochastic values are values in the program affected by random vari- +ables. Stochastic values initially originate at assume and then propagate through +programs via function applications and if expressions. For example, a1 (line 10) +is stochastic because of assume. We subsequently use a1 to define v2 via n1 +(line 12), which is then also stochastic. Similarly, a1 is the condition for the if +resulting in f5 (line 14), and the function f5 is therefore also stochastic. When +we apply f5, it results in yet another stochastic value, v4 (line 18). In conclusion, +the stochastic values are a1, v2, f5, and v4. +Consider the flow of unalignment in Fig. 4. We mark expressions that may +execute due to stochastic branching as unaligned. From our analysis of stochastic +values, the program’s only stochastic if condition is at line 15, and we determine +that all expressions directly within the branches are unaligned. That is, the +expression labeled by t5 is unaligned. Furthermore, we apply the variable f4 +when defining t5. Thus, all expressions in bodies of lambdas that flow to f4 are +unaligned. Here, it implies that t4 is unaligned. Finally, we established that the +function f5 produced at line 15 is stochastic. Due to the application at line 18, all +names bound by lets in bodies of lambdas that flow to f5 are unaligned. Here, +it implies that t2 and t3 are unaligned. In conclusion, the unaligned expressions + +Automatic Alignment in Higher-Order PPLs +13 +Algorithm 1 Constraint generation function for t ∈ TANF. We denote the power +set of a set E with P(E). +function generateConstraints(t): TANF → P(R) = +1 match t with +2 | x → ∅ +3 | let x = t1 in t2 → +4 +generateConstraints(t2) ∪ +5 +match t1 with +6 +| y → {Sy ⊆ Sx} +7 +| c → if |c| > 0 then {const |c| ∈ Sx} +8 +else ∅ +9 +| λy. ty → generateConstraints(ty) +10 +∪ {λy. name(ty) ∈ Sx} +11 +∪ {unalignedy ⇒ unalignedn +12 +| n ∈ names(ty)} +13 +| lhs rhs → { +14 +∀z∀y λz.y ∈ Slhs +15 +⇒ (Srhs ⊆ Sz) ∧ (Sy ⊆ Sx), +16 +∀n (const n ∈ Slhs) ∧ (n > 1) +17 +⇒ const n − 1 ∈ Sx, +18 +stoch ∈ Slhs ⇒ stoch ∈ Sx, +19 +const _ ∈ Slhs +20 +⇒ (stoch ∈ Srhs ⇒ stoch ∈ Sx), +21 +unalignedx +22 +⇒ (∀y λy._ ∈ Slhs ⇒ unalignedy), +23 +stoch ∈ Slhs +24 +⇒ (∀y λy._ ∈ Slhs ⇒ unalignedy) +25 +} +26 +| if y then tt else te → +27 +generateConstraints(tt) +28 +∪ generateConstraints(te) +29 +∪ {Sname(tt) ⊆ Sx, Sname(te) ⊆ Sx, +30 +stoch ∈ Sy ⇒ stoch ∈ Sx} +31 +∪ {unalignedx ⇒ unalignedn +32 +| n ∈ names(tt) ∪ names(te)} +33 +∪ {stoch ∈ Sy ⇒ unalignedn +34 +| n ∈ names(tt) ∪ names(te)} +35 +| assume _ → {stoch ∈ Sx} +36 +| weight _ → ∅ +37 +38 function name(t): TANF → X = +39 +match t with +40 +| x → x +41 +| let x = t1 in t2 → name(t2) +42 +43 function names(t): TANF → P(X) = +44 +match t with +45 +| x → ∅ +46 +| let x = _ in t2 → {x} ∪ names(t2) +47 +48 +49 +50 +are named by t2, t3, t4, and t5. For example, aligned SMC therefore resamples +at the weight at t1, but not at the weights at t2, t3, and t4. +Consider the program in Fig. 1a again, and assume it is transformed to ANF. +The alignment analysis must mark all names bound within the stochastic if at +line 3 as unaligned because a stochastic value flows to its condition. In particular, +the weight expressions at lines 5 and 8 are unaligned (and the weight at line 12 +is aligned). Thus, aligned SMC resamples only at line 12. +To formalize the flow of stochastic values, we define abstract values a ∈ A, +that flow within the program, as follows. +Definition 7 (Abstract values). a ::= λx.y | stoch | const n where x, y ∈ +X and n ∈ N. +The stoch abstract value is new and represents stochastic values. The λx.y +and const n abstract values are standard and represent abstract closures and +intrinsics, respectively. For each variable name x in the program, we define a +set Sx containing abstract values that may occur at x. For example, in Fig. 4, +we have stoch ∈ Sa1, (λx2.t2) ∈ Sf2, and (const 1) ∈ Sn1. The abstract value +λx2.t2 represents all closures originating at λx2, and const 1 represents intrinsic +functions in C of arity 1 (in our example, ¬). The body of the abstract lambda is +the variable name labeling the body, not the body itself. For example, t2 labels +the body let t2 = one in t2 of λx2. Due to ANF, all terms have a label, which +the function name in Algorithm 1 formalizes. + +14 +D. Lundén et al. +We also define booleans unaligned x that state whether or not the expression +labeled by x is unaligned. For example, we previously reasoned that unalignedx = +true for x ∈ {t2, t3, t4, t5} in Fig. 4. The alignment analysis aims to deter- +mine minimal sets Sx and boolean assignments of unalignedx for every pro- +gram variable x ∈ X. A trivial solution is that all abstract values (there is a +finite number of them in the program) flow to each program variable and that +unalignedx = true for all x ∈ X. This solution is sound but useless. To compute +a more precise solution, we follow the rules given by constraints c ∈ R (see +Appendix B for a formal definition). +We present the constraints through the generateConstraints function in +Algorithm 1 and for the example in Fig. 4. There are no constraints for variables +that occur at the end of ANF let sequences (line 2 in Algorithm 1), and the +case for let expressions (lines 3–36) instead produces all constraints. The cases +for aliases (line 6), intrinsics (line 7), assume (line 35), and weight (line 36) are +the most simple. Aliases of the form let x = y in t2 establish Sy ⊆ Sx. That +is, all abstract values at y are also in x. Intrinsic operations results in a const +abstract value. For example, the definition of n1 at line 1 in Fig. 4 results in the +constraint const 1 ∈ Sn1. Applications of assume are the source of stochastic +values. For example, the definition of a1 at line 10 results in the constraint stoch +∈ Sa1. Note that assume cannot produce any other abstract values, as we only +allow distributions over intrinsics with arity 0 (see Definition 3). Finally, we use +weight only for its side effect (likelihood updating), and therefore weights do +not produce any abstract values and consequently no constraints. +The cases for abstractions (line 9), applications (line 13), and ifs (line 26) +are more complex. The abstraction at line 4 in Fig. 4 generates (omitting the +recursively generated constraints for the abstraction body ty) the constraints +{λx1.x1 ∈ Sf1} ∪ {unalignedx1 ⇒ unalignedt1}. The first constraint is standard: +the abstract lambda λx1.x1 flows to Sf1. The second constraint states that if the +abstraction is unaligned, all expressions in its body (here, only t1) are unaligned. +We define the sets of expressions within abstraction bodies and if branches +through the names function in Algorithm 1 (line 43). +The application f5 one at line 18 in Fig. 4 generates the constraints +{∀z∀y λz.y ∈ Sf5 ⇒ (Sone ⊆ Sz) ∧ (Sy ⊆ Sv4), +∀n (const n ∈ Sf5) ∧ (n > 1) ⇒ const n − 1 ∈ Sv4, +stoch ∈ Sf5 ⇒ stoch ∈ Sv4, +const _ ∈ Sf5 ⇒ (stoch ∈ Sone ⇒ stoch ∈ Sv4), +unalignedv4 ⇒ (∀y λy._ ∈ Sf5 ⇒ unalignedy), +stoch ∈ Sf5 ⇒ (∀y λy._ ∈ Slhs ⇒ unalignedy)} +(4) +The first constraint is standard: if an abstract value λz.y flows to f5, the abstract +values of one (the right-hand side) flow to z. Furthermore, the result of the appli- +cation, given by the body name y, must flow to the result v4 of the application. +The second constraint is also relatively standard: if an intrinsic function of arity +n is applied, it produces a const of arity n − 1. The other constraints are new + +Automatic Alignment in Higher-Order PPLs +15 +and specific for stochastic values and unalignment. The third constraint states +that if the function is stochastic, the result is stochastic. The fourth constraint +states that if we apply an intrinsic function to a stochastic argument, the result is +stochastic. We could also make the analysis of intrinsic applications less conser- +vative through intrinsic-specific constraints. The fifth and sixth constraints state +that if the expression (labeled by v4) is unaligned or the function is stochastic, +all abstract lambdas that flow to the function are unaligned. +The if resulting in f5 at line 14 in Fig. 4 generates (omitting the recursively +generated constraints for the branches tt and te) the constraints +{Sname(f2) ⊆ Sf5, Sname(f3) ⊆ Sf5, stoch ∈ Sa1 ⇒ stoch ∈ Sf5} +∪ {unalignedf5 ⇒ unalignedt5} ∪ {stoch ∈ Sa1 ⇒ unalignedt5} +(5) +The first two constraints are standard and state that the result of the branches +flows to the result of the if expression. The remaining constraints are new. The +third constraint states that if the condition is stochastic, the result is stochastic. +The last two constraints state that if the if is unaligned or if the condition is +stochastic, all names in the branches (here, only t5) are unaligned. +Given constraints for a program, we need to compute a solution satisfying all +constraints. We do this by repeatedly iterating through all the constraints and +propagating abstract values accordingly. We terminate when we reach a fixed +point, i.e., when no constraint results in an update of either Sx or unalignedx +for any x in the program. Algorithm 4 in Appendix B.1 formalizes our extension +of the 0-CFA constraint propagation algorithm that also handles the constraints +generated for tracking stochastic values and unalignment. The analysis function +analyzeAlign: TANF → ((X → P(A))×P(X)) returns a map associating each +variable to a set of abstract values and a set of unaligned variables. In other +words, analyzeAlign computes a solution to Sx and unalignedx for each x in +the analyzed program. For example, analyzeAlign(texample) results in +Sn1 = {const 1} Sn2 = {const 1} Sf1 = {λx1.x1} Sf2 = {λx2.t2} +Sf3 = {λx3.t3} Sf4 = {λx4.t4} Sa1 = {stoch} Sv2 = {stoch} +Sf5 = {λx2.t2, λx3.t3, stoch} Sv4 = {stoch} Sn = ∅ | other n ∈ X +unalignedn = true | n ∈ {t2, t3, t4, t5} unaligned n = false | other n ∈ X. +(6) +The example confirms our earlier intuition: an intrinsic (¬) flows to n1, stoch +flows to a1, f5 is stochastic and originates at either (λx2.t2) or (λx3.t3), and the +unaligned variables are t2, t3, t4, and t5. We now give soundness results. +Lemma 1 (0-CFA soundness). For every t ∈ TANF, the solution produced by +analyzeAlign(t) satisfies the constraints generateConstraints(t). +Proof. The well-known soundness of 0-CFA extends to the new alignment con- +straints. See, e.g., Nielson et al. [33, Chapter 3] and Shivers [41]. +⊓⊔ +Theorem 1 (Alignment analysis soundness). Assume t ∈ TANF, At from +Definition 6, and an assignment to Sx and unalignedx for x ∈ X according + +16 +D. Lundén et al. +to analyzeAlign(t). Let �At = {x | ¬unalignedx} and take arbitrary ∅ ⊢ +t s1⇓w1 +l1 v1 and ∅ ⊢ t s2⇓w2 +l2 v2. Then, l1| � +At = l2| � +At and consequently �At ⊆ At. +Proof. Follows by Lemma 3 in Appendix B.2 with t′ = t and ρ1 = ρ2 = ∅. The +proof uses simultaneous structural induction over the derivations ∅ ⊢ t s1⇓w1 +l1 v1 +and ∅ ⊢ t s2⇓w2 +l2 v2. At corresponding stochastic branches or stochastic function +applications in the two derivations, a separate structural induction argument +shows that, for the let-sequences l′ +1 and l′ +2 of the two stochastic subderivations, +l′ +1| � +At = l′ +2| � +At = []. Combined, the two arguments give the result. +⊓⊔ +The result �At ⊆ At (cf. Definition 6) shows that the analysis is conservative. +4.3 +Dynamic Alignment +An alternative to static alignment is dynamic alignment, which we explored +in early stages when developing the alignment analysis. Dynamic alignment is +fully context sensitive and amounts to introducing variables in programs that +track (at runtime) when evaluation enters stochastic branching. To identify these +stochastic branches, dynamic alignment also requires a runtime data structure +that keeps track of the stochastic values. Similarly to k-CFA, dynamic alignment +is potentially more precise than the 0-CFA approach. However, we discovered +that dynamic alignment introduces significant runtime overhead. Again, we note +that the models in Section 7 do not require a context-sensitive analysis, justifying +the choice of 0-CFA over dynamic alignment and k-CFA. +5 +Aligned SMC and MCMC +This section presents detailed algorithms for aligned SMC (Section 5.1) and +aligned lightweight MCMC (Section 5.2). For a more pedagogical introduction +to the algorithms, see Section 2. We assume a basic understanding of SMC and +Metropolis–Hastings MCMC algorithms (see, e.g., Bishop [4]). +5.1 +Aligned SMC +We saw in Section 2.1 that SMC operates by executing many instances of t +concurrently, and resampling them at calls to weight. Critically, resampling +requires that the inference algorithm can both suspend and resume executions. +Here, we assume that we can create execution instances e of the probabilistic +program t, and that we can arbitrarily suspend and resume the instances. The +technical details of suspension are beyond the scope of this paper. See Goodman +and Stuhlmüller [14], Wood et al. [47], and Lundén et al. [25] for further details. +Algorithm 2 presents all steps for the aligned SMC inference algorithm. Af- +ter running the alignment analysis and setting up the n execution instances, +the algorithm iteratively executes and resamples the instances. Note that the +algorithm resamples only at aligned weights (see Section 2.1). + +Automatic Alignment in Higher-Order PPLs +17 +Algorithm 2 Aligned SMC. The input is a program t ∈ TANF and the number +of execution instances n. +1. Run the alignment analysis on t, resulting in � +At (see Theorem 1). +2. Initiate n execution instances {ei | i ∈ N, 1 ≤ i ≤ n} of t. +3. Execute all ei and suspend execution upon reaching an aligned weight (i.e., let +x = weight w in t and x ∈ � +At) or when the execution terminates naturally. The +result is a new set of execution instances e′ +i with weights w′ +i accumulated from +unaligned weights and the single final aligned weight during execution. +4. If all e′ +i = v′ +i (i.e., all executions have terminated and returned a value), terminate +inference and return the set of weighted samples (v′ +i, w′ +i). The samples approximate +the posterior probability distribution encoded by t. +5. Resample the e′ +i according to their weights w′ +i. The result is a new set of unweighted +execution instances e′′ +i . Set ei ← e′′ +i . Go to 3. +1 if assume Bernoulli(0.5) +2 then weight 1; weight 100; true +3 else weight 100; weight 1; false +(a) Aligned better than unaligned. +1 if assume Bernoulli(0.5) +2 then weight 1; true +3 else weight 100; false +(b) Unaligned better than aligned. +Fig. 5: Programs illustrating properties of aligned and unaligned SMC. Fig. (a) +shows a program better suited for aligned SMC. Fig. (b) shows a program better +suited for aligned SMC. +Aligned SMC is preferable over unaligned SMC for all practically relevant +models, as the evaluation in Section 7 justifies. However, it is possible to con- +struct contrived programs in which unaligned SMC has the advantage. Consider +the programs in Fig. 5, both encoding simple Bernoulli distributions in a con- +trived way. Unaligned SMC resamples at the first weights in Fig. 5a, which +are not indicative of the final weights. The result is reduced inference accuracy. +Aligned SMC does not resample at all as the weights are within a stochastic +branch, and avoids the problem. Aligned SMC, however, does not resample at +all in Fig. 5b, where it is beneficial to do so. Consequently, the results are poorer +compared to unaligned SMC. We are not aware of any real model with the prop- +erty in Fig. 5b. In practice, it is always best to resample when using weight +to condition on observed data. Such conditioning is, in practice, always done +outside of stochastic branches, justifying the benefit of aligned SMC. +5.2 +Aligned Lightweight MCMC +Aligned lightweight MCMC is a version of lightweight MCMC [46], where the +alignment analysis provides information about how to reuse random draws be- +tween executions. Algorithm 3, a Metropolis–Hastings algorithm in the context +of PPLs, presents the details. Essentially, the algorithm executes the program +repeatedly using the Run function, and redraws one aligned random draw in +each step, while reusing all other aligned draws and as many unaligned draws as + +18 +D. Lundén et al. +Algorithm 3 Aligned lightweight MCMC. The input is a program t ∈ TANF, +the number of steps n, and the global step probability g > 0. +1. Run the alignment analysis on t, resulting in � +At (see Theorem 1). +2. Set i ← 0, k ← 1, and l ← 1. Call Run. +3. Set i ← i + 1. If i = n, terminate inference and return the samples {vj | j ∈ N, 0 ≤ +j < n}. They approximate the probability distribution encoded by t. +4. Uniformly draw an index 1 ≤ j ≤ |si−1| at random. Set global ← true with +probability g, and global ← false otherwise. Set w′ +−1 ← 1, w′ ← 1, k ← 1, l ← 1, +and reuse ← true. Call Run. +5. Compute the Metropolis–Hastings acceptance ratio A = min +� +1, +wi +wi−1 +w′ +w′ +−1 +� +. +6. With probability A, accept vi and go to 3. Otherwise, set vi ← vi−1, wi ← wi−1, +si ← si−1, pi ← pi−1, s′ +i ← s′ +i−1, p′ +i ← p′ +i−1, and n′ +i ← n′ +i−1. Go to 3. +function run() = Run t and do the following: +– Record the total weight wi accumulated from calls to weight. +– Record the final value vi. +– At unaligned terms let c = assume d in t (c ̸∈ � +At), do the following. +1. If reuse = false, global = true, n′ +i−1,k,l ̸= c, or if s′ +i−1,k,l does not exist, sample +a value x from d and set reuse ← false. Otherwise, reuse the sample x = s′ +i−1,k,l +and set w′ +−1 ← w′ +−1 · p′ +i−1,k,l and w′ ← w′ · fd(c). +2. Set s′ +i,k,l ← x, p′ +i,k,l ← fd(x), and n′ +i,k,l ← c. +3. Set l ← l + 1. In the program, bind c to the value x and resume execution. +– At aligned terms let c = assume d in t (c ∈ � +At), do the following. +1. If j = k, global = true, or if si−1,k does not exist, sample a value x from d +normally. Otherwise, reuse the sample x = si−1,k. Set w′ +−1 ← w′ +−1 · pi−1,k and +w′ ← w′ · fd(x). +2. Set si,k ← x and pi,k ← fd(x). +3. Set k ← k + 1, l ← 1, and reuse ← true. In the program, bind c to the value +x and resume execution. +possible (illustrated in Section 2.2). We provide a derivation of the Metropolis– +Hastings acceptance ratio in step 5 in Appendix E. A key property in Algorithm 3 +due to alignment (Definition 6) is that the length of si (and pi) is constant, as +executing t always results in the same number of aligned random draws. +In addition to redrawing only one aligned random draw, each step has a +probability g > 0 of being global—meaning that inference redraws every random +draw in the program. Occasional global steps fix problems related to slow mixing +and ergodicity of lightweight MCMC identified by Kiselyov [21]. In a global step, +the Metropolis–Hastings acceptance ratio reduces to A = min +� +1, +wi +wi−1 +� +. +6 +Implementation +We implement the alignment analysis (Section 4), aligned SMC (Section 5.1), +and aligned lightweight MCMC (Section 5.2) for the functional PPL Miking +CorePPL [25], implemented as part of the Miking framework [7]. We implement + +Automatic Alignment in Higher-Order PPLs +19 +the alignment analysis as a core component in the Miking CorePPL compiler, +and then use the analysis when compiling to two Miking CorePPL backends: +RootPPL and Miking Core. RootPPL is a low-level PPL with built-in highly +efficient SMC inference [25], and we extend the CorePPL to RootPPL compiler +introduced by Lundén et al. [25] to support aligned SMC inference. Furthermore, +we implement aligned lightweight MCMC inference standalone as a translation +from Miking CorePPL to Miking Core. Miking Core is the general-purpose pro- +gramming language of the Miking framework, currently compiling to OCaml. +The idealized calculus in (1) does not capture all features of Miking CorePPL. +In particular, the alignment analysis implementation must support records, vari- +ants, sequences, and pattern matching over these. Extending 0-CFA to such lan- +guage features is not new, but it does introduce a critical challenge for the align- +ment analysis: identifying all possible stochastic branches. Determining stochas- +tic ifs is straightforward, as we simply check if stoch flows to the condition. +However, complications arise when we add a match construct (and, in general, +any type of branching construct). Consider the extension +t ::= . . . | match t with p then t else t | {k1 = x1, . . ., kn = xn} +p ::= x | true | false | {k1 = p, . . ., kn = p} +x, x1, . . . , xn ∈ X +k1, . . . , kn ∈ K +n ∈ N +(7) +of (1), adding records and simple pattern matching. K is a set of record keys. As- +sume we also extend the abstract values as a ::= . . . | {k1 = X1, . . . , kn = Xn}, +where X1, . . . , Xn ⊆ X. That is, we add an abstract record tracking the names +in the program that flow to its entries. Consider the program match t1 with { +a = x1, b = false } then t2 else t3. This match is, similar to ifs, stochastic +if stoch ∈ St1. It is also, however, stochastic in other cases. Assume we have +two program variables, x and y, such that stoch ∈ Sx and stoch ̸∈ Sy. Now, +the match is stochastic if, e.g., {a = {y}, b = {x}} ∈ St1, because the random +value flowing from x to the pattern false may not match because of randomness. +However, it is not stochastic if, instead, St1 = {{a = {x}, b = {y}}}. The ran- +domness of x does not influence whether or not the branch is stochastic—the +variable pattern x1 for label a always matches. +Our alignment analysis implementation handles the intricacies of identify- +ing stochastic match cases for nested record, variant, and sequence patterns. In +total, the alignment analysis, aligned SMC, and aligned lightweight MCMC im- +plementations consist of approximately 1000 lines of code directly contributed +as part of this paper. The code is available on GitHub [2]. +7 +Evaluation +This section evaluates aligned SMC and aligned lightweight MCMC on a set +of models encoded in Miking CorePPL: CRBD [32,38] in Sections 7.1 and 7.5, +ClaDS [27,38] in Section 7.2, state-space aircraft localization in Section 7.3, +and latent Dirichlet allocation in Section 7.4. CRBD and ClaDS are non-trivial + +20 +D. Lundén et al. +models of considerable interest in evolutionary biology and phylogenetics [38]. +Similarly, LDA is a non-trivial topic model [5]. Running the alignment analysis +took approximately 5 ms–30 ms for all models considered in the experiment, +justifying that the time complexity is not a problem in practice. +We compare aligned SMC with standard unaligned SMC [14], which is iden- +tical to Algorithm 2, except that it resamples at every call to weight (see Ap- +pendix C). We carefully checked that automatic alignment corresponds to previ- +ous manual alignments of each model. For all SMC experiments, we estimate the +normalizing constant produced as a by-product of SMC inference rather than +the complete posterior distributions. The normalizing constant, also known as +marginal likelihood or model evidence, frequently appears in Bayesian inference +and gives the probability of the observed data averaged over the prior. The +normalizing constant is useful for model comparison as it measures how well dif- +ferent probabilistic models fit the data (a larger normalizing constant indicates +a better fit). +We ran aligned and unaligned SMC with Miking CorePPL and the RootPPL +backend configured for a single-core (compiled with GCC 7.5.0). Lundén et +al. [25] shows that the RootPPL backend is significantly more efficient than other +state-of-the-art PPL SMC implementations. We ran aligned and unaligned SMC +inference 300 times (and with 3 warmup runs) for each experiment for 104, 105, +and 106 executions (also known as particles in SMC literature). +We compare aligned lightweight MCMC to lightweight MCMC (see Ap- +pendix D). We implement both versions as compilers from Miking CorePPL to +Miking Core in the Miking framework, which in turn compiles to OCaml (ver- +sion 4.12). The lightweight MCMC databases are functional-style maps from +the OCaml standard Map library. We set g (the global step probability in Al- +gorithm 3) to 0.1 for both aligned lightweight MCMC and lightweight MCMC. +We ran aligned lightweight and lightweight MCMC inference 300 times for each +experiment. We burned 10% of samples in all MCMC runs. +For all experiments, we used an Intel Xeon 656 Gold 6136 CPU (12 cores) +and 64 GB of memory running Ubuntu 18.04.5. +7.1 +SMC: Constant Rate Birth-Death (CRBD) +This experiment considers the CRBD diversification model from [38] applied to +the Alcedinidae phylogeny (Kingfisher birds, 54 extant species) [19]. We use fixed +diversification rates to simplify the model, as unaligned SMC inference accuracy +is too poor for the full model with priors over diversification rates. Aligned SMC +is accurate for both the full and simplified models. We provide the source code +for the complete model in Listing 1 of Appendix A.1 (130 lines of code). The +total experiment execution time was 16 hours. +Fig. 6 presents the experiment results. Aligned SMC is roughly twice as fast +and produces superior estimates of the normalizing constant. Unaligned SMC +has not yet converged to the correct value −304.75 (available for this particular +model due to the fixing the diversification rates) for 106 particles, while aligned +SMC produces precise estimates already at 104 particles. Excess resampling is a + +Automatic Alignment in Higher-Order PPLs +21 +106 +105 +104 +57.49 +5.41 +0.4 +122.53 +11.91 +0.82 +(a) Execution times. +104 +105 +106 +−315 +−330 +−304.75 +(b) Log normalizing constant estimates. +Fig. 6: SMC experiment results for CRBD. The x-axes give the number of parti- +cles. Fig. (a) shows execution times (in seconds) for aligned (gray) and unaligned +(white) SMC. Error bars show one standard deviation. Fig. (b) shows box plot log +normalizing constant estimates for aligned (gray) and unaligned (white) SMC. +The analytically computed log normalizing constant is −304.75. +106 +105 +104 +92.41 +8.88 +0.6 +634.07 +59.3 +3.56 +(a) Execution times. +104 +105 +106 +−400 +−500 +−314.35 +(b) Log normalizing constant estimates. +Fig. 7: SMC experiment results for ClaDS. The x-axes give the number of parti- +cles. Fig. (a) shows execution times (in seconds) for aligned (gray) and unaligned +(white) SMC. Error bars show one standard deviation. Fig. (b) shows box plot log +normalizing constant estimates for aligned (gray) and unaligned (white) SMC. +The average estimate for aligned SMC with 106 particles is −314.35. +significant factor in the increase in execution time for unaligned SMC, as each +execution encounters far more resampling checkpoints than in aligned SMC. +7.2 +SMC: Cladogenetic Diversification Rate Shift (ClaDS) +A limitation of CRBD is that the diversification rates are constant. ClaDS [27,38] +is a set of diversification models that allow shifting rates over phylogenies. We +evaluate the ClaDS2 model for the Alcedinidae phylogeny. As in CRBD, we +use fixed (initial) diversification rates to simplify the model on account of un- +aligned SMC. The source code for the complete model is available in Listing 2 +of Appendix A.2 (147 lines of code). Automatic alignment simplifies the ClaDS2 +model significantly, as manual alignment requires collecting and passing weights +around in unaligned parts of the program, which are later consumed by aligned +weights. The total experiment execution time was 67 hours. +Fig. 7 presents the experiment results. 12 unaligned runs for 106 particles +and nine runs for 105 particles ran out of the preallocated stack memory for +each particle (10 kB). We omit these runs from Fig. 7. The consequence of not +aligning SMC is more severe than for CRBD. Aligned SMC is now almost seven +times faster than unaligned SMC and the unaligned SMC normalizing constant + +22 +D. Lundén et al. +106 +105 +104 +4.22 +0.42 +0.05 +6.07 +0.59 +0.06 +(a) Execution times. +104 +105 +106 +−55 +−65 +−61.26 +(b) Log normalizing constant estimates. +Fig. 8: SMC experiment results for the state-space aircraft localization model. +The x-axes give the number of particles. Fig. (a) shows execution times (in +seconds) for aligned (gray) and unaligned (white) SMC. Error bars show one +standard deviation. Fig. (b) shows box plot log normalizing constant estimates on +the y-axis for aligned (gray) and unaligned (white) SMC. The average estimate +for aligned SMC with 106 particles is −61.26. +estimates are significantly worse compared to the aligned SMC estimates. The +unaligned SMC estimates do not even improve when moving from 104 to 106 +particles (we need even more particles to see improvements). Again, aligned +SMC produces precise estimates already at 104 particles. +7.3 +SMC: State-Space Aircraft Localization +This experiment considers an artificial but non-trivial state-space model for air- +craft localization. Appendix A.3 presents the model as well as the source code +in Listing 3 (62 lines of code). The total experiment execution time was 1 hour. +Fig. 8 presents the experiment results. The execution time difference is not as +significant as for CRBD and ClaDS. However, the unaligned SMC normalizing +constant estimates are again much less precise. Aligned SMC is accurate (cen- +tered at approximately −61.26) already at 104 particles. The model’s straightfor- +ward control flow explains the less dramatic difference in execution time—there +are at most ten unaligned likelihood updates in the aircraft model, while the +number is, in theory, unbounded for CRBD and ClaDS. Therefore, the cost of +extra resampling compared to aligned SMC is not as significant. +7.4 +MCMC: Latent Dirichlet Allocation (LDA) +This experiment considers latent Dirichlet allocation (LDA), a topic model used +in the evaluations by Wingate et al. [46] and Ritchie et al. [37]. We use a synthetic +data set, comparable in size to the data set used by Ritchie et al. [37], with a +vocabulary of 100 words, 10 topics, and 25 documents each containing 30 words. +Note that we are not using methods based on collapsed Gibbs sampling [17], and +the inference task is therefore computationally challenging even with a rather +small number of words and documents. The source code for the complete model +is available in Listing 4 of Appendix A.4 (31 lines of code). The total experiment +execution time was 41 hours. + +Automatic Alignment in Higher-Order PPLs +23 +105 +104 +103 +125.24 +11.82 +1.17 +325.25 +32.47 +3.23 +Fig. 9: MCMC experiment results for LDA showing execution time (in seconds) +for aligned lightweight MCMC (gray) and lightweight MCMC (white). Error bars +show one standard deviation and the x-axis the number of MCMC iterations. +The LDA model consists of only aligned random draws. As a consequence, +aligned lightweight and lightweight MCMC reduces to the same inference algo- +rithm, and we can compare the algorithms by just considering the execution +times. We justify the correctness of both algorithms in Appendix A.4. +Fig. 9 presents the experiment results. Aligned lightweight MCMC is al- +most three times faster than lightweight MCMC. To justify the execution times +with our implementations, we also implemented and ran the experiment with +lightweight MCMC in WebPPL [14] for 105 iterations, repeated 50 times (and +with 3 warmup runs). The mean execution time was 383 s with standard devia- +tion 5 s. We used WebPPL version 0.9.15 and Node version 16.18.0. +7.5 +MCMC: Constant Rate Birth-Death (CRBD) +This experiment again considers CRBD. MCMC is not as suitable for CRBD as +SMC, and therefore we use a simple synthetic phylogeny with six leaves and an +age span of 5 age units (Alcedinidae used for the SMC experiment has 54 leaves +and an age span of 35 age units). The source code for the complete model is the +same as in Section 7.1, but we now allow the use of proper prior distributions +for the diversification rates. The total experiment execution time was 7 hours. +Unlike LDA, the CRBD model contains both unaligned and aligned random +draws. Because of this, aligned lightweight MCMC and standard lightweight +MCMC do not reduce to the same algorithm. To judge the difference in infer- +ence accuracy, we consider the mean estimates of the birth diversification rate +produced by the two algorithms, in addition to execution times. The experiment +results shows that the posterior distribution over the birth rate is unimodal +(see Appendix A.5), which motivates using the posterior mean as a measure of +accuracy. +Fig. 10 presents the experiment results. Aligned lightweight MCMC is ap- +proximately 3.5 times faster than lightweight MCMC. There is no obvious dif- +ference in accuracy. To justify the execution times and correctness of our im- +plementations, we also implemented and ran the experiment with lightweight +MCMC in WebPPL [14] for 3 · 106 iterations, repeated 50 times (and with 3 +warmup runs). The mean estimates agreed with Fig. 10. The mean execution +time was 37.1 s with standard deviation 0.8 s. The speedup compared to stan- +dard lightweight MCMC in Miking CorePPL is likely explained by the use of + +24 +D. Lundén et al. +3 · 106 +3 · 105 +3 · 104 +18.54 +1.82 +0.2 +63.95 +6.21 +0.63 +(a) Execution times. +3 · 104 +3 · 105 +3 · 106 +0.4 +0.45 +0.33 +(b) Birth rate mean estimates. +Fig. 10: MCMC experiment results for CRBD. The x-axes give the number of +iterations. Fig. (a) shows execution times (in seconds) for aligned lightweight +MCMC (gray) and lightweight MCMC (white). Error bars show one standard +deviation. Fig. (b) shows box plot posterior mean estimates of the birth rate for +aligned lightweight MCMC (gray) and lightweight MCMC (white). The average +estimate for aligned lightweight MCMC with 3 · 106 iterations is 0.33. +early termination in WebPPL, which benefits CRBD. Early termination easily +combines with alignment but relies on execution suspension, which we do not +currently use in our implementations. Note that aligned lightweight MCMC is +faster than WebPPL even without early termination. +In conclusion, the experiments clearly demonstrate the need for alignment. +8 +Related Work +The approach by Wingate et al. [46] is closely related to ours. A key similarity +with alignment is that executions reaching the same aligned checkpoint also +have matching stack traces according to Wingate et al.’s addressing transform. +However, Wingate et al. do not consider the separation between unaligned and +aligned parts of the program, their approach is not static, and they do not +generalize to other inference algorithms such as SMC. +Ronquist et al. [38], Turing [12], Anglican [47], Paige and Wood [35], and van +de Meent et al. [45] consider the alignment problem. Manual alignment is critical +for the models in Ronquist et al. [38] to make SMC inference tractable, which +strongly motivates the automatic alignment approach. The documentation of +Turing states that: “The observe statements [i.e., likelihood updates] should be +arranged so that every possible run traverses all of them in exactly the same +order. This is equivalent to demanding that they are not placed inside stochastic +control flow” [1]. Turing does not include any automatic checks for this property. +Anglican [47] checks, at runtime (resulting in overhead), that all SMC executions +encounter the same number of likelihood updates, and thus resamples the same +number of times. If not, Anglican reports an error: “some observe directives [i.e., +likelihood updates] are not global”. This error refers to the alignment problem, +but the documentation does not explain it further. Probabilistic C, introduced by +Paige and Wood [35], similarly assumes that the number of likelihood updates +is the same in all executions. Van de Meent et al. [45] state, in reference to +SMC: “Each breakpoint [i.e., checkpoint] needs to occur at an expression that + +Automatic Alignment in Higher-Order PPLs +25 +is evaluated in every execution of a program”. Again, they do not provide any +formal definition of alignment nor an automatic solution to enforce it. +Lundén et al. [24] briefly mention the general problem of selecting optimal +resampling locations in PPLs for SMC but do not consider the alignment problem +in particular. They also acknowledge the overhead resulting from not all SMC +executions resampling the same number of times, which alignment avoids. +The PPLs Birch [30], Pyro [3], and WebPPL [14] support SMC inference. +Birch and Pyro enforce alignment for SMC as part of model construction. Note +that this is only true for SMC in Pyro—other Pyro inference algorithms use +other modeling approaches. The approaches in Birch and Pyro are sound but +demand more of their users compared to the alignment approach. WebPPL does +not consider alignment and resamples at all likelihood updates for SMC. +Ritchie et al. [37] and Nori et al. [34] present MCMC algorithms for proba- +bilistic programs. Ritchie et al. [37] optimize lightweight MCMC by Wingate et +al. [46] through execution suspensions and callsite caching. The optimizations are +independent of and potentially combines well with aligned lightweight MCMC. +Another MCMC optimization which potentially combines well with alignment +is due to Nori et al. [34]. They use static analysis to propagate observations +backwards in programs to improve inference. +Information flow analyses [39] may determine if particular parts of a program +execute as a result of different program inputs. Specifically, if program input is +random, such approaches have clear similarities to the alignment analysis. +Many other PPLs exist, such as Gen [10], Venture [28], Edward [43], Stan [8], +and AugurV2 [18]. Gen, Venture, and Edward focus on simplifying the joint +specification of a model and its inference to give users more low-level control, +and do not consider automatic alignment specifically. However, the incremen- +tal inference approach [9] in Gen does use the addressing approach by Wingate +et al. [46]. Stan and AugurV2 have less expressive modeling languages to al- +low more powerful inference. Alignment is by construction due to the reduced +expressiveness. +Borgström et al. [6], Staton et al. [42], Ścibior et al. [40], and Vákár et al. [44] +treat PPL topics related to semantics and correctness, but do not specifically +consider alignment. +9 +Conclusion +This paper gives, for the first time, a formal definition of alignment in PPLs. +Furthermore, we introduce a static analysis technique and use it to align check- +points in PPLs and apply it to SMC and MCMC inference. We formalize the +alignment analysis, prove its correctness, and implement it in Miking CorePPL. +We also implement aligned SMC and aligned lightweight MCMC, and evaluate +the implementations on non-trivial CRBD and ClaDS models from phylogenet- +ics, the LDA topic model, and a state-space model, demonstrating significant +improvements compared to standard SMC and lightweight MCMC. + +26 +D. Lundén et al. +Acknowledgments We thank Lawrence Murray, Johannes Borgström, and Jan +Kudlicka for early discussions on the alignment idea, and Viktor Senderov for im- +plementing ClaDS in Miking CorePPL. We also thank the anonymous reviewers +at ESOP for their valuable comments. + +Automatic Alignment in Higher-Order PPLs +27 +References +1. Turing.jl. https://turing.ml/dev/ (2022), accessed: 2022-02-24 +2. Miking DPPL. https://github.com/miking-lang/miking-dppl (2023), accessed: +2023-01-02 +3. 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The code for the analysis itself and all inference +algorithms are available on GitHub [2]. +A.1 +SMC: Constant Rate Birth-Death (CRBD) +Listing 1 gives the Miking CorePPL source code used for the case study model +in Section 7.1. +Listing 1: The source code for the experiment in Sections 7.1 and 7.5 +1 ------------------------------------------------ +2 -- The Constant-Rate Birth-Death (CRBD) model -- +3 ------------------------------------------------ +4 +5 -- The prelude includes a few PPL helper functions +6 include "pplprelude.mc" +7 +8 -- The tree.mc file defines the general tree structure +9 include "tree.mc" +10 +11 -- The tree-instance.mc file includes the actual tree and the rho constant +12 include "tree-instance.mc" +13 +14 mexpr +15 +16 -- CRBD goes undetected, including iterations. Mutually recursive functions. +17 recursive +18 +let iter: Int -> Float -> Float -> Float -> Float -> Float -> Bool = +19 +lam n: Int. +20 +lam startTime: Float. +21 +lam branchLength: Float. +22 +lam lambda: Float. +23 +lam mu: Float. +24 +lam rho: Float. +25 +if eqi n 0 then +26 +true +27 +else +28 +let eventTime = assume (Uniform (subf startTime branchLength) startTime) in +29 +if crbdGoesUndetected eventTime lambda mu rho then +30 +iter (subi n 1) startTime branchLength lambda mu rho +31 +else +32 +false +33 +34 +let crbdGoesUndetected: Float -> Float -> Float -> Float -> Bool = +35 +lam startTime: Float. +36 +lam lambda: Float. +37 +lam mu: Float. +38 +lam rho: Float. +39 +let duration = assume (Exponential mu) in +40 +let cond = +41 +-- ‘and‘ does not use short-circuiting: using ‘if‘ as below is more +42 +-- efficient +43 +if (gtf duration startTime) then +44 +(eqBool (assume (Bernoulli rho)) true) +45 +else false +46 +in +47 +if cond then +48 +false +49 +else +50 +let branchLength = if ltf duration startTime then duration else startTime in +51 +let n = assume (Poisson (mulf lambda branchLength)) in +52 +iter n startTime branchLength lambda mu rho +53 in +54 +55 -- Simulation of branch +56 recursive +57 let simBranch: Int -> Float -> Float -> Float -> Float -> Float -> () = +58 +lam n: Int. +59 +lam startTime: Float. +60 +lam stopTime: Float. +61 +lam lambda: Float. +62 +lam mu: Float. +63 +lam rho: Float. +64 +if eqi n 0 then () +65 +else + +Automatic Alignment in Higher-Order PPLs +31 +66 +let currentTime = assume (Uniform stopTime startTime) in +67 +if crbdGoesUndetected currentTime lambda mu rho then +68 +let w1 = weight (log 2.) in +69 +simBranch (subi n 1) startTime stopTime lambda mu rho +70 +else +71 +let w2 = weight (negf inf) in +72 +() +73 in +74 +75 -- Simulating along the tree structure +76 recursive +77 let simTree: Tree -> Tree -> Float -> Float -> Float -> () = +78 +lam tree: Tree. +79 +lam parent: Tree. +80 +lam lambda: Float. +81 +lam mu: Float. +82 +lam rho: Float. +83 +let lnProb1 = mulf (negf mu) (subf (getAge parent) (getAge tree)) in +84 +let lnProb2 = match tree with Node _ then log lambda else log rho in +85 +86 +let startTime = getAge parent in +87 +let stopTime = getAge tree in +88 +let n = assume (Poisson (mulf lambda (subf startTime stopTime))) in +89 +simBranch n startTime stopTime lambda mu rho; +90 +91 +let w3 = weight (addf lnProb1 lnProb2) in +92 +93 +match tree with Node { left = left, right = right } then +94 +simTree left tree lambda mu rho; +95 +simTree right tree lambda mu rho +96 +else () +97 in +98 +99 -- Fixed priors used for the SMC experiment +100 -- let lambda = 0.2 in +101 -- let mu = 0.1 in +102 +103 -- +104 let lambda = assume (Gamma 1.0 1.0) in +105 let mu = assume (Gamma 1.0 0.5) in +106 +107 -- Adjust for normalizing constant +108 let numLeaves = countLeaves tree in +109 let corrFactor = +110 +subf (mulf (subf (int2float numLeaves) 1.) (log 2.)) (lnFactorial numLeaves) in +111 weight corrFactor; +112 +113 -- Start of the simulation along the two branches +114 (match tree with Node { left = left, right = right } then +115 +simTree left tree lambda mu rho; +116 +simTree right tree lambda mu rho +117 else ()); +118 +119 -- Compute the joint posterior over lambda and mu ... +120 (lambda,mu) +121 +122 -- ... or the marginal posterior over lambda (MCMC experiment) ... +123 -- lambda +124 +125 -- ... or nothing for just estimating the normalizing constant (SMC experiment) +126 -- () +A.2 +SMC: Cladogenetic Diversification Rate Shift (ClaDS) +Listing 2 gives the Miking CorePPL source code used for the case study model +in Section 7.2. +Listing 2: The source code for the experiment in Section 7.2 +1 ------------------------------------------------------------------ +2 -- The ClaDogenetic Diversification Shifts model (ClaDS2) model -- +3 ------------------------------------------------------------------ +4 +5 -- The prelude includes a few PPL helper functions +6 include "pplprelude.mc" +7 +8 -- The tree.mc file defines the general tree structure +9 include "tree.mc" +10 +11 -- The tree-instance.mc file includes the actual tree and the rho constant +12 include "tree-instance.mc" +13 +14 mexpr + +32 +D. Lundén et al. +15 +16 -- Multiplier guards +17 let maxM = 10e5 in +18 let minM = 0. in +19 +20 -- Clads2 goes undetected. +21 recursive +22 let clads2GoesUndetected: Float -> Float -> Float -> Float +23 +-> Float -> Float -> Float -> Bool = +24 +lam startTime_Mya: Float. +25 +lam lambda0: Float. +26 +lam mu0: Float. +27 +lam m: Float. -- Multiplier +28 +lam logAlpha: Float. -- Logarithm of alpha +29 +lam sigma: Float. -- Standard deviation +30 +lam rho: Float. +31 +32 +-- Guard: m is not allowed to exceed maxM or be 0. +33 +if or (gtf m maxM) (leqf m minM) then false +34 +else +35 +let eventTime_My = +36 +assume (Exponential (addf (mulf m lambda0) (mulf m mu0))) in +37 +let currentTime_Mya = subf startTime_Mya eventTime_My in +38 +if ltf currentTime_Mya 0. then +39 +if assume (Bernoulli rho) then false +40 +else true +41 +else +42 +let extinction = +43 +assume (Bernoulli (divf (mulf m mu0) +44 +(addf (mulf m lambda0) (mulf m mu0)))) in +45 +if extinction then true +46 +else +47 +let m1 = mulf m (exp (assume (Gaussian logAlpha sigma))) in +48 +let m2 = mulf m (exp (assume (Gaussian logAlpha sigma))) in +49 +if clads2GoesUndetected currentTime_Mya +50 +lambda0 mu0 m1 logAlpha sigma rho then +51 +clads2GoesUndetected currentTime_Mya +52 +lambda0 mu0 m2 logAlpha sigma rho +53 +else false +54 in +55 +56 -- Simulation of branch +57 recursive +58 let simBranch: Float -> Float -> Float -> Float +59 +-> Float -> Float -> Float -> Float -> Float = +60 +lam startTime_Mya: Float. +61 +lam stopTime_Mya: Float. +62 +lam lambda0: Float. +63 +lam mu0: Float. +64 +lam m: Float. -- multiplier +65 +lam logAlpha: Float. +66 +lam sigma: Float. +67 +lam rho: Float. +68 +69 +-- Guard: m is not allowed to exceed maxM or be 0. +70 +if or (gtf m maxM) (ltf m minM) then +71 +let w0 = weight (negf inf) in +72 +m +73 +else +74 +let tSpeciation_My = assume (Exponential (mulf m lambda0)) in +75 +let currentTime_Mya = subf startTime_Mya tSpeciation_My in +76 +let branchLength_My = subf startTime_Mya stopTime_Mya in +77 +if (ltf currentTime_Mya stopTime_Mya) then +78 +let w1 = weight (mulf (negf (mulf m mu0)) branchLength_My) in +79 +m +80 +else +81 +let m1 = mulf m (exp (assume (Gaussian logAlpha sigma))) in +82 +if clads2GoesUndetected currentTime_Mya lambda0 +83 +mu0 m1 logAlpha sigma rho then +84 +let m2 = mulf m (exp (assume (Gaussian logAlpha sigma))) in +85 +let w2 = weight (log 2.) in +86 +let w3 = weight (mulf (negf (mulf m mu0)) tSpeciation_My) in +87 +simBranch currentTime_Mya stopTime_Mya +88 +lambda0 mu0 m2 logAlpha sigma rho +89 +else -- side branch detected +90 +let w4 = weight (negf inf) in +91 +m +92 in +93 +94 -- Simulating along the tree structure +95 recursive +96 let simTree: Tree -> Tree -> Float -> Float +97 +-> Float -> Float -> Float -> Float -> () = +98 +lam tree: Tree. +99 +lam parent: Tree. +100 +lam lambda0: Float. +101 +lam mu0: Float. +102 +lam m: Float. +103 +lam logAlpha: Float. +104 +lam sigma: Float. + +Automatic Alignment in Higher-Order PPLs +33 +105 +lam rho: Float. +106 +107 +let startTime_Mya = getAge parent in +108 +let stopTime_Mya = getAge tree in +109 +110 +let mEnd = +111 +simBranch startTime_Mya stopTime_Mya lambda0 mu0 m logAlpha sigma rho in +112 +(match tree with Node _ +113 +then weight (log (mulf mEnd lambda0)) else weight (log rho)); +114 +115 +let m1 = mulf mEnd (exp (assume (Gaussian logAlpha sigma))) in +116 +let m2 = mulf mEnd (exp (assume (Gaussian logAlpha sigma))) in +117 +match tree with Node { left = left, right = right } then +118 +simTree left tree lambda0 mu0 m1 logAlpha sigma rho; +119 +simTree right tree lambda0 mu0 m2 logAlpha sigma rho +120 +else () +121 in +122 +123 -- Priors +124 let lambda0 = 0.2 in +125 let mu0 = 0.1 in +126 let logAlpha = negf 0.3 in +127 let sigma = sqrt 0.1 in +128 let m = 1.0 in +129 +130 -- Adjust for normalizing constant +131 let numLeaves = countLeaves tree in +132 let corrFactor = +133 +subf (mulf (subf (int2float numLeaves) 1.) (log 2.)) (lnFactorial numLeaves) in +134 weight corrFactor; +135 +136 let m1 = mulf m (exp (assume (Gaussian logAlpha sigma))) in +137 let m2 = mulf m (exp (assume (Gaussian logAlpha sigma))) in +138 +139 -- Start of the simulation along the two branches +140 (match tree with Node { left = left, right = right } then +141 +simTree left tree lambda0 mu0 m1 logAlpha sigma rho; +142 +simTree right tree lambda0 mu0 m2 logAlpha sigma rho +143 +else ()); +144 +145 -- Returns nothing, as the current model is only used to compute the +146 -- normalizing constant +147 () +A.3 +SMC: State-Space Aircraft Localization +Fig. 11 presents the aircraft model used for the experiment in Section 7.3. An +aircraft flies along a one-dimensional axis in discrete time steps, and the crew +needs to estimate the aircraft’s current position using noisy satellite position +data available for the ten most recent time steps (defined at line 1). A second +model component—the aircraft’s altitude—further complicates the model as the +crew cannot observe it (the altimeter is not functioning). The aircraft’s velocity +and the precision of the satellite observations depend on the altitude, as dictated +by the functions velocity (defined at line 13) and positionObsStDev (defined at +line 18). The velocity (in meters per second) increases linearly with increasing +altitude (less air resistance) but is capped to the range [100, 500]. On the other +hand, the observation standard deviation (in meters) decreases linearly with +increasing altitude (less interference between the satellites and the aircraft) but +is never less than ten. +Lines 25 to 44 define the main function simulate iterating over the ten data +items. The critical component illustrating the need for alignment is the weight +0.5 at line 32. This weight encodes that the pilot adjusts the aircraft’s pitch +when air traffic control signals altitude deviations more than 100 feet from the +assigned altitude of 35 000 feet. Each time step where the actual altitude deviates +more than 100 feet from the assigned altitude thus gives a penalty factor of 0.5. +Unlike the weight at line 29, this weight is unaligned. + +34 +D. Lundén et al. +1 let data = [ +2 +603.57, 860.42, 1012.07, 1163.53, +3 +1540.29, 1818.10, 2045.38, 2363.49, +4 +2590.77, 2801.91 +5 ] +6 let holdingAltitude = 35 000 in +7 let altitudeRange = 100 in +8 let position = assume Uniform(0, 1000) in +9 let altitude = +10 +assume N(holdingAltitude, 2002) in +11 let positionStDev = 50 in +12 let baseVelocity = 250 in +13 let velocity = λaltitude. +14 +let k = +baseVelocity +holdingAltitude in +15 +min (500, max (100, (k · altitude))) +16 in +17 let basePositionObsStDev = 50 in +18 let positionObsStDev : = λaltitude. +19 +let m = 100 in +20 +let k = − basePositionObsStDev +holdingAltitude +in +21 +max (10, m + k · altitude) +22 in +23 let altitudeStDev = 100 in +24 let rec simulate = +25 +λdata. λposition. λaltitude. +26 +match data with d :: ds then +27 +let σ = +28 +positionObsStDev altitude in +29 +weight fN (position,σ2)(d) +30 +if |altitude − holdingAltitude| +31 +> altitudeRange then +32 +weight 0.5 +33 +else (); +34 +let position = +35 +assume N( +36 +position + velocity altitude, +37 +positionStDev 2 +38 +) in +39 +let altitude = +40 +assume N(altitude, altitudeStDev 2) +41 +in +42 +simulate ds position altitude +43 +else position +44 in +45 simulate data position altitude +Fig. 11: A state-space model for estimating an aircraft’s position given a set of +noisy position estimates. The text contains further details. The program uses +the syntax (1), extended with sequences, pattern matching over sequences, and +the pattern :: for sequence deconstruction. The function fN(µ,σ2) is the PDF of +the normal distribution at µ with variance σ2. +The simulation also accounts for variations in, e.g., wind resistance when +updating the position at line 34 through a standard deviation of positionStDev +meters. Similarly, the altitude varies with a standard deviation of altitudeStDev +feet when updating the altitude at line 39. +We generated the ten data points used for the experiment in Section 7.3 +by running the model (ignoring line 32) and sampling from N(position, σ2) at +line 29. +Listing 3 gives the Miking CorePPL source code used for the case study +model in Section 7.3. +A.4 +MCMC: Latent Dirichlet Allocation (LDA) +Listing 4 gives the Miking CorePPL source code used for the case study model +in Section 7.4. Furthermore, we conduct an additional LDA experiment justi- +fying the correctness of the aligned lightweight MCMC and lightweight MCMC +implementations. The experiment uses a simplified generated data set with only +two topics, a vocabulary of two words, and three documents with 10 words each. +To generate the data, we use the true values θ1 = 0.95, θ2 = 0.05, and θ3 = 0.5 +for the document topic distributions, and φ1 = 0.99 and φ2 = 0.01 for the +word distribution within the two topics. Note that the true proportions above +are uniquely determined by the proportion of the first topic and first word, as +there are only two topics and two words in the vocabulary. The simplicity of the + +Automatic Alignment in Higher-Order PPLs +35 +Listing 3: The source code for the experiment in Section 7.3 +1 --------------------------------------------------- +2 -- A state-space model for aircraft localization -- +3 --------------------------------------------------- +4 +5 include "math.mc" +6 +7 mexpr +8 +9 -- Noisy satellite observations of position (accuracy is improved at higher +10 -- altitude) +11 let ysPos: [Float] = [ +12 +603.5736741666899, 860.4207338929477, 1012.0766100484578, 1163.5339974878366, +13 +1540.2972028551385, 1818.1023092741882, 2045.3888580253108, +14 +2363.4902615131796, 2590.773153142429, 2801.9143537470927 +15 ] in +16 +17 let holdingAltitude = 35000. in +18 let altitudeRange = 100. in +19 let position: Float = assume (Uniform 0. 1000.) in +20 let altitude: Float = assume (Gaussian holdingAltitude 200.) in +21 +22 let positionStDev = 50. in +23 +24 let baseVelocity = 250. in +25 let velocity: Float -> Float = lam altitude. +26 +let k = divf baseVelocity holdingAltitude in +27 +minf 500. (maxf 100. (mulf k altitude)) +28 in +29 +30 let basePositionObsStDev = 50. in +31 let positionObsStDev: Float -> Float = lam altitude. +32 +let m = 100. in +33 +let k = negf (divf basePositionObsStDev holdingAltitude) in +34 +maxf 10. (addf m (mulf k altitude)) +35 in +36 +37 let altitudeStDev = 100. in +38 +39 recursive let simulate: Int -> Float -> Float -> Float = +40 +lam t: Int. lam position: Float. lam altitude: Float. +41 +42 +-- Observe position +43 +let dataPos: Float = get ysPos t in +44 +observe dataPos (Gaussian position (positionObsStDev altitude)); +45 +let t = addi t 1 in +46 +47 +-- Penalize altitude divergence of more than ‘altitudeRange‘ feet from +48 +-- holding altitude +49 +(if gtf (absf (subf altitude holdingAltitude)) altitudeRange then +50 +weight (log 0.5) +51 +else ()); +52 +53 +-- Transition +54 +let position: Float = +55 +assume (Gaussian (addf position (velocity altitude)) positionStDev) in +56 +let altitude: Float = assume (Gaussian altitude altitudeStDev) in +57 +58 +if eqi (length ysPos) t then position +59 +else simulate t position altitude +60 in +61 +62 simulate 0 position altitude + +36 +D. Lundén et al. +0 +0.5 +1 +θ1 +0 +0.5 +1 +θ2 +0 +0.5 +1 +θ3 +(a) Aligned lightweight MCMC posteriors. +0 +0.5 +1 +θ1 +0 +0.5 +1 +θ2 +0 +0.5 +1 +θ3 +(b) Lightweight MCMC posteriors. +Fig. 12: Fig. (a) and (b) plots aligned lightweight MCMC and lightweight MCMC +posterior distributions for the three documents θ1, θ2, and θ3 in the simplified +LDA data set in Section A.4. The posteriors are the combined samples of 300 +independent MCMC runs, each with 3 · 106 iterations and 10% burn. +model and rather extreme true values used to generate the data allows for easy +visualization of the document topic posteriors and justification of their correct- +ness. Fig. 12 presents the posterior topic distributions for the three documents +for a very large number of MCMC iterations. As expected, aligned lightweight +MCMC and lightweight MCMC produce identical results agreeing with the true +values for θ1, θ2, and θ3. The bimodal posteriors for θ1 and θ2 are due to the +interchangeability of topics in LDA. +A.5 +MCMC: Constant Rate Birth-Death (CRBD) +Listing 1 gives the Miking CorePPL source code used for the case study model in +Section 7.5. Furthermore, Fig 13 shows the posterior distributions over lambda, +justifying the use of the mean as a measure of accuracy as the posterior is clearly +unimodal. +B +Alignment Analysis, Continued +This section presents the full alignment constraint propagation algorithm (Sec- +tion B.1) and proof of soundness of the alignment analysis (Section B.2). +B.1 +Algorithm +Algorithm 4 presents the full alignment algorithm that produces a solution +to the constraints generated by Algorithm 1. For reference, we now also give a +more formal definition of constraints c. + +Automatic Alignment in Higher-Order PPLs +37 +Listing 4: The source code for the experiment in Section 7.4 +1 include "common.mc" +2 include "string.mc" +3 include "seq.mc" +4 include "ext/dist-ext.mc" +5 +6 -- The data.mc file contains the generated data +7 include "data.mc" +8 +9 mexpr +10 +11 let alpha: [Float] = make numtopics 1. in +12 let beta: [Float] = make vocabsize 1. in +13 let phi = create numtopics (lam. assume (Dirichlet beta)) in +14 let theta = create numdocs (lam. assume (Dirichlet alpha)) in +15 repeati (lam w. +16 +let word = get docs w in +17 +let counts = assume (Multinomial word.1 (get theta (get docids w))) in +18 +iteri (lam z. lam e. +19 +weight (mulf (int2float e) +20 +(bernoulliLogPmf (get (get phi z) word.0) true)) +21 +) counts +22 +) (length docs); +23 +24 -- Returns the joint posterior distribution over theta and phi ... +25 (theta, phi) +26 +27 -- ... or the marginal posterior over theta (simple data set) ... +28 -- theta +29 +30 -- ... or nothing if only comparing exeuction time (C3 data set) +31 -- () +0 +0.2 0.4 0.6 0.8 +1 +0 +0.2 0.4 0.6 0.8 +1 +0 +0.2 0.4 0.6 0.8 +1 +(a) Aligned lightweight MCMC. +0 +0.2 0.4 0.6 0.8 +1 +0 +0.2 0.4 0.6 0.8 +1 +0 +0.2 0.4 0.6 0.8 +1 +(b) Lightweight MCMC +Fig. 13: One iteration of the CRBD experiment in Section 7.5. Fig. (a) shows +posteriors for aligned lightweight MCMC (gray). From left to right: 3 · 104 iter- +ations, 3 · 105 iterations, and 3 · 106 iterations. Fig. (b) shows the corresponding +posteriors for lightweight MCMC (white). + +38 +D. Lundén et al. +Algorithm 4 Alignment analysis. +function analyzeAlign(t): TANF → ((X → P(A)) × P(X)) = +1 worklist: [X] := [] +2 data: X → P(A) := {(x, ∅) | x ∈ X} +3 unaligned: P(X) := ∅ +4 edges: X → P(R) := {(x, ∅) | x ∈ X} +5 for c ∈ generateConstraints(t): +6 +initializeConstraint(c) +7 iter(); +return (data, unaligned) +8 +9 function iter: () → () = match worklist with +10 +| [] → () +11 +| x :: worklist’ → +12 +worklist := worklist’ +13 +for c ∈ edges(x): +14 +propagateConstraint(c) +15 +iter () +16 +17 function initializeConstraint(c): R → () = +18 +match c with +19 +| a ∈ Sx → addData(x, {a}) +20 +| Sx ⊆ Sy → initializeConstraint′(x, c) +21 +| a1 ∈ Sx ⇒ a2 ∈ Sy → +22 +initializeConstraint′(x, c) +23 +| ∀x∀y λx.y ∈ Slhs +24 +⇒ (Srhs ⊆ Sx) ∧ (Sy ⊆ Sapp) → +25 +initializeConstraint′(lhs, c) +26 +| ∀n (const n ∈ Slhs) ∧ (n > 1) +27 +⇒ const n − 1 ∈ Sapp → +28 +initializeConstraint′(lhs, c) +29 +| const _ ∈ Slhs +30 +⇒ (stoch ∈ rhs ⇒ stoch ∈ app) → +31 +initializeConstraint′(lhs, c) +32 +| unalignedx ⇒ unalignedy → +33 +initializeConstraint′(x, c) +34 +| stoch ∈ Sx ⇒ unalignedy → +35 +initializeConstraint′(x, c) +36 +| ∀x λx._ ∈ Slhs ⇒ unalignedx → +37 +initializeConstraint′(lhs, c) +38 +| unalignedres ⇒ +39 +(∀x λx._ ∈ Slhs ⇒ unalignedx) → +40 +initializeConstraint′(res, c) +41 +| stoch ∈ Slhs ⇒ +42 +(∀x λx._ ∈ Slhs ⇒ unalignedx) → +43 +initializeConstraint′(lhs, c) +44 +45 function initializeConstraint′(x,c) +46 +: X → () = +47 +edges(x) := edges(x) ∪ {c}; +48 +propagateConstraint(c) +49 +50 function addData(x, A): X × P(A) → () = +51 +if A ̸⊆ data(x) then +52 +data(x) := data(x) ∪ A +53 +worklist := x :: worklist +54 +55 function addUnaligned(x): X → () = +56 +if x ̸∈ unaligned then +57 +unaligned := unaligned ∪{x} +58 +worklist := x :: worklist +59 +60 function propagateConstraint(c): R → () = +61 +match c with +62 +| a ∈ Sx → () +63 +| Sx ⊆ Sy → addData(y, data(x)) +64 +| a1 ∈ Sx ⇒ a2 ∈ Sy → +65 +if a1 ∈ data(x) then addData(y,{a2}) +66 +| ∀x∀y λx.y ∈ Slhs +67 +⇒ (Srhs ⊆ Sx) ∧ (Sy ⊆ Sapp) → +68 +for λx.y ∈ data(lhs): +69 +initializeConstraint(Srhs ⊆ Sx) +70 +initializeConstraint(Sy ⊆ Sapp) +71 +| ∀n (const n ∈ Slhs) ∧ (n > 1) +72 +⇒ const n − 1 ∈ Sapp → +73 +for const n ∈ data(lhs): +74 +if n > 1 then +75 +addData(app, {const n − 1}) +76 +| const _ ∈ Slhs +77 +⇒ (stoch ∈ rhs ⇒ stoch ∈ app) → +78 +if ∃n const n ∈ Slhs then +79 +initializeConstraint( +80 +stoch ∈ rhs ⇒ stoch ∈ app +81 +) +82 +| unalignedx ⇒ unalignedy → +83 +if x ∈ unaligned then addUnaligned(y) +84 +| stoch ∈ Sx ⇒ unalignedy → +85 +if stoch ∈ data(x) then addUnaligned(y) +86 +| ∀x λx._ ∈ Slhs ⇒ unalignedx → +87 +for λx._ ∈ data(lhs): addUnaligned(x) +88 +| unalignedres ⇒ +89 +(∀x λx._ ∈ Slhs ⇒ unalignedx) → +90 +if res ∈ unaligned then +91 +initializeConstraint( +92 +∀x λx._ ∈ Slhs ⇒ unalignedx +93 +) +94 +| stoch ∈ Slhs ⇒ +95 +(∀x λx._ ∈ Slhs ⇒ unalignedx) → +96 +if stoch ∈ data(lhs) then +97 +initializeConstraint( +98 +∀x λx._ ∈ Slhs ⇒ unalignedx +99 +) + +Automatic Alignment in Higher-Order PPLs +39 +Definition 8 (Constraints). +c ::= a ∈ Sx | Sx ⊆ Sy | a ∈ Sx ⇒ a ∈ Sy +| ∀x∀y λx.y ∈ Slhs ⇒ (Srhs ⊆ Sx) ∧ (Sy ⊆ Sapp) +| ∀n (const n ∈ Slhs) ∧ (n > 1) ⇒ const n − 1 ∈ Sapp +| const _ ∈ Slhs ⇒ (stoch ∈ Srhs ⇒ stoch ∈ Sapp) +| unalignedx ⇒ unalignedy | stoch ∈ Sx ⇒ unalignedy +| ∀x λx._ ∈ Slhs ⇒ unalignedx +| unalignedres ⇒ (∀x λx._ ∈ Slhs ⇒ unalignedx) +| stoch ∈ Slhs ⇒ (∀x λx._ ∈ Slhs ⇒ unalignedx) +x, y, lhs, rhs, app, res ∈ X. +(8) +The main function analyzeAlign consists of two steps: initialization and itera- +tion. In the initialization step, generateConstraints provides constraints to +the initializeConstraint function, which initializes the maps data and edges, +and the set unaligned. The map data contains the sets of abstract values for +all program variables and is initially empty. At termination, data(x) is a sound +approximation of Sx for each x (Lemma 1). The map edges associates a set of +constraints with each variable in the program. Specifically, we must propagate +the constraints associated with a variable x after updating data(x) with new +information. Finally, the set unaligned tracks unaligned expressions and is ini- +tially empty. At termination, unaligned contains the set of all unaligned variables +identified by the analysis. This set is sound according to Lemma 1. +The iteration step iter propagates constraints with propagateConstraint +for all variables updated with new abstract values or unalignment since their last +propagation. We store these updated variables in the sequence worklist, which, +when empty, signals fixpoint and termination. Note that, e.g., the lambda ap- +plication constraint at line 67 initializes new constraints dynamically during +propagation, depending on which abstract lambdas flow to the left-hand side of +the application. +B.2 +Correctness Proof +This section presents the correctness proof that is ultimately used to prove The- +orem 1. +Throughout this section, t1 = t2 means that the terms t1 and t2 are alpha +equivalent. For constant comparisons c1 = c2, we assume the prior existence of +an equality function over constants. We first require a specific equality relation +on values. +Definition 9 (Value equality). v1 +V= v2 iff +– v1 = ⟨λx.t1, ρ1⟩, v2 = ⟨λx.t2, ρ2⟩, and t1 = t2, or +– v1 = c1, v2 = c2, and c1 = c2. + +40 +D. Lundén et al. +Note, in particular, that +V= treats closures as equal even if their environments +differ. As we will see, this property is critical in the proof of Lemma 2. +Next, we formally define subterms. +Definition 10 (Subterms). We say that t′ is a subterm of t iff +(1) t′ = t, or +(2) either +t = λx. t1, +t = t1 t2, +t = let x = t1 in t2, +t = if t1 then t2 else t3, +t = assume t1, +or t = weight t1, +and t′ is a subterm of either t1, t2, or t3. +In the below, we assume a +– fixed t ∈ TANF, +– an assignment to Sx and unalignedx for x ∈ X from analyzeAlign(t), and +– �At = {x | ¬unaligned x}. +We begin with a lemma concerning unaligned expressions in single evaluations +of ⇓. +Lemma 2 (Unaligned evaluations). Let +– t′ be a subterm of t, t′ ∈ TANF, and +– ρ ⊢ t′ s⇓w +l v +with ρ such that, for each x ∈ X, +(C1) ρ(x) = ⟨λy.ty, ρy⟩ implies that (λy.ty) is a subterm of t, λy.name(ty) ∈ +Sx, and that (C1) holds for ρy. Also, ρ(x) = c such that |c| > 1 implies +const |c| ∈ Sx. +Then, +(R1) if unalignedn for all n ∈ names(t′), then l| � +At = [], and +(R2) v = ⟨λy.ty, ρy⟩ implies (λy.ty) is a subterm of t and λy.name(ty) ∈ +Sname(t′), and that (C1) holds for ρy. Furthermore, v = c such that |c| > 1 +implies const |c| ∈ Sname(t′). +Proof. We proceed by structural induction over ρ ⊢ t′ s⇓w +l v. +Case t′ = x: +The derivation is +ρ ⊢ x []⇓1 +[] ρ(x) +(Var) +(R1) Immediate as l = [] = l| �At. +(R2) By definition, name(t′) = x and ρ(x) = v. The result follows from (C1). + +Automatic Alignment in Higher-Order PPLs +41 +Case t′ = (let x = t1 in t2): +The derivation is +ρ ⊢ t1 +s1⇓w1 +l1 v′ +ρ, x �→ v′ ⊢ t2 +s2⇓w2 +l2 v +ρ ⊢ let x = t1 in t2 +s1∥s2⇓w1·w2 +l1∥[x]∥l2 v +(Let) +Note that unalignedn for all n ∈ names(t′) and the definition of �At implies +[x]| � +At = []. Also, +l| � +At = (l1 ∥ [x] ∥ l2)| � +At = l1| � +At ∥ [x]| � +At ∥ l2| � +At. +To show (R1), we therefore only need l1| � +At = l2| � +At = []. Now, let ρ′ = ρ, x �→ +ρ(y). To apply the induction hypothesis, we must establish (C1) for ρ′, denoted +(C1′). To prove (C1′), note that we only need to consider ρ′(x). For all other +x′ ∈ X, ρ′(x′) = ρ(x′) and (C1′) follows directly as a result of (C1). We denote +the induction hypothesis results (R1)–(R2) for ρ′ ⊢ t2 +s2⇓w2 +l2 v with (R1′)– +(R2′). Next, we consider each case for t1 (according to t′ +ANF in (3), p. 10). Note +that (R2) follows directly from (R2′) as name(t2) = name(t′). Thus, we only +need to consider (C1′) and (R1). +Subcase t1 = y +The derivation for t1 is +ρ ⊢ y []⇓1 +[] ρ(y) +(Var) +Clearly ρ′(x) = ρ(y). Also, Sy ⊆ Sx from Lemma 1. +(C1′) If ρ′(x) = ⟨λz.tz, ρz⟩, then λz.tz is a subterm of t by ρ′(x) = ρ(y) +and (C1). Also, clearly (C1) holds for ρz by (C1) for ρ. Furthermore, +λz.name(tz) ∈ Sy by (C1) and Sy ⊆ Sx implies λz.name(tz) ∈ Sx. By a +similar argument, if ρ′(x) = c such that |c| > 0, const |c| ∈ Sx. +We now apply the induction hypothesis and get (R1′)–(R2′). +(R1) We clearly have l1 = []. The result now follows immediately from (R1′). +Subcase t1 = c +The derivation for t1 is +ρ ⊢ c []⇓1 +[] c +(Const) +Clearly, ρ′(x) = c. +(C1′) If |c| > 0, we have const |c| ∈ Sx as a result of Lemma 1. +We now apply the induction hypothesis and get (R1′)–(R2′). +(R1) We clearly have l1 = []. The result now follows immediately from (R1′). + +42 +D. Lundén et al. +Subcase t1 = λy.ty +The derivation for t1 is +ρ ⊢ λy.ty +[]⇓1 +[] ⟨λy.ty, ρ⟩ +(Lam) +Clearly, ρ′(x) = ⟨λy.ty, ρ⟩. +(C1′) First, it is clear that λy.ty is a subterm of t and that (C1) holds for ρ. +Lastly, Lemma 1 also gives λy.name(ty) ∈ Sx. +We now apply the induction hypothesis and get (R1′)–(R2′). +(R1) We clearly have l1 = []. The result now follows immediately from (R1′). +Subcase t1 = y z +The possible derivations are +ρ ⊢ y []⇓1 +[] ⟨λy′.ty′, ρy′⟩ +ρ ⊢ z []⇓1 +[] ρ(z) +ρy′, y′ �→ ρ(z) ⊢ ty′ s1⇓w1 +l1 v′ +ρ ⊢ y z s1⇓w1 +l1 v′ +(App) +ρ ⊢ y []⇓1 +[] c +ρ ⊢ z []⇓1 +[] ρ(z) +|c| > 0 +ρ ⊢ y z []⇓1 +[] δ(c, ρ(z)) +(Const-App) +(C1′) We first consider the case (App). Let ρ′′ = ρy′, y′ �→ ρ(z) and consider +the derivation ρ′′ ⊢ ty′ s1⇓w1 +l1 v′. To apply the induction hypothesis, we must +establish (C1) for ρ′′. First, (C1) holds for ρy′ by (C1) for ρ. (C1) therefore +also holds for ρ′′ by a similar argument to ρ′ in the subcase t1 = y above. +From the induction hypothesis, we then get (R1′′)–(R2′′). From Lemma 1, +we have Sname(ty′ ) ⊆ Sx. Combined with (R2′′), the result follows. +Now, consider the case (Const-App). From Lemma 1, ∀n const n ∈ Sy ∧ +n > 1 ⇒ const n − 1 ∈ Sx. From Definition 2, we also have |δ(c, ρ(z))| = +|c| − 1. The result now follows. +We now apply the induction hypothesis and get (R1′)–(R2′). +(R1) The (Const-App) case is immediate by l1 = [] and (R1′). Therefore, +assume the derivation is (App) and that unalignedn for all n ∈ names(t′) (in +particular unalignedx). By Lemma 1, we have unalignedy′ and unalignedn′ +for n′ ∈ names(ty′). By (R1′′), we then have l1| � +At = []. From (R1′), we +also have l2| � +At = [] and the result follows. +Subcase t1 = if y then tt else te +The possible derivations are +ρ ⊢ y []⇓1 +[] true +ρ ⊢ tt +s1⇓w1 +l1 vt +ρ ⊢ if y then tt else te +s1⇓w1 +l1 vt +(If-True) +ρ ⊢ y []⇓1 +[] false +ρ ⊢ te +s1⇓w1 +l1 ve +ρ ⊢ if y then tt else te +s1⇓w1 +l1 ve +(If-False) + +Automatic Alignment in Higher-Order PPLs +43 +Without loss of generality, we only consider (If-True). Note that for the sub- +derivation ρ ⊢ tt +s1⇓w1 +l1 vt, (R1)–(R2), denoted (R1t)–(R2t) below, holds im- +mediately by the induction hypothesis as (C1) holds for ρ. +(C1′) Follows from (R2t) and name(tt) ⊆ Sx from Lemma 1. +We now apply the induction hypothesis and get (R1′)–(R2′). +(R1) Assume we have unalignedn for all n ∈ names(t′) (including unalignedx). +Then, by Lemma 1, unalignedn′ for n′ ∈ names(tt). Therefore, l1| � +At = [] by +(R1t). From (R1′), we also have l2| � +At = [] and the result follows. +Subcase t1 = assume y +The derivation is +ρ ⊢ y []⇓1 +[] d +w = fd(c) +ρ ⊢ assume y [c]⇓w +[] c +(Assume) +(C1′) By Definition 3, |c| = 0. The result follows immediately. +We now apply the induction hypothesis and get (R1′)–(R2′). +(R1) We clearly have l1 = []. The result now follows immediately from (R1′). +Subcase t1 = weight y +The derivation is +ρ ⊢ y []⇓1 +[] w +ρ ⊢ weight y []⇓w +[] () +(Weight) +(C1′) We have w ∈ R and |w| = 0. The result follows immediately. +We now apply the induction hypothesis and get (R1′)–(R2′). +(R1) We clearly have l1 = []. The result now follows immediately from (R1′). +⊓⊔ +With Lemma 2 established, we now give the main lemma used to prove Theo- +rem 1. +Lemma 3 (Aligned evaluations). Let +– t′ ∈ t, t′ ∈ TANF, +– ρ1 ⊢ t′ s1⇓w1 +l1 v1, and +– ρ2 ⊢ t′ s2⇓w2 +l2 v2 +with ρ1 and ρ2 such that, for each x ∈ X, +(C2) for ρ ∈ {ρ1, ρ2}, (C1) holds, +(C3) if ρ1(x) = ⟨λy.ty, ρ′ +1⟩, ρ2(x) = ⟨λy.ty, ρ′ +2⟩, and stoch ̸∈ Sx, then ρ′ +1 and +ρ′ +2 fulfill (C2)–(C4), and +(C4) If ρ1(x) ̸ +V= ρ2(x), then stoch ∈ Sx. + +44 +D. Lundén et al. +Then, +(R3) l1| � +At = l2| � +At, +(R4) if v1 = ⟨λy.ty, ρ′ +1⟩, v2 = ⟨λy.ty, ρ′ +2⟩, and stoch ̸∈ Sname(t′), then ρ′ +1 and +ρ′ +2 fulfill (C2)–(C4), and +(R5) If v1 ̸ +V= v2, then stoch ∈ Sname(t′). +Proof. We proceed by simultaneous structural induction over ρ1 ⊢ t′ s1⇓w1 +l1 v1 +and ρ2 ⊢ t′ s2⇓w2 +l2 v2. +Case t′ = x: +The possible derivations are +ρ1 ⊢ x []⇓1 +[] ρ1(x) +(Var) +ρ2 ⊢ x []⇓1 +[] ρ2(x) +(Var) +(R3) We have l1 = l2 = [] = l1| � +At = l2| � +At. +(R4) By name(t′) = x and (C3). +(R5) By name(t′) = x and (C4). +Case t′ = (let x = t1 in t2): +The possible derivations are +ρ1 ⊢ t1 +s11⇓w11 +l11 v′ +1 +ρ1, x �→ v′ +1 ⊢ t2 +s12⇓w12 +l12 v1 +ρ1 ⊢ let x = t1 in t2 +s11∥s12⇓w11·w12 +l11∥[x]∥l12 v1 +(Let) +ρ2 ⊢ t1 +s21⇓w21 +l21 v′ +2 +ρ2, x �→ v′ +2 ⊢ t2 +s22⇓w22 +l22 v2 +ρ2 ⊢ let x = t1 in t2 +s21∥s22⇓w21·w22 +l21∥[x]∥l22 v2 +(Let) +Assume l11| � +At = l21| � +At and l12| � +At = l22| � +At. Then, +l1| � +At = (l11 ∥ [x] ∥ l12)| � +At += l11| � +At ∥ [x]| � +At ∥ l12| � +At = l21| � +At ∥ [x]| � +At ∥ l22| � +At += (l21 ∥ [x] ∥ l22)| � +At = l2| � +At. +That is, for (R3), we only need l11| � +At = l21| � +At and l12| � +At = l22| � +At. Let ρ′ +1 = +ρ1, x �→ v′ +1) and ρ′ +2 = ρ2, x �→ v′ +2. To apply the induction hypothesis, we must +establish (C2)–(C4) for ρ′ +1 and ρ′ +2. To avoid confusion with the original as- +sumptions (C2)–(C4) for ρ1 and ρ2, we use the notation (C2′)–(C4′) for the +ρ′ +1 and ρ′ +2 conditions. To prove (C2′)–(C4′), note that we only need to consider +ρ′ +1(x) and ρ′ +2(x). For all other x′ ∈ X, ρ′ +1(x′) = ρ1(x′) and ρ′ +2(x′) = ρ2(x′), and +(C2′)–(C4′) follow directly as a result of (C2)–(C4). We denote the induc- +tion hypothesis results (R3)–(R5) for ρ′ +1 ⊢ t2 +s12⇓w12 +l12 v1 and ρ′ +2 ⊢ t2 +s22⇓w22 +l22 v2 +with (R3′)–(R5′). Next, we consider each case for t1 (according to t′ +ANF in +(3), p. 10). Note that (R4) and (R5) follow directly from (R4′) and (R5′) as +name(t2) = name(t′). Thus, we only need to consider (C2′)–(C4′) and (R3). + +Automatic Alignment in Higher-Order PPLs +45 +Subcase t1 = y +The derivations for t1 are +ρ1 ⊢ y []⇓1 +[] ρ1(y) +(Var) +ρ2 ⊢ y []⇓1 +[] ρ2(y) +(Var) +We first establish (C2′)–(C4′). Clearly ρ′ +1(x) = ρ1(y) and ρ′ +2(x) = ρ2(y). Also, +Sy ⊆ Sx from Lemma 1. +(C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for both +ρ′ +1(x) and ρ′ +2(x). +(C3′) Assume ρ′ +1(x) = ⟨λz.tz, ρ′′ +1⟩, ρ′ +2(x) = ⟨λz.tz, ρ′′ +2⟩, and stoch ̸∈ Sx. By +Sy ⊆ Sx, stoch ̸∈ Sy. Because ρ′ +1(x) = ρ1(y) and ρ′ +2(x) = ρ2(y), the result +follows from (C3). +(C4′) If ρ′ +1(x) ̸ +V= ρ′ +2(x), then clearly ρ1(y) ̸ +V= ρ2(y). Hence, stoch ∈ Sy by (C4) +and the result follows by Sy ⊆ Sx. +We now apply the induction hypothesis and get (R3′)–(R5′). +(R3) The result follows from l11 = l21 = [] and (R3′). +Subcase t1 = c +The derivations for t1 are +ρ1 ⊢ c []⇓1 +[] c +(Const) +ρ2 ⊢ c []⇓1 +[] c +(Const) +We first establish (C2′)–(C4′). +(C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for +ρ′ +1(x) = ρ′ +2(x). +(C3′) Follows directly as ρ′ +1(x) = ρ′ +2(x) = c. +(C4′) Follows directly because ρ′ +1(x) = ρ′ +2(x) = c. +We now apply the induction hypothesis and get (R3′)–(R5′). +(R3) The result follows from l11 = l21 = [] and (R3′). +Subcase t1 = λy.ty +The derivations are +ρ1 ⊢ λy.ty +[]⇓1 +[] ⟨λy.ty, ρ1⟩ +(Lam) +ρ2 ⊢ λy.ty +[]⇓1 +[] ⟨λy.ty, ρ2⟩ +(Lam) +We first establish (C2′)–(C4′). +(C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for both +ρ′ +1(x) and ρ′ +2(x). + +46 +D. Lundén et al. +(C3′) Follows because ρ1 and ρ2 fulfills (C2)–(C4). +(C4′) Follows because ρ′ +1(x) +V= ρ′ +2(x). +We now apply the induction hypothesis and get (R3′)–(R5′). +(R3) The result follows from l11 = l21 = [] and (R3′). +Subcase t1 = y z +The possible derivations are +ρ1 ⊢ y []⇓1 +[] ⟨λy1.ty1, ρy1⟩ +ρ1 ⊢ z []⇓1 +[] ρ1(z) +ρy1, y1 �→ ρ1(z) ⊢ ty1 +s11⇓w11 +l11 v′ +1 +ρ1 ⊢ y z s11⇓w11 +l11 v′ +1 +(App) +ρ2 ⊢ y []⇓1 +[] ⟨λy2.ty2, ρy2⟩ +ρ2 ⊢ z []⇓1 +[] ρ2(z) +ρy2, y2 �→ ρ2(z) ⊢ ty2 +s21⇓w21 +l21 v′ +2 +ρ2 ⊢ y z s21⇓w21 +l21 v′ +2 +(App) +ρ1 ⊢ y []⇓1 +[] c1 +ρ1 ⊢ z []⇓1 +[] ρ1(z) +|c1| > 0 +ρ1 ⊢ y z []⇓1 +[] δ(c1, ρ1(z)) +(Const-App) +ρ2 ⊢ y []⇓1 +[] c2 +ρ2 ⊢ z []⇓1 +[] ρ2(z) +|c2| > 0 +ρ2 ⊢ y z []⇓1 +[] δ(c2, ρ2(z)) +(Const-App) +We first establish (C2′)–(C4′). +(C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for both +ρ′ +1(x) and ρ′ +2(x). +(C3′) Assume stoch ̸∈ Sx. For (Const-App), ρ′ +1(x) ∈ C and ρ′ +2(x) ∈ C, and +the result follows immediately. Therefore, assume that both derivations are +(App). By Lemma 1, stoch ∈ Sy ⇒ stoch ∈ Sx, and consequently stoch ̸∈ +Sy. By (C4), this leads to ρ1(y) = ⟨λy1.ty1, ρy1⟩ +V= ρ2(y) = ⟨λy2.ty2, ρy2⟩. +That is, λy1.ty1 = λy2.ty2 = λy′.ty′. By (C3), ρy1 and ρy2 fulfill (C2)– +(C4). Let ρ′′ +1 = ρy1, y′ �→ ρ1(z) and ρ′′ +2 = ρy2, y′ �→ ρ2(z) and consider the +derivations ρ′′ +1 ⊢ ty′ s11⇓w11 +l11 v′ +1 and ρ′′ +2 ⊢ ty′ s21⇓w21 +l21 v′ +2. It is straightforward +to check that ρ′′ +1 and ρ′′ +2 fulfill (C1), and we apply the induction hypothe- +sis and get the results (R3′′)–(R5′′). Now, by Lemma 1, Sname(ty′ ) ⊆ Sx. +Combined with (R4′′), the result follows. +(C4′) Assume ρ′ +1(x) ̸ +V= ρ′ +2(x), and consider first the case where ρ1(y) ̸ +V= ρ2(y). +Then, by (C4), stoch ∈ Sy and by stoch ∈ Sy ⇒ stoch ∈ Sx from +Lemma 1, stoch ∈ Sx and we are done. Therefore, assume ρ1(y) +V= ρ2(y). +Consequently, both derivations are either (Const-App) or (App). If both +derivations are (Const-App), c1 +V= c2 +V= c and ρ′ +1(x) ̸ +V= ρ′ +2(x) implies ρ1(z) ̸ +V= +ρ2(z). By (C4), this implies stoch ∈ Sz. Lemma 1 gives const _ ∈ Sy ⇒ +(stoch ∈ Sz ⇒ stoch ∈ Sx). Clearly, const |c| ∈ Sy by (C2) and |c| > 1. + +Automatic Alignment in Higher-Order PPLs +47 +It follows that stoch ∈ Sx. If both derivations are instead (App), we repeat +the argument for (C3′) and get (R3′′)–(R5′′). Furthermore, we must have +v′ +1 = ρ′ +1(x) ̸ +V= ρ′ +2(x) = v′ +2. By Lemma 1, Sname(ty′ ) ⊆ Sx. The result now +follows from (R5′′). +We now apply the induction hypothesis and get (R3′)–(R5′). +(R3) First, by (R3′), we have l12| � +At = l22| � +At. We now show l11| � +At = l21| � +At. +Assume that stoch ∈ Sy. Then, in all cases we have l11| � +At = l21| � +At = [] +and the result follows. To see this, note first that for the (Const-App) +derivations, the result holds immediately. Therefore, assume both deriva- +tions are (App). Now, by Lemma 1, we have stoch ∈ Sy ⇒ (∀y′ λy′._ ∈ +Sy ⇒ unaligned ′ +y). In other words, unalignedy1 and unalignedy2. Again by +Lemma 1, unalignedn1 for all n1 ∈ names(ty1) and unaligned n2 for all +n2 ∈ names(ty2). Let ρ′′ +1 = ρy1, y1 �→ ρ1(z) and ρ′′ +2 = ρy2, y2 �→ ρ2(z) and +consider the derivations ρ′′ +1 ⊢ ty1 +s11⇓w11 +l11 v′ +1 and ρ′′ +2 ⊢ ty2 +s21⇓w21 +l21 v′ +2. It is +straightforward to check that ρ′′ +1 and ρ′′ +2 fulfill (C1) and double applications +of Lemma 2 give the required result l11| � +At = l21| � +At = []. +Now, assume that stoch ̸∈ Sy. Clearly, both derivations are either (Const- +App) or (App). The (Const-App) case is trivial, because l11| � +At = l21| � +At = []. +Therefore, assume both derivations are (App). By repeating the reasoning in +(C3′), we get (R3′′)–(R5′′) by the induction hypothesis for the derivations +ρ′′ +1 ⊢ ty′ s11⇓w11 +l11 v′ +1 and ρ′′ +2 ⊢ ty′ s21⇓w21 +l21 v′ +2. In other words, l11| � +At = l21| � +At +by (R3′′) and we are done. +Subcase t1 = if y then tt else te +The possible derivations are +ρ1 ⊢ y []⇓1 +[] true +ρ1 ⊢ tt +s11⇓w11 +l11 vt1 +ρ1 ⊢ if y then tt else te +s11⇓w11 +l11 vt1 +(If-True) +ρ2 ⊢ y []⇓1 +[] true +ρ2 ⊢ tt +s21⇓w21 +l21 vt2 +ρ2 ⊢ if y then tt else te +s21⇓w21 +l21 vt2 +(If-True) +ρ1 ⊢ y []⇓1 +[] false +ρ1 ⊢ te +s11⇓w11 +l11 ve1 +ρ1 ⊢ if y then tt else te +s11⇓w11 +l11 ve1 +(If-False) +ρ2 ⊢ y []⇓1 +[] false +ρ2 ⊢ te +s21⇓w21 +l21 ve2 +ρ2 ⊢ if y then tt else te +s21⇓w21 +l21 ve2 +(If-False) +We first establish (C2′)–(C4′). +(C2′) Holds in all four cases by repeating the corresponding argument for (C1′) +in Lemma 2. +(C3′) Assume stoch ̸∈ Sx. By Lemma 1, clearly stoch ̸∈ Sy and both deriva- +tions are either (If-True) or (If-False). Without loss of generality, assume +both derivations are (If-True). The induction hypothesis directly applies +to ρ1 ⊢ tt +s11⇓w11 +l11 vt1 and ρ2 ⊢ tt +s21⇓w21 +l21 vt2, and we get the result (R3t)– +(R5t). By Lemma 1, name(tt) ⊆ Sx. The result now follows from (R4t). + +48 +D. Lundén et al. +(C4′) Assume first that stoch ∈ Sy. Then stoch ∈ Sx by Lemma 1, and the +result is immediate. Therefore, assume stoch ̸∈ Sy. Again, both derivations +are either (If-True) or (If-False) and we assume, without loss of gener- +ality, that both are (If-True). The induction hypothesis directly applies to +ρ1 ⊢ tt +s11⇓w11 +l11 vt1 and ρ2 ⊢ tt +s21⇓w21 +l21 vt2, and we get the result (R3t)– +(R5t). By Lemma 1, name(tt) ⊆ Sx. The result now follows from (R5t). +We now apply the induction hypothesis and get (R3′)–(R5′). +(R3) First, by (R3′), we have l12| � +At = l22| � +At. We now show l11| � +At = l21| � +At. +If stoch ∈ Sy, then by Lemma 1, unaligned nt for all nt ∈ names(tt) and +unalignedne for all ne ∈ names(te). By repeating Lemma 2 twice, we get +l11| � +At = l21| � +At = [] and the result follows. +Assume stoch ̸∈ Sy. Again, both derivations are either (If-True) or (If- +False) and we assume, without loss of generality, that both are (If-True). +The induction hypothesis directly applies to ρ1 ⊢ tt +s11⇓w11 +l11 vt1 and ρ2 ⊢ +tt +s21⇓w21 +l21 vt2, and we get the result (R3t)–(R5t). By (R3t), l11| � +At = l21| � +At. +Subcase t1 = assume y +The derivations are +ρ1 ⊢ y []⇓1 +[] d1 +w1 = fd1(c1) +ρ1 ⊢ assume y [c1]⇓w1 +[] +c1 +(Assume) +ρ2 ⊢ y []⇓1 +[] d2 +w2 = fd2(c2) +ρ2 ⊢ assume y [c2]⇓w2 +[] +c2 +(Assume) +We first establish (C2′)–(C4′). +(C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for both +ρ′ +1(x) and ρ′ +2(x). +(C3′) Immediate as ρ′ +1(x) = c1 and ρ′ +2(x) = c2. +(C4′) By Lemma 1, stoch ∈ Sx. +We now apply the induction hypothesis and get (R3′)–(R5′). +(R3) The result follows from l11 = l21 = [] and (R3′). +Subcase t1 = weight y +The derivations are +ρ1 ⊢ y []⇓1 +[] w1 +ρ1 ⊢ weight y []⇓w1 +[] +() +(Weight) +ρ2 ⊢ y []⇓1 +[] w2 +ρ2 ⊢ weight y []⇓w2 +[] +() +(Weight) +We first establish (C2′)–(C4′). + +Automatic Alignment in Higher-Order PPLs +49 +Algorithm 5 Unaligned SMC. The input is a program t ∈ TANF and the number +of execution instances n. +1. Initiate n execution instances {ei | i ∈ N, 1 ≤ i ≤ n} of t. +2. Execute all ei (for already terminated ei, do nothing) and suspend execution upon +reaching a weight (i.e., let x = weight w in t) or when the execution terminates +naturally. The result is a new set of execution instances e′ +i with weights w′ +i (from +w, or 1 if already terminated). +3. If all e′ +i = v′ +i (i.e., all executions have terminated and returned a value), terminate +inference and return the set of samples v′ +i. The samples approximate the probability +distribution encoded by t. +4. Resample the e′ +i according to their weights w′ +i. The result is a new set of unweighted +execution instances e′′ +i . Set ei ← e′′ +i . Go to 2. +(C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for both +ρ′ +1(x) and ρ′ +2(x). +(C3′) Immediate as ρ′ +1(x) = ρ′ +2(x) = (). +(C4′) Immediate as ρ′ +1(x) +V= ρ′ +2(x). +We now apply the induction hypothesis and get (R3′)–(R5′). +(R3) The result follows from l11 = l21 = [] and (R3′). +⊓⊔ +C +Unaligned SMC +Algorithm 5 presents the unaligned SMC algorithm. It is in many ways similar +to Algorithm 2. +D +Lightweight MCMC +Algorithm 6 presents the lightweight MCMC algorithm. The algorithm is in +many ways similar to Algorithm 3, but relies on databases represented with +Di (random draws) and pi (probability densities/masses of the draws) to reuse +random draws. The Run function keeps track of the current stack trace t at all +times and uses it to index the databases. +E +Metropolis–Hastings Acceptance Ratio +This section derives the Metropolis–Hastings acceptance ratio used in Algo- +rithm 3 and Algorithm 6. We assume basic familiarity with Bayesian statistics +and the Metropolis–Hastings algorithm. +Bayes’ theorem on probability density/mass functions is usually written as +p(x|y) = p(y|x)p(x) +p(y) +(10) + +50 +D. Lundén et al. +Algorithm 6 Lightweight MCMC. The input is a program t ∈ TANF, the num- +ber of steps n, and the global step probability g > 0. +1. Set i ← 0. Call Run. +2. Set i ← i + 1. If i = n, terminate inference and return the samples {vj | j ∈ N, 0 ≤ +j < n}. They approximate the probability distribution encoded by t. +3. Uniformly draw a trace t′ from dom(Di−1) at random. Set global ← true with +probability g, and global ← false otherwise. Set w′ +−1 ← 1, and w′ ← 1. Call Run. +4. Compute the Metropolis–Hastings acceptance ratio +A = min +� +1, +wi +wi−1 +w′ +w′ +−1 +|dom(Di−1)| +|dom(Di)| +� +. +(9) +5. With probability A, accept vi and go to 2. Otherwise, set vi ← vi−1, wi ← wi−1, +Di ← Di−1, and pi ← pi−1. Go to 2. +function run() = Let t represent the current stack trace throughout execution. Run t +and do the following: +– Record the total weight wi accumulated from calls to weight. +– Record the final value vi. +– At terms let c = assume d in t, do the following. +1. If t = t′, global = true, or if t ̸∈ dom(Di−1), sample a value x from d. +Otherwise, reuse the sample x = Di−1(t) and set w′ +−1 ← w′ +−1 · pi−1(t) and +w′ ← w′ · fd(c). +2. Set Di(t) ← x and pi(t) ← fd(x). +3. In the program, bind c to the value x and resume execution. +where y is some fixed observed random variable. The standard Metropolis– +Hastings ratio for a proposal distribution with probability density/mass q(x′|x) +is then +A(x, x′) = min +� +1, p(x′|y) +p(x|y) +q(x|x′) +q(x′|x) +� += min +� +1, p(y|x′)p(x′) +p(y|x)p(x) +q(x|x′) +q(x′|x) +� +. +(11) +Assume a fixed program t ∈ T in the remainder of this section. For such a +program, Bayes’ theorem takes a generalized form +ˆp(s) = L(s)p(s) +Z +. +(12) +Here, we have replaced x with a trace s (a sequence of random values during +evaluation of a probabilistic program) and removed the dependence on y entirely. +We use the notation ˆp and p to differentiate between the posterior and prior. The +likelihood function is denoted L. Z is a normalizing constant that disappears in +the Metropolis–Hastings ratio. +One can view (12) in the context of the semantics in Fig. 3. We (very infor- +mally) have ˆp(s) = w up to normalization iff ∅ ⊢ t l⇓w +s v for some l and v. L(s) +is then the contribution to w from (Weight), and p(s) from (Assume). The +PPL version of the Metropolis–Hastings ratio is +A(s, s′) = min +� +1, ˆp(s′) +ˆp(s) +q(s|s′) +q(s′|s) +� += min +� +1, L(s′)p(s′) +L(s)p(s) +q(s|s′) +q(s′|s) +� +. +(13) + +Automatic Alignment in Higher-Order PPLs +51 +The most trivial proposal, amounting to not reusing any draws, is +q(s′|s) = p(s′) +(14) +This directly gives the ratio +A(s, s′) = min +� +1, L(s′) +L(s) +� +. +(15) +To derive the ratio for aligned lightweight MCMC and lightweight MCMC, we +need to first capture the proposal q used in Algorithm 3 and Algorithm 6. We +capture the reuse mechanisms (alignment and the stack trace database) in both +algorithms through functions D1 : S × S → P(N) and D2 : S × S → P(N) such +that |D1(s, s′)| = |D2(s, s′)|, D1(s, s′) = D2(s′, s), and D2(s, s′) = D1(s′, s). +Intuitively, D1(s, s′) gives the indices in s that match the indices D2(s, s′) in s′. +We now define the proposal q as +q(s′, i|s) = [s′|A′ = s|A]p|A′C(s′)pi(i|s) +(16) +and make the following definitions. +– A′ = D2(s, s′) \ f(i, s′) and A = D1(s, s′) \ f(i, s). +– The function f(i, s) transforms the index i in the context of s (explained +further below). +– The function pi(i|s) is the density for selecting an i given s. +– The trace s|A is the restriction of s to A (cf. Definition 5). +– [· · · ] is the Iverson bracket (i.e., evaluates to zero if the predicate · · · is false +and to one if the predicate is true). +– We denote the contribution to p(s) from the indices A in s with p|A(s). +Importantly, p|A(s) · p|AC(s) = p(s). +Note that q also proposes an auxiliary variable i—the trace index that we choose +to redraw in the proposal. Due to the auxiliary variable i, the acceptance ratio +is now a function of three arguments. +A(s, s′, i) = min +� +1, L(s′)p(s′) +L(s)p(s) +q(s, i|s′) +q(s′, i|s) +� += min +� +1, L(s′)p(s′) +L(s)p(s) +[s|A = s′|A′] +[s′|A′ = s|A] +p|AC(s) +p|A′C(s′) +pi(i|s′) +pi(i|s) +� += min +� +1, L(s′) +L(s) +p(s′) +p|A′C(s′) +p|AC(s) +p(s) +pi(i|s′) +pi(i|s) +� += min +� +1, L(s′) +L(s) +p|A′(s′) +p|A(s) +pi(i|s′) +pi(i|s) +� +(17) +This acceptance ratio is equivalent to the ratio derived by van de Meent et al. [45, +Equation 4.21]. +We first view (17) in the context of aligned lightweight MCMC in Algo- +rithm 3. Here, f(i, s) returns the i-th aligned index in s (not index i in s). In + +52 +D. Lundén et al. +aligned lightweight MCMC, we only select what to redraw among the aligned +draws. As we know, the number of aligned draws is fixed across all possible +executions. pi(i|s) is thus a constant, and (17) reduces to +A(s, s′, i) = min +� +1, L(s′) +L(s) +p|A′(s′) +p|A(s) +� +(18) +This is the ratio computed in step 5 of Algorithm 3. +Next, we consider lightweight MCMC in Algorithm 6. Here, we simply choose +f(i, s) = i (the identity function), and select an element to redraw uniformly over +the previous trace s. Thus, pi(i|s) = 1/|s| and (17) reduces to +A(s, s′, i) = min +� +1, L(s′) +L(s) +p|A′(s′) +p|A(s) +|s| +|s′| +� +(19) +This is the ratio computed in step 4 of Algorithm 6. + diff --git a/V9FJT4oBgHgl3EQf4C1z/content/tmp_files/load_file.txt b/V9FJT4oBgHgl3EQf4C1z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b36285dc0226f8bb8d88bb06136f5ad03dbbddf2 --- /dev/null +++ b/V9FJT4oBgHgl3EQf4C1z/content/tmp_files/load_file.txt @@ -0,0 +1,2291 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf,len=2290 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='11664v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='PL] 27 Jan 2023 Automatic Alignment in Higher-Order Probabilistic Programming Languages⋆ Daniel Lundén1(�) , Gizem Çaylak1 , Fredrik Ronquist2,3 , and David Broman1 1 EECS and Digital Futures, KTH Royal Institute of Technology, Stockholm, Sweden, {dlunde,caylak,dbro}@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='se 2 Department of Bioinformatics and Genetics, Swedish Museum of Natural History, Stockholm, Sweden, fredrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ronquist@nrm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='se 3 Department of Zoology, Stockholm University, Stockholm, Sweden Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Probabilistic Programming Languages (PPLs) allow users to encode statistical inference problems and automatically apply an infer- ence algorithm to solve them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Popular inference algorithms for PPLs, such as sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC), are built around checkpoints—relevant events for the inference algorithm during the execution of a probabilistic program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Deciding the location of checkpoints is, in current PPLs, not done optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To solve this problem, we present a static analysis technique that automatically determines checkpoints in programs, relieving PPL users of this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The analysis identifies a set of checkpoints that execute in the same order in every program run—they are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We formalize alignment, prove the correctness of the analysis, and implement the analysis as part of the higher-order functional PPL Miking CorePPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By utilizing the align- ment analysis, we design two novel inference algorithm variants: aligned SMC and aligned lightweight MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We show, through real-world ex- periments, that they significantly improve inference execution time and accuracy compared to standard PPL versions of SMC and MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Keywords: Probabilistic programming · Operational semantics · Static analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1 Introduction Probabilistic programming languages (PPLs) are languages used to encode sta- tistical inference problems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' common in research fields such as phylogenetics [38],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' ⋆ This project is financially supported by the Swedish Foundation for Strategic Re- search (FFL15-0032 and RIT15-0012),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' and also partially supported by the Wallen- berg Al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' and the Swedish Research Council (grants 2018- 04620 and 2021-04830).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The research has also been carried out as part of the Vinnova Competence Center for Trustworthy Edge Computing Systems and Applications at KTH Royal Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' computer vision [16], topic modeling [5], data cleaning [23], and cognitive sci- ence [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' PPL implementations automatically solve encoded problems by ap- plying an inference algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In particular, automatic inference allows users to solve inference problems without having in-depth knowledge of inference al- gorithms and how to apply them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Some examples of PPLs are WebPPL [14], Birch [30], Anglican [47], Miking CorePPL [25], Turing [12], and Pyro [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) are general-purpose families of inference algorithms often used for PPL implemen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' These algorithms share the concept of checkpoints: relevant execution events for the inference algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For SMC, the checkpoints are likelihood up- dates [47,14] and determine the resampling of executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Alternatively, users must sometimes manually annotate or write the probabilistic program in a cer- tain way to make resampling explicit [25,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For MCMC, checkpoints are instead random draws, which allow the inference algorithm to manipulate these draws to construct a Markov chain over program executions [46,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' When designing SMC and MCMC algorithms for universal PPLs4, both the placement and handling of checkpoints are critical to making the inference both efficient and accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For SMC, a standard inference approach is to resample at all likelihood updates [14,47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This approach produces correct results asymptotically [24] but is highly problematic for certain models [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Such models require non-trivial and SMC-specific manual program rewrites to force good resampling locations and make SMC tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Overall, choosing the likelihood updates at which to resample significantly affects SMC execution time and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For MCMC, a standard approach for inference in universal PPLs is lightweight MCMC [46], which constructs a Markov chain over random draws in programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The key idea is to use an addressing transformation and a runtime database of random draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Specifically, the database enables matching and reusing random draws between executions according to their stack traces, even if the random draws may or may not occur due to randomness during execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' However, the dynamic approach of looking up random draws in the database through their stack traces is expensive and introduces significant runtime overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To overcome the SMC and MCMC problems in universal PPLs, we present a static analysis technique for higher-order functional PPLs that automatically determines checkpoints in a probabilistic program that always occur in the same order in every program execution—they are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We formally define align- ment, formalize the alignment analysis, and prove the soundness of the analysis with respect to the alignment definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The novelty and challenge in developing the static analysis technique is to capture alignment properties through the iden- tification of expressions in programs that may evaluate to stochastic values and expressions that may evaluate due to stochastic branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Stochastic branching results from if expressions with stochastic values as conditions or function ap- plications where the function itself is stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Stochastic values and branches pose a significant challenge when proving the soundness of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4 A term coined by Goodman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Essentially, it means that the types and numbers of random variables cannot be determined statically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 3 We design two new inference algorithms that improve accuracy and execu- tion time compared to current approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Unlike the standard SMC algorithm for PPLs [47,14], aligned SMC only resamples at aligned likelihood updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Re- sampling only at aligned likelihood updates guarantees that each SMC execution resamples the same number of times, which makes expensive global termination checks redundant [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We evaluate aligned SMC on two diversification models from Ronquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [38] and a state-space model for aircraft localization, demon- strating significantly improved inference accuracy and execution time compared to traditional SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Both models—constant rate birth-death (CRBD) and clado- genetic diversification rate shift (ClaDS)—are used in real-world settings and are of considerable interest to evolutionary biologists [32,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The documentations of both Anglican [47] and Turing [12] acknowledge the importance of alignment for SMC and state that all likelihood updates must be aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' However, Turing and Anglican neither formalize nor enforce this property—it is up to the users to manually guarantee it, often requiring non-standard program rewrites [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We also design aligned lightweight MCMC, a new version of lightweight MCMC [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned lightweight MCMC constructs a Markov chain over the program using the aligned random draws as synchronization points to match and reuse aligned random draws and a subset of unaligned draws between execu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned lightweight MCMC does not require a runtime database of random draws and therefore reduces runtime overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We evaluate aligned lightweight MCMC for latent Dirichlet allocation (LDA) [5] and CRBD [38], demonstrat- ing significantly reduced execution times and no decrease in inference accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, automatic alignment is orthogonal to and easily combines with the lightweight MCMC optimizations introduced by Ritchie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We implement the analysis, aligned SMC, and aligned lightweight MCMC in Miking CorePPL [25,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In addition to analyzing stochastic if-branching, the implementation analyzes stochastic branching at a standard pattern-matching construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Compared to if expressions, the pattern-matching construct requires a more sophisticated analysis of the pattern and the value matched against it to determine if the pattern-matching causes a stochastic branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In summary, we make the following contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – We invent and formalize alignment for PPLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned parts of a program occur in the same order in every execution (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – We formalize and prove the soundness of a novel static analysis technique that determines stochastic value flow and stochastic branching, and in turn alignment, in higher-order probabilistic programs (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – We design aligned SMC inference that only resamples at aligned likelihood updates, improving execution time and inference accuracy (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – We design aligned lightweight MCMC inference that only reuses aligned random draws, improving execution time (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – We implement the analysis and inference algorithms in Miking CorePPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The implementation extends the alignment analysis to identify stochastic branching resulting from pattern matching (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Section 7 describes the evaluation and discusses its results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The paper also has an accompanying artifact that supports the evaluation [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Section 8 discusses related work and Section 9 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Next, Section 2 considers a simple mo- tivating example to illustrate the key ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Section 3 introduces syntax and semantics for the calculus used to formalize the alignment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2 A Motivating Example This section presents a motivating example that illustrates the key alignment ideas in relation to aligned SMC (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1) and aligned lightweight MCMC (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We assume basic knowledge of probability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Knowledge of PPLs is helpful, but not a strict requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The book by van de Meent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [45] provides a good introduction to PPLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Probabilistic programs encode Bayesian statistical inference problems with two fundamental constructs: assume and weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The assume construct defines random variables, which make execution nondeterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Intuitively, a proba- bilistic program then encodes a probability distribution over program executions (the prior distribution), and it is possible to sample from this distribution by executing the program with random sampling at assumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The weight construct updates the likelihood of individual executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Updating likelihoods for execu- tions modifies the probability distribution induced by assumes, and the inference problem encoded by the program is to determine or approximate this modified distribution (the posterior distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The main purpose of weight in real- world models is to condition executions on observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 Consider the probabilistic program in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The program is contrived and purposefully constructed to compactly illustrate alignment, but the real- world diversification models in Ronquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [38] that we also consider in Section 7 inspired the program’s general structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The program defines (line 1) and returns (line 18) a Gamma-distributed random variable rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Figure 1b illustrates the Gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To modify the likelihood for values of rate, the program executes the iter function (line 10) three times, and the survives function (line 2) a random number of times n (line 13) within each iter call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Conceptually, to infer the posterior distribution of the program, we execute the program infinitely many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In each execution, we draw samples for the random variables defined at assume, and accumulate the likelihood at weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The return value of the execution, weighted by the accumulated likelihood, rep- resents one sample from the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1c shows a histogram of such weighted samples of rate resulting from a large number of executions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The fundamental inference algorithm that produces such weighted samples is called likelihood weighting (a type of importance sampling [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We see that, compared to the prior distribution for rate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1b, the posterior is more sharply peaked due to the likelihood modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5 A number of more specialized constructs for likelihood updating are also available in various PPLs, for example observe [47,14] and condition [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 5 1 let rate = assume Gamma(2, 2) in 2 let rec survives = λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3 if n = 0 then () else 4 if assume Bernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='9) then 5 weight 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 6 survives (n − 1) 7 else 8 weight 0 9 in 10 let rec iter = λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 11 if i = 0 then () else 12 weight rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 13 let n = assume Poisson(rate) in 14 survives n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 15 iter (i − 1) 16 in 17 iter 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 18 rate (a) Probabilistic program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 0 5 10 15 (b) Gamma(2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 0 5 10 15 (c) Histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' w1 12 12 5 5 5 12 5 5 w2 12 5 5 12 8 12 5 w1 12 12 5 5 5 12 5 5 w2 12 5 5 12 8 12 5 (d) Aligning weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' s1 1 13 4 13 4 4 4 13 s2 1 13 4 4 13 4 4 13 4 s1 1 13 4 13 4 4 4 13 s2 1 13 4 4 13 4 4 13 4 (e) Aligning assume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1: A simple example illustrating alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (a) gives a probabilis- tic program using functional-style PPL pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (b) illustrates the Gamma(2, 2) probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (c) illustrates a histogram over weighted rate samples produced by running the program in (a) a large num- ber of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (d) shows two line number sequences w1 and w2 of weights encountered in two program runs (top) and how to align them (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (e) shows two line number sequences s1 and s2 of assumes encountered in two program runs (top) and how to align them (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 Aligned SMC Likelihood weighting can only handle the simplest of programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1a, a problem with likelihood weighting is that we assign the weight 0 to many exe- cutions at line 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' These executions contribute nothing to the final distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' SMC solves this by executing many program instances concurrently and occa- sionally resampling them (with replacement) based on their current likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Resampling discards executions with lower weights (in the worst case, 0) and re- places them with executions with higher weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The most common approach in popular PPLs is to resample just after likelihood updates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', calls to weight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Resampling at all calls to weight in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1a is suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The best option is instead to only resample at line 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This is because executions encounter lines 5 and 8 a random number of times due to the stochastic branch at line 3, while they encounter line 12 a fixed number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' As a result of resampling at lines 5 and 8, executions become unaligned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in each resampling, executions can have reached either line 5, line 8, or line 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' On the other hand, if we resample only at line 12, all executions will always have reached line 12 for the same iteration of iter in every resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Intuitively, this is a sensible approach since, when resampling, executions have progressed the same distance through the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We say that the weight at line 12 is aligned, and resampling only at aligned weights results in our new inference approach called aligned SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1d visualizes the weight alignment for two sample executions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 Aligned Lightweight MCMC Another improvement over likelihood weighting is to construct a Markov chain over program executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' It is beneficial to propose new executions in the Markov chain by making small, rather than large, modifications to the previous execu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The lightweight MCMC [46] algorithm does this by redrawing a single random draw in the previous execution, and then reusing as many other ran- dom draws as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Random draws in the current and previous executions match through stack traces—the sequences of applications leading up to a ran- dom draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Consider the random draw at line 13 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' It is called exactly three times in every execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If we identify applications and assumes by line numbers, we get the stack traces [17, 13], [17, 15, 13], and [17, 15, 15, 13] for these three assumes in every execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Consequently, lightweight MCMC can reuse these draws by storing them in a database indexed by stack traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The stack trace indexing in lightweight MCMC is overly complicated when reusing aligned random draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that the assumes at lines 1 and 13 in Fig 1a are aligned, while the assume at line 4 is unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1e visualizes the assume alignment for two sample executions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned random draws occur in the same same order in every execution, and are therefore trivial to match and reuse between executions through indexing by counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The appeal with stack trace indexing is to additionally allow reusing a subset of unaligned draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A key insight in this paper is that aligned random draws can also act as synchronization points in the program to allow reusing unaligned draws without a stack trace database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' After an aligned draw, we reuse unaligned draws occurring up until the next aligned draw, as long as they syntactically originate at the same assume as the corresponding unaligned draws in the previous execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' As soon as an unaligned draw does not originate from the same assume as in the previous execution, we redraw all remaining unaligned draws up until the next aligned draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Instead of a trace-indexed database, this approach requires storing a list of unaligned draws (tagged with identifiers of the assumes at which they originated) for each execution segment in between aligned random draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, for the execution s1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1e, we store lists of unaligned Bernoulli random draws from line 4 for each execution segment in between the three aligned random draws at line 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If a Poisson random draw n at line 13 does not change or decreases, we can reuse the stored unaligned Bernoulli draws up until the next Poisson random draw as survives executes n or fewer times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If the drawn n instead increases to n′, we can again reuse all stored Bernoulli draws, but must supplement them with new Bernoulli draws to reach n′ draws in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' As we show in Section 7, using aligned draws as synchronization points works very well in practice and avoids the runtime overhead of the lightweight MCMC database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' However, manually identifying aligned parts of programs and rewrit- ing them so that inference can make use of alignment is, if even possible, te- dious, error-prone, and impractical for large programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This paper presents an automated approach to identifying aligned parts of programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Combining static alignment analysis and using aligned random draws as synchronization points form the key ideas of the new algorithm that we call aligned lightweight MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 7 3 Syntax and Semantics In preparation for the alignment analysis in Section 4, we require an idealized base calculus capturing the key features of expressive PPLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This section intro- duces such a calculus with a formal syntax (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1) and semantics (Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We assume a basic understanding of the lambda calculus (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Pierce [36] for a complete introduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Section 6 further describes extending the idealized calculus and the analysis in Section 4 to a full-featured PPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 Syntax We use the untyped lambda calculus as the base for our calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We also add let expressions for convenience, and if expressions to allow intrinsic booleans to affect control flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The calculus is a subset of the language used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We inductively define terms t and values v as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Definition 1 (Terms and values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' t ::= x | c | λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' t | t t | let x = t in t v ::= c | ⟨λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' t, ρ⟩ | if t then t else t | assume t | weight t x, y ∈ X ρ ∈ P c ∈ C {false, true, ()} ∪ R ∪ D ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (1) X is a countable set of variable names, C a set of intrinsic values and operations, and D ⊂ C a set of probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The set P contains all evaluation environments ρ, that is, partial functions mapping names in X to values v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We use T and V to denote the set of all terms and values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Values v are intrinsics or closures, where closures are abstractions with an en- vironment binding free variables in the abstraction body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We require that C include booleans, the unit value (), and real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The reason is that weight takes real numbers as argument and returns () and that if expression conditions are booleans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, probability distributions are often over booleans and real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, we can include the normal distribution constructor N ∈ C that takes real numbers as arguments and produces normal distributions over real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, N 0 1 ∈ D, the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We often write functions in C in infix position or with standard function appli- cation syntax for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, 1 + 2 with + ∈ C means + 1 2, and N(0, 1) means N 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Additionally, we use the shorthand t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' t2 for let _ = t1 in t2, where _ is the do-not-care symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' That is, t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' t2 evaluates t1 for side effects only before evaluating t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Finally, the untyped lambda calculus supports recursion through fixed-point combinators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We encapsulate this in the shorthand let rec f = λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t1 in t2 to conveniently define recursive functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The assume and weight constructs are PPL-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We define random vari- ables from intrinsic probability distributions with assume (also known as sam- ple in PPLs with sampling-based inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, the term let x = assume N(0, 1) in t defines x as a random variable with a standard normal distribution in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Boolean random variables combined with if expressions result 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1 let rec geometric = λ_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2 let x = assume Bernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5) in 3 if x then 4 weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5 1 + geometric () 6 else 1 7 in geometric () (a) Probabilistic program tgeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Standard geometric 1 2 3 4 5 6 7 8 9 Weighted geometric (b) Probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2: A probabilistic program tgeo [25], illustrating (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (a) gives the pro- gram, and (b) the corresponding probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In (b), the y-axis gives the probability, and the x-axis gives the outcome (the number of coin flips).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The upper part of (b) excludes the shaded weight at line 4 in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in stochastic branching—causing the alignment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lastly, weight (also known as factor or score) is a standard construct for likelihood updating (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Borgström et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Next, we illustrate and formalize a semantics for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 Semantics Consider the small probabilistic program tgeo ∈ T in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The program encodes the standard geometric distribution via a function geometric, which recursively flips a fair coin (a Bernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5) distribution) at line 2 until the outcome is false (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' At that point, the program returns the total number of coin flips, including the last tails flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2b illustrates the result distribution for an infinite number of program runs with line 4 ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To illustrate the effect of weight, consider tgeo with line 4 included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This weight modifies the likelihood with a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 each time the flip outcome is true (or, heads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Intuitively, this emphasizes larger return values, illustrated in the lower part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Specifically, the (unnormalized) probability of seeing n coin flips is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5n · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5n−1, compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5n for the unweighted version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5n−1 is the result of the calls to weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now introduce a big-step operational semantics for single runs of programs t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Such a semantics is essential to formalize the probability distributions encoded by probabilistic programs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2b for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2a) and to prove the correctness of PPL inference algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, Borgström et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [6] define a PPL calculus and semantics similar to this paper and formally proves the correctness of an MCMC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Another example is Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [24], who also define a similar calculus and semantics and prove the correctness of PPL SMC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In particular, the correctness of our aligned SMC algorithm (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1) follows from this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The purpose of the semantics in this paper is to formalize alignment and prove the soundness of our analysis in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We use a big- step semantics as the finer granularity in a small-step semantics is redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We begin with a definition for intrinsics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 9 ρ ⊢ x []⇓1 [] ρ(x) (Var) ρ ⊢ c []⇓1 [] c (Const) ρ ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t []⇓1 [] ⟨λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t, ρ⟩ (Lam) ρ ⊢ t1 s1⇓w1 l1 ⟨λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' ρ′⟩ ρ ⊢ t2 s2⇓w2 l2 v2 ρ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' x �→ v2 ⊢ t s3⇓w3 l3 v ρ ⊢ t1 t2 s1∥s2∥s3⇓w1·w2·w3 l1∥l2∥l3 v (App) ρ ⊢ t1 s1⇓w1 l1 c1 ρ ⊢ t2 s2⇓w2 l2 c2 ρ ⊢ t1 t2 s1∥s2⇓w1·w2 l1∥l2 δ(c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' c2) (Const-App) ρ ⊢ t s⇓w l d w′ = fd(c) ρ ⊢ assume t s∥[c]⇓w·w′ l c (Assume) ρ ⊢ t1 s1⇓w1 l1 v1 ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' x �→ v1 ⊢ t2 s2⇓w2 l2 v ρ ⊢ let x = t1 in t2 s1∥s2⇓w1·w2 l1∥[x]∥l2 v (Let) ρ ⊢ t s⇓w l w′ ρ ⊢ weight t s⇓w·w′ l () (Weight) ρ ⊢ t1 s1⇓w1 l1 true ρ ⊢ t2 s2⇓w2 l2 v2 ρ ⊢ if t1 then t2 else t3 s1∥s2⇓w1·w2 l1∥l2 v2 (If-True) ρ ⊢ t1 s1⇓w1 l1 false ρ ⊢ t3 s3⇓w3 l3 v3 ρ ⊢ if t1 then t2 else t3 s1∥s3⇓w1·w3 l1∥l3 v3 (If-False) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3: A big-step operational semantics for terms, formalizing single runs of pro- grams t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The operation ρ, x �→ v produces a new environment extending ρ with a binding v for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For each distribution d ∈ D, fd is its probability density or probability mass function—encoding the relative probability of drawing par- ticular values from the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, fBernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3)(true) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3 and fBernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3)(false) = 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We use · to denote multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Definition 2 (Intrinsic functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For every c ∈ C, we attach an arity |c| ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We define a partial function δ : C × C → C such that δ(c, c1) = c2 is defined for |c| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For all c, c1, and c2, such that δ(c, c1) = c2, |c2| = |c| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Intrinsic functions are curried and produce intrinsic or intrinsic functions of one arity less through δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, for + ∈ C, we have δ(δ(+, 1), 2) = 3, |+| = 2, |δ(+, 1)| = 1, and |δ(δ(+, 1), 2)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Next, randomness in our semantics is deterministic via a trace of random draws in the style of Kozen [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Definition 3 (Traces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The set S of traces is the set such that, for all s ∈ S, s is a sequence of intrinsics from C with arity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In the following, we use the notation [c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' , cn] for sequences and ∥ for sequence concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, [c1, c2] ∥ [c2, c4] = [c1, c2, c3, c4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We also use subscripts to select elements in a sequence, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', [c1, c2, c3, c4]2 = c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In practice, traces are often sequences of real numbers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4] ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3 presents the semantics as a relation ρ ⊢ t s⇓w l v over P × T × S × R × L × V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' L is the set of sequences over X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', sequences of names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, [x, y, z] ∈ L, where x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We use l ∈ L to track the sequence of let- bindings during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, evaluating let x = 1 in let y = 2 in x + y results in l = [x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In Section 4, we use the sequence of encountered let-bindings to define alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For simplicity, from now on we assume that bound variables are always unique (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', variable shadowing is impossible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 10 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' It is helpful to think of ρ, t, and s as the input to ⇓, and l, w and v as the out- put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In the environment ρ, t, with trace s, evaluates to v, encounters the sequence of let bindings l, and accumulates the weight w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The trace s is the sequence of all random draws, and each random draw in (Assume) consumes precisely one element of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The rule (Let) tracks the sequence of bindings by adding x at the correct position in l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The number w is the likelihood of the execution—the prob- ability density of all draws in the program, registered at (Assume), combined with direct likelihood modifications, registered at (Weight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The remaining as- pects of the semantics are standard (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Kahn [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To give an example of the semantics, we have ∅ ⊢ tgeo [true,true,true,false]⇓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5·1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5·0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5·1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5·0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5·1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5·0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 [geometric,x,x,x,x] 4 (2) for the particular execution of tgeo making three recursive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Next, we for- malize and apply the alignment analysis to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4 Alignment Analysis This section presents the main contribution of this paper: automatic alignment in PPLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 introduces A-normal form and gives a precise definition of alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 formalizes and proves the correctness of the alignment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lastly, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3 discusses a dynamic version of alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 A-Normal Form and Alignment To reason about all subterms t′ of a program t and to enable the analysis in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2, we need to uniquely label all subterms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A straightforward approach is to use variable names within the program itself as labels (remember that we assume bound variables are always unique).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This leads us to the standard A-normal form (ANF) representation of programs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Definition 4 (A-normal form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' tANF ::= x | let x = t′ ANF in tANF t′ ANF ::= x | c | λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' tANF | x y | if x then tANF else tANF | assume x | weight x (3) We use TANF to denote the set of all terms tANF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Unlike t ∈ T , tANF ∈ TANF enforces that a variable bound by a let labels each subterm in the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, we can automatically transform any program in T to a semantically equivalent TANF program, and TANF ⊂ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Therefore, we assume in the remainder of the paper that all terms are in ANF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Given the importance of alignment in universal PPLs, it is somewhat surpris- ing that there are no previous attempts to give a formal definition of its meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Here, we give a first such formal definition, but before defining alignment, we require a way to restrict, or filter, sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 11 Definition 5 (Restriction of sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For all l ∈ L and Y ⊆ X, l|Y (the restriction of l to Y ) is the subsequence of l with all elements not in Y removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, [x, y, z, y, x]|{x,z} = [x, z, x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now formally define alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Definition 6 (Alignment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For t ∈ TANF, the set At ⊆ X is the largest set such that, for arbitrary ∅ ⊢ t s1⇓w1 l1 v1 and ∅ ⊢ t s2⇓w2 l2 v2, l1|At = l2|At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The aligned expressions—the expressions in a program bound by a let to a variable name in At—are those that occur in the same order in every program execution, regardless of random draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that we seek the largest set, as At = ∅ is always a trivial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Assume we have a program such that l = [x, y, x, z, x] or l = [x, y, x, z, x, y] are the only possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Then, At = {x, z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Next, assume that we transform the programs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1a to ANF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The expression labeled by x in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2a is then clearly not aligned, as random draws determine how many times it executes (l could be, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', [x, x] or [x, x, x, x]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' On the other hand, the expression n (line 13) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1a is aligned, as its number and order of evaluations do not depend on any random draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Definition 6 is context insensitive: for each name x in the program, the ex- pression bound by x is either aligned or unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' One could also consider a context-sensitive definition of alignment in which x can be aligned in some con- texts and unaligned in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A context could, for example, be the sequence of function applications (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', the call stack) leading up to an expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Consider- ing different contexts for x is complicated and difficult to take full advantage of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We justify the choice of context-insensitive alignment with the real-world models in Section 7, neither of which requires a context-sensitive alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' With alignment defined, we now move on to the static alignment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 Alignment Analysis The basis for the alignment analysis is 0-CFA [33,41]—a static analysis frame- work for higher-order functional programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The prefix 0 indicates that 0-CFA is context insensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' There is also a set of analyses k-CFA [29] that adds increas- ing amounts (with k ∈ N) of context sensitivity to 0-CFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We could use such analyses with a context-sensitive version of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' However, the potential benefit of k-CFA is also offset by the worst-case exponential time complexity, already at k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In contrast, the time complexity of 0-CFA is polynomial (cu- bic in the worst-case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The alignment analysis for the models in Section 7 runs instantaneously, justifying that the time complexity is not a problem in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The extensions to 0-CFA required to analyze alignment are non-trivial to design, but the resulting formalization is surprisingly simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The challenge is instead to prove that the extensions correctly capture the alignment property from Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We extend 0-CFA to analyze stochastic values and alignment in programs t ∈ TANF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' As with most static analyses, our analysis is sound but conservative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', sound but incomplete)—the analysis may mark aligned expressions of programs as unaligned, but not vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' That the analysis is conservative does not degrade the alignment analysis results for any model in 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1 let n1 = ¬ in let n2 = ¬ in 2 let one = 1 in 3 let half = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 in let c = true in 4 let f1 = λx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' let t1 = weight one in x1 in 5 let f2 = λx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' let t2 = weight one in t2 in 6 let f3 = λx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' let t3 = weight one in t3 in 7 let f4 = λx4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' let t4 = weight one in t4 in 8 let bern = Bernoulli in 9 let d1 = bern half in 10 let a1 = assume d1 11 let v1 = f1 one in 12 let v2 = n1 a1 in 13 let v3 = n2 c in 14 let f5 = 15 if a1 then let t5 = f4 one in f2 16 else f3 17 in 18 let v4 = f5 one in 19 let i1 = 20 if c then let t6 = f1 one in t6 21 else one 22 in i1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4: A program texample ∈ TANF illustrating the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Section 7, which justifies the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We divide the formal analysis into two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Algorithm 1 generates constraints for t that a valid analysis solution must satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This section describes Algorithm 1 and the generated constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 provides the second algorithm, Algorithm 4, that computes a solution satisfying the generated constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We provide examples of applying Algorithm 4 here, but defer the complete description to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For soundness of the analysis, we require ⟨λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' t, ρ⟩ ̸∈ C (recall that C is the set of intrinsics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' That is, closures are not in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By Definition 3, this im- plies that closures are not in the sample space of probability distributions in D and that evaluating intrinsics never produces closures (this would unnecessarily complicate the analysis without any benefit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In addition to standard 0-CFA constraints, Algorithm 1 generates new con- straints for stochastic values and unalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We use the contrived but illus- trative program in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4 as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that, while omitted from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4 for ease of presentation, the analysis also supports recursion introduced through let rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Stochastic values are values in the program affected by random vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Stochastic values initially originate at assume and then propagate through programs via function applications and if expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, a1 (line 10) is stochastic because of assume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We subsequently use a1 to define v2 via n1 (line 12), which is then also stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Similarly, a1 is the condition for the if resulting in f5 (line 14), and the function f5 is therefore also stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' When we apply f5, it results in yet another stochastic value, v4 (line 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In conclusion, the stochastic values are a1, v2, f5, and v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Consider the flow of unalignment in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We mark expressions that may execute due to stochastic branching as unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' From our analysis of stochastic values, the program’s only stochastic if condition is at line 15, and we determine that all expressions directly within the branches are unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' That is, the expression labeled by t5 is unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, we apply the variable f4 when defining t5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Thus, all expressions in bodies of lambdas that flow to f4 are unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Here, it implies that t4 is unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Finally, we established that the function f5 produced at line 15 is stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Due to the application at line 18, all names bound by lets in bodies of lambdas that flow to f5 are unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Here, it implies that t2 and t3 are unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In conclusion, the unaligned expressions Automatic Alignment in Higher-Order PPLs 13 Algorithm 1 Constraint generation function for t ∈ TANF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We denote the power set of a set E with P(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' function generateConstraints(t): TANF → P(R) = 1 match t with 2 | x → ∅ 3 | let x = t1 in t2 → 4 generateConstraints(t2) ∪ 5 match t1 with 6 | y → {Sy ⊆ Sx} 7 | c → if |c| > 0 then {const |c| ∈ Sx} 8 else ∅ 9 | λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' ty → generateConstraints(ty) 10 ∪ {λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' name(ty) ∈ Sx} 11 ∪ {unalignedy ⇒ unalignedn 12 | n ∈ names(ty)} 13 | lhs rhs → { 14 ∀z∀y λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='y ∈ Slhs 15 ⇒ (Srhs ⊆ Sz) ∧ (Sy ⊆ Sx), 16 ∀n (const n ∈ Slhs) ∧ (n > 1) 17 ⇒ const n − 1 ∈ Sx, 18 stoch ∈ Slhs ⇒ stoch ∈ Sx, 19 const _ ∈ Slhs 20 ⇒ (stoch ∈ Srhs ⇒ stoch ∈ Sx), 21 unalignedx 22 ⇒ (∀y λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedy), 23 stoch ∈ Slhs 24 ⇒ (∀y λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedy) 25 } 26 | if y then tt else te → 27 generateConstraints(tt) 28 ∪ generateConstraints(te) 29 ∪ {Sname(tt) ⊆ Sx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Sname(te) ⊆ Sx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 30 stoch ∈ Sy ⇒ stoch ∈ Sx} 31 ∪ {unalignedx ⇒ unalignedn 32 | n ∈ names(tt) ∪ names(te)} 33 ∪ {stoch ∈ Sy ⇒ unalignedn 34 | n ∈ names(tt) ∪ names(te)} 35 | assume _ → {stoch ∈ Sx} 36 | weight _ → ∅ 37 38 function name(t): TANF → X = 39 match t with 40 | x → x 41 | let x = t1 in t2 → name(t2) 42 43 function names(t): TANF → P(X) = 44 match t with 45 | x → ∅ 46 | let x = _ in t2 → {x} ∪ names(t2) 47 48 49 50 are named by t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' t4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' and t5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, aligned SMC therefore resamples at the weight at t1, but not at the weights at t2, t3, and t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Consider the program in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1a again, and assume it is transformed to ANF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The alignment analysis must mark all names bound within the stochastic if at line 3 as unaligned because a stochastic value flows to its condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In particular, the weight expressions at lines 5 and 8 are unaligned (and the weight at line 12 is aligned).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Thus, aligned SMC resamples only at line 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To formalize the flow of stochastic values, we define abstract values a ∈ A, that flow within the program, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Definition 7 (Abstract values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' a ::= λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='y | stoch | const n where x, y ∈ X and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The stoch abstract value is new and represents stochastic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='y and const n abstract values are standard and represent abstract closures and intrinsics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For each variable name x in the program, we define a set Sx containing abstract values that may occur at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4, we have stoch ∈ Sa1, (λx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t2) ∈ Sf2, and (const 1) ∈ Sn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The abstract value λx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t2 represents all closures originating at λx2, and const 1 represents intrinsic functions in C of arity 1 (in our example, ¬).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The body of the abstract lambda is the variable name labeling the body, not the body itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, t2 labels the body let t2 = one in t2 of λx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Due to ANF, all terms have a label, which the function name in Algorithm 1 formalizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 14 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We also define booleans unaligned x that state whether or not the expression labeled by x is unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, we previously reasoned that unalignedx = true for x ∈ {t2, t3, t4, t5} in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The alignment analysis aims to deter- mine minimal sets Sx and boolean assignments of unalignedx for every pro- gram variable x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A trivial solution is that all abstract values (there is a finite number of them in the program) flow to each program variable and that unalignedx = true for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This solution is sound but useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To compute a more precise solution, we follow the rules given by constraints c ∈ R (see Appendix B for a formal definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We present the constraints through the generateConstraints function in Algorithm 1 and for the example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' There are no constraints for variables that occur at the end of ANF let sequences (line 2 in Algorithm 1), and the case for let expressions (lines 3–36) instead produces all constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The cases for aliases (line 6), intrinsics (line 7), assume (line 35), and weight (line 36) are the most simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aliases of the form let x = y in t2 establish Sy ⊆ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' That is, all abstract values at y are also in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Intrinsic operations results in a const abstract value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, the definition of n1 at line 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4 results in the constraint const 1 ∈ Sn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Applications of assume are the source of stochastic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, the definition of a1 at line 10 results in the constraint stoch ∈ Sa1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that assume cannot produce any other abstract values, as we only allow distributions over intrinsics with arity 0 (see Definition 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Finally, we use weight only for its side effect (likelihood updating), and therefore weights do not produce any abstract values and consequently no constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The cases for abstractions (line 9), applications (line 13), and ifs (line 26) are more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The abstraction at line 4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4 generates (omitting the recursively generated constraints for the abstraction body ty) the constraints {λx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='x1 ∈ Sf1} ∪ {unalignedx1 ⇒ unalignedt1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The first constraint is standard: the abstract lambda λx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='x1 flows to Sf1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The second constraint states that if the abstraction is unaligned, all expressions in its body (here, only t1) are unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We define the sets of expressions within abstraction bodies and if branches through the names function in Algorithm 1 (line 43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The application f5 one at line 18 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4 generates the constraints {∀z∀y λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='y ∈ Sf5 ⇒ (Sone ⊆ Sz) ∧ (Sy ⊆ Sv4), ∀n (const n ∈ Sf5) ∧ (n > 1) ⇒ const n − 1 ∈ Sv4, stoch ∈ Sf5 ⇒ stoch ∈ Sv4, const _ ∈ Sf5 ⇒ (stoch ∈ Sone ⇒ stoch ∈ Sv4), unalignedv4 ⇒ (∀y λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Sf5 ⇒ unalignedy), stoch ∈ Sf5 ⇒ (∀y λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedy)} (4) The first constraint is standard: if an abstract value λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='y flows to f5, the abstract values of one (the right-hand side) flow to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, the result of the appli- cation, given by the body name y, must flow to the result v4 of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The second constraint is also relatively standard: if an intrinsic function of arity n is applied, it produces a const of arity n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The other constraints are new Automatic Alignment in Higher-Order PPLs 15 and specific for stochastic values and unalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The third constraint states that if the function is stochastic, the result is stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The fourth constraint states that if we apply an intrinsic function to a stochastic argument, the result is stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We could also make the analysis of intrinsic applications less conser- vative through intrinsic-specific constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The fifth and sixth constraints state that if the expression (labeled by v4) is unaligned or the function is stochastic, all abstract lambdas that flow to the function are unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The if resulting in f5 at line 14 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4 generates (omitting the recursively generated constraints for the branches tt and te) the constraints {Sname(f2) ⊆ Sf5, Sname(f3) ⊆ Sf5, stoch ∈ Sa1 ⇒ stoch ∈ Sf5} ∪ {unalignedf5 ⇒ unalignedt5} ∪ {stoch ∈ Sa1 ⇒ unalignedt5} (5) The first two constraints are standard and state that the result of the branches flows to the result of the if expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The remaining constraints are new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The third constraint states that if the condition is stochastic, the result is stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The last two constraints state that if the if is unaligned or if the condition is stochastic, all names in the branches (here, only t5) are unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Given constraints for a program, we need to compute a solution satisfying all constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We do this by repeatedly iterating through all the constraints and propagating abstract values accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We terminate when we reach a fixed point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', when no constraint results in an update of either Sx or unalignedx for any x in the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Algorithm 4 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 formalizes our extension of the 0-CFA constraint propagation algorithm that also handles the constraints generated for tracking stochastic values and unalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The analysis function analyzeAlign: TANF → ((X → P(A))×P(X)) returns a map associating each variable to a set of abstract values and a set of unaligned variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In other words, analyzeAlign computes a solution to Sx and unalignedx for each x in the analyzed program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For example, analyzeAlign(texample) results in Sn1 = {const 1} Sn2 = {const 1} Sf1 = {λx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='x1} Sf2 = {λx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t2} Sf3 = {λx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t3} Sf4 = {λx4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t4} Sa1 = {stoch} Sv2 = {stoch} Sf5 = {λx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t2, λx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t3, stoch} Sv4 = {stoch} Sn = ∅ | other n ∈ X unalignedn = true | n ∈ {t2, t3, t4, t5} unaligned n = false | other n ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (6) The example confirms our earlier intuition: an intrinsic (¬) flows to n1, stoch flows to a1, f5 is stochastic and originates at either (λx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t2) or (λx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t3), and the unaligned variables are t2, t3, t4, and t5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now give soundness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lemma 1 (0-CFA soundness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For every t ∈ TANF, the solution produced by analyzeAlign(t) satisfies the constraints generateConstraints(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The well-known soundness of 0-CFA extends to the new alignment con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Nielson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [33, Chapter 3] and Shivers [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' ⊓⊔ Theorem 1 (Alignment analysis soundness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Assume t ∈ TANF, At from Definition 6, and an assignment to Sx and unalignedx for x ∈ X according 16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' to analyzeAlign(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Let �At = {x | ¬unalignedx} and take arbitrary ∅ ⊢ t s1⇓w1 l1 v1 and ∅ ⊢ t s2⇓w2 l2 v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Then, l1| � At = l2| � At and consequently �At ⊆ At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Follows by Lemma 3 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 with t′ = t and ρ1 = ρ2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The proof uses simultaneous structural induction over the derivations ∅ ⊢ t s1⇓w1 l1 v1 and ∅ ⊢ t s2⇓w2 l2 v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' At corresponding stochastic branches or stochastic function applications in the two derivations, a separate structural induction argument shows that, for the let-sequences l′ 1 and l′ 2 of the two stochastic subderivations, l′ 1| � At = l′ 2| � At = [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Combined, the two arguments give the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' ⊓⊔ The result �At ⊆ At (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Definition 6) shows that the analysis is conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3 Dynamic Alignment An alternative to static alignment is dynamic alignment, which we explored in early stages when developing the alignment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Dynamic alignment is fully context sensitive and amounts to introducing variables in programs that track (at runtime) when evaluation enters stochastic branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To identify these stochastic branches, dynamic alignment also requires a runtime data structure that keeps track of the stochastic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Similarly to k-CFA, dynamic alignment is potentially more precise than the 0-CFA approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' However, we discovered that dynamic alignment introduces significant runtime overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Again, we note that the models in Section 7 do not require a context-sensitive analysis, justifying the choice of 0-CFA over dynamic alignment and k-CFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5 Aligned SMC and MCMC This section presents detailed algorithms for aligned SMC (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1) and aligned lightweight MCMC (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For a more pedagogical introduction to the algorithms, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We assume a basic understanding of SMC and Metropolis–Hastings MCMC algorithms (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Bishop [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 Aligned SMC We saw in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 that SMC operates by executing many instances of t concurrently, and resampling them at calls to weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Critically, resampling requires that the inference algorithm can both suspend and resume executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Here, we assume that we can create execution instances e of the probabilistic program t, and that we can arbitrarily suspend and resume the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The technical details of suspension are beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' See Goodman and Stuhlmüller [14], Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [47], and Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [25] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Algorithm 2 presents all steps for the aligned SMC inference algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Af- ter running the alignment analysis and setting up the n execution instances, the algorithm iteratively executes and resamples the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that the algorithm resamples only at aligned weights (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 17 Algorithm 2 Aligned SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The input is a program t ∈ TANF and the number of execution instances n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Run the alignment analysis on t, resulting in � At (see Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Initiate n execution instances {ei | i ∈ N, 1 ≤ i ≤ n} of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Execute all ei and suspend execution upon reaching an aligned weight (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', let x = weight w in t and x ∈ � At) or when the execution terminates naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result is a new set of execution instances e′ i with weights w′ i accumulated from unaligned weights and the single final aligned weight during execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If all e′ i = v′ i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', all executions have terminated and returned a value), terminate inference and return the set of weighted samples (v′ i, w′ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The samples approximate the posterior probability distribution encoded by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Resample the e′ i according to their weights w′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result is a new set of unweighted execution instances e′′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set ei ← e′′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Go to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1 if assume Bernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5) 2 then weight 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' weight 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' true 3 else weight 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' weight 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' false (a) Aligned better than unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1 if assume Bernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5) 2 then weight 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' true 3 else weight 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' false (b) Unaligned better than aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5: Programs illustrating properties of aligned and unaligned SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (a) shows a program better suited for aligned SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (b) shows a program better suited for aligned SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned SMC is preferable over unaligned SMC for all practically relevant models, as the evaluation in Section 7 justifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' However, it is possible to con- struct contrived programs in which unaligned SMC has the advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Consider the programs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5, both encoding simple Bernoulli distributions in a con- trived way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Unaligned SMC resamples at the first weights in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5a, which are not indicative of the final weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result is reduced inference accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned SMC does not resample at all as the weights are within a stochastic branch, and avoids the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned SMC, however, does not resample at all in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5b, where it is beneficial to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Consequently, the results are poorer compared to unaligned SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We are not aware of any real model with the prop- erty in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In practice, it is always best to resample when using weight to condition on observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Such conditioning is, in practice, always done outside of stochastic branches, justifying the benefit of aligned SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 Aligned Lightweight MCMC Aligned lightweight MCMC is a version of lightweight MCMC [46], where the alignment analysis provides information about how to reuse random draws be- tween executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Algorithm 3, a Metropolis–Hastings algorithm in the context of PPLs, presents the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Essentially, the algorithm executes the program repeatedly using the Run function, and redraws one aligned random draw in each step, while reusing all other aligned draws and as many unaligned draws as 18 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Algorithm 3 Aligned lightweight MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The input is a program t ∈ TANF, the number of steps n, and the global step probability g > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Run the alignment analysis on t, resulting in � At (see Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set i ← 0, k ← 1, and l ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Call Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set i ← i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If i = n, terminate inference and return the samples {vj | j ∈ N, 0 ≤ j < n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' They approximate the probability distribution encoded by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Uniformly draw an index 1 ≤ j ≤ |si−1| at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set global ← true with probability g, and global ← false otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set w′ −1 ← 1, w′ ← 1, k ← 1, l ← 1, and reuse ← true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Call Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Compute the Metropolis–Hastings acceptance ratio A = min � 1, wi wi−1 w′ w′ −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' With probability A, accept vi and go to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Otherwise, set vi ← vi−1, wi ← wi−1, si ← si−1, pi ← pi−1, s′ i ← s′ i−1, p′ i ← p′ i−1, and n′ i ← n′ i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Go to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' function run() = Run t and do the following: – Record the total weight wi accumulated from calls to weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – Record the final value vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – At unaligned terms let c = assume d in t (c ̸∈ � At), do the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If reuse = false, global = true, n′ i−1,k,l ̸= c, or if s′ i−1,k,l does not exist, sample a value x from d and set reuse ← false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Otherwise, reuse the sample x = s′ i−1,k,l and set w′ −1 ← w′ −1 · p′ i−1,k,l and w′ ← w′ · fd(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set s′ i,k,l ← x, p′ i,k,l ← fd(x), and n′ i,k,l ← c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set l ← l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In the program, bind c to the value x and resume execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – At aligned terms let c = assume d in t (c ∈ � At), do the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If j = k, global = true, or if si−1,k does not exist, sample a value x from d normally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Otherwise, reuse the sample x = si−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set w′ −1 ← w′ −1 · pi−1,k and w′ ← w′ · fd(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set si,k ← x and pi,k ← fd(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set k ← k + 1, l ← 1, and reuse ← true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In the program, bind c to the value x and resume execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' possible (illustrated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We provide a derivation of the Metropolis– Hastings acceptance ratio in step 5 in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A key property in Algorithm 3 due to alignment (Definition 6) is that the length of si (and pi) is constant, as executing t always results in the same number of aligned random draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In addition to redrawing only one aligned random draw, each step has a probability g > 0 of being global—meaning that inference redraws every random draw in the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Occasional global steps fix problems related to slow mixing and ergodicity of lightweight MCMC identified by Kiselyov [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In a global step, the Metropolis–Hastings acceptance ratio reduces to A = min � 1, wi wi−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 6 Implementation We implement the alignment analysis (Section 4), aligned SMC (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1), and aligned lightweight MCMC (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2) for the functional PPL Miking CorePPL [25], implemented as part of the Miking framework [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We implement Automatic Alignment in Higher-Order PPLs 19 the alignment analysis as a core component in the Miking CorePPL compiler, and then use the analysis when compiling to two Miking CorePPL backends: RootPPL and Miking Core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' RootPPL is a low-level PPL with built-in highly efficient SMC inference [25], and we extend the CorePPL to RootPPL compiler introduced by Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [25] to support aligned SMC inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, we implement aligned lightweight MCMC inference standalone as a translation from Miking CorePPL to Miking Core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Miking Core is the general-purpose pro- gramming language of the Miking framework, currently compiling to OCaml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The idealized calculus in (1) does not capture all features of Miking CorePPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In particular, the alignment analysis implementation must support records, vari- ants, sequences, and pattern matching over these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Extending 0-CFA to such lan- guage features is not new, but it does introduce a critical challenge for the align- ment analysis: identifying all possible stochastic branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Determining stochas- tic ifs is straightforward, as we simply check if stoch flows to the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' However, complications arise when we add a match construct (and, in general, any type of branching construct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Consider the extension t ::= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' | match t with p then t else t | {k1 = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', kn = xn} p ::= x | true | false | {k1 = p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', kn = p} x, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' , xn ∈ X k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' , kn ∈ K n ∈ N (7) of (1), adding records and simple pattern matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' K is a set of record keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' As- sume we also extend the abstract values as a ::= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' | {k1 = X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' , kn = Xn}, where X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' , Xn ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' That is, we add an abstract record tracking the names in the program that flow to its entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Consider the program match t1 with { a = x1, b = false } then t2 else t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This match is, similar to ifs, stochastic if stoch ∈ St1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' It is also, however, stochastic in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Assume we have two program variables, x and y, such that stoch ∈ Sx and stoch ̸∈ Sy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Now, the match is stochastic if, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', {a = {y}, b = {x}} ∈ St1, because the random value flowing from x to the pattern false may not match because of randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' However, it is not stochastic if, instead, St1 = {{a = {x}, b = {y}}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The ran- domness of x does not influence whether or not the branch is stochastic—the variable pattern x1 for label a always matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Our alignment analysis implementation handles the intricacies of identify- ing stochastic match cases for nested record, variant, and sequence patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In total, the alignment analysis, aligned SMC, and aligned lightweight MCMC im- plementations consist of approximately 1000 lines of code directly contributed as part of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The code is available on GitHub [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 7 Evaluation This section evaluates aligned SMC and aligned lightweight MCMC on a set of models encoded in Miking CorePPL: CRBD [32,38] in Sections 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5, ClaDS [27,38] in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2, state-space aircraft localization in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3, and latent Dirichlet allocation in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' CRBD and ClaDS are non-trivial 20 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' models of considerable interest in evolutionary biology and phylogenetics [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Similarly, LDA is a non-trivial topic model [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Running the alignment analysis took approximately 5 ms–30 ms for all models considered in the experiment, justifying that the time complexity is not a problem in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We compare aligned SMC with standard unaligned SMC [14], which is iden- tical to Algorithm 2, except that it resamples at every call to weight (see Ap- pendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We carefully checked that automatic alignment corresponds to previ- ous manual alignments of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For all SMC experiments, we estimate the normalizing constant produced as a by-product of SMC inference rather than the complete posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The normalizing constant, also known as marginal likelihood or model evidence, frequently appears in Bayesian inference and gives the probability of the observed data averaged over the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The normalizing constant is useful for model comparison as it measures how well dif- ferent probabilistic models fit the data (a larger normalizing constant indicates a better fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We ran aligned and unaligned SMC with Miking CorePPL and the RootPPL backend configured for a single-core (compiled with GCC 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [25] shows that the RootPPL backend is significantly more efficient than other state-of-the-art PPL SMC implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We ran aligned and unaligned SMC inference 300 times (and with 3 warmup runs) for each experiment for 104, 105, and 106 executions (also known as particles in SMC literature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We compare aligned lightweight MCMC to lightweight MCMC (see Ap- pendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We implement both versions as compilers from Miking CorePPL to Miking Core in the Miking framework, which in turn compiles to OCaml (ver- sion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The lightweight MCMC databases are functional-style maps from the OCaml standard Map library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We set g (the global step probability in Al- gorithm 3) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 for both aligned lightweight MCMC and lightweight MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We ran aligned lightweight and lightweight MCMC inference 300 times for each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We burned 10% of samples in all MCMC runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For all experiments, we used an Intel Xeon 656 Gold 6136 CPU (12 cores) and 64 GB of memory running Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 SMC: Constant Rate Birth-Death (CRBD) This experiment considers the CRBD diversification model from [38] applied to the Alcedinidae phylogeny (Kingfisher birds, 54 extant species) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We use fixed diversification rates to simplify the model, as unaligned SMC inference accuracy is too poor for the full model with priors over diversification rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned SMC is accurate for both the full and simplified models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We provide the source code for the complete model in Listing 1 of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 (130 lines of code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The total experiment execution time was 16 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 6 presents the experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned SMC is roughly twice as fast and produces superior estimates of the normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Unaligned SMC has not yet converged to the correct value −304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='75 (available for this particular model due to the fixing the diversification rates) for 106 particles, while aligned SMC produces precise estimates already at 104 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Excess resampling is a Automatic Alignment in Higher-Order PPLs 21 106 105 104 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='49 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='53 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='82 (a) Execution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 104 105 106 −315 −330 −304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='75 (b) Log normalizing constant estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 6: SMC experiment results for CRBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The x-axes give the number of parti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (a) shows execution times (in seconds) for aligned (gray) and unaligned (white) SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Error bars show one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (b) shows box plot log normalizing constant estimates for aligned (gray) and unaligned (white) SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The analytically computed log normalizing constant is −304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 106 105 104 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='41 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='6 634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='07 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='56 (a) Execution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 104 105 106 −400 −500 −314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='35 (b) Log normalizing constant estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 7: SMC experiment results for ClaDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The x-axes give the number of parti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (a) shows execution times (in seconds) for aligned (gray) and unaligned (white) SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Error bars show one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (b) shows box plot log normalizing constant estimates for aligned (gray) and unaligned (white) SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The average estimate for aligned SMC with 106 particles is −314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' significant factor in the increase in execution time for unaligned SMC, as each execution encounters far more resampling checkpoints than in aligned SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 SMC: Cladogenetic Diversification Rate Shift (ClaDS) A limitation of CRBD is that the diversification rates are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' ClaDS [27,38] is a set of diversification models that allow shifting rates over phylogenies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We evaluate the ClaDS2 model for the Alcedinidae phylogeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' As in CRBD, we use fixed (initial) diversification rates to simplify the model on account of un- aligned SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The source code for the complete model is available in Listing 2 of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 (147 lines of code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic alignment simplifies the ClaDS2 model significantly, as manual alignment requires collecting and passing weights around in unaligned parts of the program, which are later consumed by aligned weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The total experiment execution time was 67 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 7 presents the experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 12 unaligned runs for 106 particles and nine runs for 105 particles ran out of the preallocated stack memory for each particle (10 kB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We omit these runs from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The consequence of not aligning SMC is more severe than for CRBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned SMC is now almost seven times faster than unaligned SMC and the unaligned SMC normalizing constant 22 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 106 105 104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='06 (a) Execution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 104 105 106 −55 −65 −61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='26 (b) Log normalizing constant estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 8: SMC experiment results for the state-space aircraft localization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The x-axes give the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (a) shows execution times (in seconds) for aligned (gray) and unaligned (white) SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Error bars show one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (b) shows box plot log normalizing constant estimates on the y-axis for aligned (gray) and unaligned (white) SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The average estimate for aligned SMC with 106 particles is −61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' estimates are significantly worse compared to the aligned SMC estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The unaligned SMC estimates do not even improve when moving from 104 to 106 particles (we need even more particles to see improvements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Again, aligned SMC produces precise estimates already at 104 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3 SMC: State-Space Aircraft Localization This experiment considers an artificial but non-trivial state-space model for air- craft localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3 presents the model as well as the source code in Listing 3 (62 lines of code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The total experiment execution time was 1 hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 8 presents the experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The execution time difference is not as significant as for CRBD and ClaDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' However, the unaligned SMC normalizing constant estimates are again much less precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned SMC is accurate (cen- tered at approximately −61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='26) already at 104 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The model’s straightfor- ward control flow explains the less dramatic difference in execution time—there are at most ten unaligned likelihood updates in the aircraft model, while the number is, in theory, unbounded for CRBD and ClaDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Therefore, the cost of extra resampling compared to aligned SMC is not as significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 MCMC: Latent Dirichlet Allocation (LDA) This experiment considers latent Dirichlet allocation (LDA), a topic model used in the evaluations by Wingate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [46] and Ritchie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We use a synthetic data set, comparable in size to the data set used by Ritchie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [37], with a vocabulary of 100 words, 10 topics, and 25 documents each containing 30 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that we are not using methods based on collapsed Gibbs sampling [17], and the inference task is therefore computationally challenging even with a rather small number of words and documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The source code for the complete model is available in Listing 4 of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 (31 lines of code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The total experiment execution time was 41 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 23 105 104 103 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='24 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='17 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='25 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='23 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 9: MCMC experiment results for LDA showing execution time (in seconds) for aligned lightweight MCMC (gray) and lightweight MCMC (white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Error bars show one standard deviation and the x-axis the number of MCMC iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The LDA model consists of only aligned random draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' As a consequence, aligned lightweight and lightweight MCMC reduces to the same inference algo- rithm, and we can compare the algorithms by just considering the execution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We justify the correctness of both algorithms in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 9 presents the experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned lightweight MCMC is al- most three times faster than lightweight MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To justify the execution times with our implementations, we also implemented and ran the experiment with lightweight MCMC in WebPPL [14] for 105 iterations, repeated 50 times (and with 3 warmup runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The mean execution time was 383 s with standard devia- tion 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We used WebPPL version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='15 and Node version 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 MCMC: Constant Rate Birth-Death (CRBD) This experiment again considers CRBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' MCMC is not as suitable for CRBD as SMC, and therefore we use a simple synthetic phylogeny with six leaves and an age span of 5 age units (Alcedinidae used for the SMC experiment has 54 leaves and an age span of 35 age units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The source code for the complete model is the same as in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1, but we now allow the use of proper prior distributions for the diversification rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The total experiment execution time was 7 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Unlike LDA, the CRBD model contains both unaligned and aligned random draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Because of this, aligned lightweight MCMC and standard lightweight MCMC do not reduce to the same algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To judge the difference in infer- ence accuracy, we consider the mean estimates of the birth diversification rate produced by the two algorithms, in addition to execution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The experiment results shows that the posterior distribution over the birth rate is unimodal (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5), which motivates using the posterior mean as a measure of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 10 presents the experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Aligned lightweight MCMC is ap- proximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 times faster than lightweight MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' There is no obvious dif- ference in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To justify the execution times and correctness of our im- plementations, we also implemented and ran the experiment with lightweight MCMC in WebPPL [14] for 3 · 106 iterations, repeated 50 times (and with 3 warmup runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The mean estimates agreed with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The mean execution time was 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 s with standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='8 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The speedup compared to stan- dard lightweight MCMC in Miking CorePPL is likely explained by the use of 24 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3 · 106 3 · 105 3 · 104 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='95 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='63 (a) Execution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3 · 104 3 · 105 3 · 106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='33 (b) Birth rate mean estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 10: MCMC experiment results for CRBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The x-axes give the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (a) shows execution times (in seconds) for aligned lightweight MCMC (gray) and lightweight MCMC (white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Error bars show one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (b) shows box plot posterior mean estimates of the birth rate for aligned lightweight MCMC (gray) and lightweight MCMC (white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The average estimate for aligned lightweight MCMC with 3 · 106 iterations is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' early termination in WebPPL, which benefits CRBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Early termination easily combines with alignment but relies on execution suspension, which we do not currently use in our implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that aligned lightweight MCMC is faster than WebPPL even without early termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In conclusion, the experiments clearly demonstrate the need for alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 8 Related Work The approach by Wingate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [46] is closely related to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A key similarity with alignment is that executions reaching the same aligned checkpoint also have matching stack traces according to Wingate et al.’s addressing transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' However, Wingate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' do not consider the separation between unaligned and aligned parts of the program, their approach is not static, and they do not generalize to other inference algorithms such as SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Ronquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [38], Turing [12], Anglican [47], Paige and Wood [35], and van de Meent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [45] consider the alignment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Manual alignment is critical for the models in Ronquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [38] to make SMC inference tractable, which strongly motivates the automatic alignment approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The documentation of Turing states that: “The observe statements [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', likelihood updates] should be arranged so that every possible run traverses all of them in exactly the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This is equivalent to demanding that they are not placed inside stochastic control flow” [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Turing does not include any automatic checks for this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Anglican [47] checks, at runtime (resulting in overhead), that all SMC executions encounter the same number of likelihood updates, and thus resamples the same number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If not, Anglican reports an error: “some observe directives [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', likelihood updates] are not global”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This error refers to the alignment problem, but the documentation does not explain it further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Probabilistic C, introduced by Paige and Wood [35], similarly assumes that the number of likelihood updates is the same in all executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Van de Meent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [45] state, in reference to SMC: “Each breakpoint [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', checkpoint] needs to occur at an expression that Automatic Alignment in Higher-Order PPLs 25 is evaluated in every execution of a program”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Again, they do not provide any formal definition of alignment nor an automatic solution to enforce it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [24] briefly mention the general problem of selecting optimal resampling locations in PPLs for SMC but do not consider the alignment problem in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' They also acknowledge the overhead resulting from not all SMC executions resampling the same number of times, which alignment avoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The PPLs Birch [30], Pyro [3], and WebPPL [14] support SMC inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Birch and Pyro enforce alignment for SMC as part of model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that this is only true for SMC in Pyro—other Pyro inference algorithms use other modeling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The approaches in Birch and Pyro are sound but demand more of their users compared to the alignment approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' WebPPL does not consider alignment and resamples at all likelihood updates for SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Ritchie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [37] and Nori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [34] present MCMC algorithms for proba- bilistic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Ritchie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [37] optimize lightweight MCMC by Wingate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [46] through execution suspensions and callsite caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The optimizations are independent of and potentially combines well with aligned lightweight MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Another MCMC optimization which potentially combines well with alignment is due to Nori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' They use static analysis to propagate observations backwards in programs to improve inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Information flow analyses [39] may determine if particular parts of a program execute as a result of different program inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Specifically, if program input is random, such approaches have clear similarities to the alignment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Many other PPLs exist, such as Gen [10], Venture [28], Edward [43], Stan [8], and AugurV2 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Gen, Venture, and Edward focus on simplifying the joint specification of a model and its inference to give users more low-level control, and do not consider automatic alignment specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' However, the incremen- tal inference approach [9] in Gen does use the addressing approach by Wingate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Stan and AugurV2 have less expressive modeling languages to al- low more powerful inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Alignment is by construction due to the reduced expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Borgström et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [6], Staton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [42], Ścibior et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [40], and Vákár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [44] treat PPL topics related to semantics and correctness, but do not specifically consider alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 9 Conclusion This paper gives, for the first time, a formal definition of alignment in PPLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, we introduce a static analysis technique and use it to align check- points in PPLs and apply it to SMC and MCMC inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We formalize the alignment analysis, prove its correctness, and implement it in Miking CorePPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We also implement aligned SMC and aligned lightweight MCMC, and evaluate the implementations on non-trivial CRBD and ClaDS models from phylogenet- ics, the LDA topic model, and a state-space model, demonstrating significant improvements compared to standard SMC and lightweight MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Acknowledgments We thank Lawrence Murray, Johannes Borgström, and Jan Kudlicka for early discussions on the alignment idea, and Viktor Senderov for im- plementing ClaDS in Miking CorePPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We also thank the anonymous reviewers at ESOP for their valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 27 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Turing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='jl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' https://turing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ml/dev/ (2022), accessed: 2022-02-24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Miking DPPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='com/miking-lang/miking-dppl (2023), accessed: 2023-01-02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Bingham, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Jankowiak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Obermeyer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Pradhan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Karalet- sos, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Singh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Szerlip, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Horsfall, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Goodman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=': Pyro: Deep universal probabilistic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Journal of Machine Learning Research 20(28), 1–6 (2019) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Bishop, C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 525–534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Association for Computing Machinery (2016) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Tran, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Hoffman, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Proceedings of the ACM on Programming Languages 3(POPL) (2019) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' van de Meent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Paige, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Wood, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=': An introduction to probabilistic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' arXiv e-prints p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='10756 (2018) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Wingate, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Stuhlmueller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Goodman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=': Lightweight implementations of probabilistic programming languages via transformational compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In: Pro- ceedings of the 14th International Conference on Artificial Intelligence and Statis- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 15, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' PMLR (2011) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Wood, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Meent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', Mansinghka, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=': A new approach to probabilistic program- ming inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In: Proceedings of the 17th International Conference on Artificial Intelligence and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1024–1032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' PMLR (2014) 30 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A Evaluation, Continued This section presents further details related to the evaluation in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In particular, we attach code listings for the experiment models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that these listings only give the model code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The code for the analysis itself and all inference algorithms are available on GitHub [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 SMC: Constant Rate Birth-Death (CRBD) Listing 1 gives the Miking CorePPL source code used for the case study model in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Listing 1: The source code for the experiment in Sections 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 1 ------------------------------------------------ 2 -- The Constant-Rate Birth-Death (CRBD) model -- 3 ------------------------------------------------ 4 5 -- The prelude includes a few PPL helper functions 6 include "pplprelude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 7 8 -- The tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc file defines the general tree structure 9 include "tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 10 11 -- The tree-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc file includes the actual tree and the rho constant 12 include "tree-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 13 14 mexpr 15 16 -- CRBD goes undetected, including iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Mutually recursive functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 17 recursive 18 let iter: Int -> Float -> Float -> Float -> Float -> Float -> Bool = 19 lam n: Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 20 lam startTime: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 21 lam branchLength: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 22 lam lambda: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 23 lam mu: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 24 lam rho: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 25 if eqi n 0 then 26 true 27 else 28 let eventTime = assume (Uniform (subf startTime branchLength) startTime) in 29 if crbdGoesUndetected eventTime lambda mu rho then 30 iter (subi n 1) startTime branchLength lambda mu rho 31 else 32 false 33 34 let crbdGoesUndetected: Float -> Float -> Float -> Float -> Bool = 35 lam startTime: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 36 lam lambda: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 37 lam mu: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 38 lam rho: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let duration = assume (Exponential mu) in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let cond = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='-- ‘and‘ does not use short-circuiting: using ‘if‘ as below is more ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='-- efficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='if (gtf duration startTime) then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='(eqBool (assume (Bernoulli rho)) true) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='else false ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='if cond then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='false ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let branchLength = if ltf duration startTime then duration else startTime in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let n = assume (Poisson (mulf lambda branchLength)) in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='iter n startTime branchLength lambda mu rho ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='53 in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='55 -- Simulation of branch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='56 recursive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='57 let simBranch: Int -> Float -> Float -> Float -> Float -> Float -> () = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='lam n: Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 59 lam startTime: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 60 lam stopTime: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 61 lam lambda: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 62 lam mu: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 63 lam rho: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 64 if eqi n 0 then () 65 else Automatic Alignment in Higher-Order PPLs 31 66 let currentTime = assume (Uniform stopTime startTime) in 67 if crbdGoesUndetected currentTime lambda mu rho then 68 let w1 = weight (log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=') in 69 simBranch (subi n 1) startTime stopTime lambda mu rho 70 else 71 let w2 = weight (negf inf) in 72 () 73 in 74 75 -- Simulating along the tree structure 76 recursive 77 let simTree: Tree -> Tree -> Float -> Float -> Float -> () = 78 lam tree: Tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 79 lam parent: Tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 80 lam lambda: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 81 lam mu: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 82 lam rho: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 83 let lnProb1 = mulf (negf mu) (subf (getAge parent) (getAge tree)) in 84 let lnProb2 = match tree with Node _ then log lambda else log rho in 85 86 let startTime = getAge parent in 87 let stopTime = getAge tree in 88 let n = assume (Poisson (mulf lambda (subf startTime stopTime))) in 89 simBranch n startTime stopTime lambda mu rho;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 90 91 let w3 = weight (addf lnProb1 lnProb2) in 92 93 match tree with Node { left = left, right = right } then 94 simTree left tree lambda mu rho;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 95 simTree right tree lambda mu rho 96 else () 97 in 98 99 -- Fixed priors used for the SMC experiment 100 -- let lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 in 101 -- let mu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 in 102 103 -- 104 let lambda = assume (Gamma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='0) in 105 let mu = assume (Gamma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5) in 106 107 -- Adjust for normalizing constant 108 let numLeaves = countLeaves tree in 109 let corrFactor = 110 subf (mulf (subf (int2float numLeaves) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=') (log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=')) (lnFactorial numLeaves) in 111 weight corrFactor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 112 113 -- Start of the simulation along the two branches 114 (match tree with Node { left = left, right = right } then 115 simTree left tree lambda mu rho;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 116 simTree right tree lambda mu rho 117 else ());' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 118 119 -- Compute the joint posterior over lambda and mu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 120 (lambda,mu) 121 122 -- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' or the marginal posterior over lambda (MCMC experiment) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 123 -- lambda 124 125 -- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' or nothing for just estimating the normalizing constant (SMC experiment) 126 -- () A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 SMC: Cladogenetic Diversification Rate Shift (ClaDS) Listing 2 gives the Miking CorePPL source code used for the case study model in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Listing 2: The source code for the experiment in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 1 ------------------------------------------------------------------ 2 -- The ClaDogenetic Diversification Shifts model (ClaDS2) model -- 3 ------------------------------------------------------------------ 4 5 -- The prelude includes a few PPL helper functions 6 include "pplprelude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 7 8 -- The tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc file defines the general tree structure 9 include "tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 10 11 -- The tree-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc file includes the actual tree and the rho constant 12 include "tree-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 13 14 mexpr 32 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 15 16 -- Multiplier guards 17 let maxM = 10e5 in 18 let minM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in 19 20 -- Clads2 goes undetected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 21 recursive 22 let clads2GoesUndetected: Float -> Float -> Float -> Float 23 > Float -> Float -> Float -> Bool = 24 lam startTime_Mya: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 25 lam lambda0: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 26 lam mu0: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 27 lam m: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' -- Multiplier 28 lam logAlpha: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' -- Logarithm of alpha 29 lam sigma: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' -- Standard deviation 30 lam rho: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 31 32 -- Guard: m is not allowed to exceed maxM or be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 33 if or (gtf m maxM) (leqf m minM) then false 34 else 35 let eventTime_My = 36 assume (Exponential (addf (mulf m lambda0) (mulf m mu0))) in 37 let currentTime_Mya = subf startTime_Mya eventTime_My in 38 if ltf currentTime_Mya 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='if assume (Bernoulli rho) then false ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='else true ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let extinction = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='assume (Bernoulli (divf (mulf m mu0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='(addf (mulf m lambda0) (mulf m mu0)))) in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='if extinction then true ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let m1 = mulf m (exp (assume (Gaussian logAlpha sigma))) in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let m2 = mulf m (exp (assume (Gaussian logAlpha sigma))) in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='if clads2GoesUndetected currentTime_Mya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='lambda0 mu0 m1 logAlpha sigma rho then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='clads2GoesUndetected currentTime_Mya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='lambda0 mu0 m2 logAlpha sigma rho ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='else false ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='54 in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='56 -- Simulation of branch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='57 recursive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='58 let simBranch: Float -> Float -> Float -> Float ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='> Float -> Float -> Float -> Float -> Float = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='lam startTime_Mya: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 61 lam stopTime_Mya: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 62 lam lambda0: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 63 lam mu0: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 64 lam m: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' -- multiplier 65 lam logAlpha: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 66 lam sigma: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 67 lam rho: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 68 69 -- Guard: m is not allowed to exceed maxM or be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='if or (gtf m maxM) (ltf m minM) then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='71 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let w0 = weight (negf inf) in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='73 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='74 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let tSpeciation_My = assume (Exponential (mulf m lambda0)) in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let currentTime_Mya = subf startTime_Mya tSpeciation_My in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let branchLength_My = subf startTime_Mya stopTime_Mya in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='77 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='if (ltf currentTime_Mya stopTime_Mya) then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='78 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let w1 = weight (mulf (negf (mulf m mu0)) branchLength_My) in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='79 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let m1 = mulf m (exp (assume (Gaussian logAlpha sigma))) in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='if clads2GoesUndetected currentTime_Mya lambda0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='83 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mu0 m1 logAlpha sigma rho then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='84 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let m2 = mulf m (exp (assume (Gaussian logAlpha sigma))) in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='let w2 = weight (log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=') in 86 let w3 = weight (mulf (negf (mulf m mu0)) tSpeciation_My) in 87 simBranch currentTime_Mya stopTime_Mya 88 lambda0 mu0 m2 logAlpha sigma rho 89 else -- side branch detected 90 let w4 = weight (negf inf) in 91 m 92 in 93 94 -- Simulating along the tree structure 95 recursive 96 let simTree: Tree -> Tree -> Float -> Float 97 > Float -> Float -> Float -> Float -> () = 98 lam tree: Tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 99 lam parent: Tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 100 lam lambda0: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 101 lam mu0: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 102 lam m: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 103 lam logAlpha: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 104 lam sigma: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 33 105 lam rho: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 106 107 let startTime_Mya = getAge parent in 108 let stopTime_Mya = getAge tree in 109 110 let mEnd = 111 simBranch startTime_Mya stopTime_Mya lambda0 mu0 m logAlpha sigma rho in 112 (match tree with Node _ 113 then weight (log (mulf mEnd lambda0)) else weight (log rho));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 114 115 let m1 = mulf mEnd (exp (assume (Gaussian logAlpha sigma))) in 116 let m2 = mulf mEnd (exp (assume (Gaussian logAlpha sigma))) in 117 match tree with Node { left = left, right = right } then 118 simTree left tree lambda0 mu0 m1 logAlpha sigma rho;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 119 simTree right tree lambda0 mu0 m2 logAlpha sigma rho 120 else () 121 in 122 123 -- Priors 124 let lambda0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 in 125 let mu0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 in 126 let logAlpha = negf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3 in 127 let sigma = sqrt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 in 128 let m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='0 in 129 130 -- Adjust for normalizing constant 131 let numLeaves = countLeaves tree in 132 let corrFactor = 133 subf (mulf (subf (int2float numLeaves) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=') (log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=')) (lnFactorial numLeaves) in 134 weight corrFactor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 135 136 let m1 = mulf m (exp (assume (Gaussian logAlpha sigma))) in 137 let m2 = mulf m (exp (assume (Gaussian logAlpha sigma))) in 138 139 -- Start of the simulation along the two branches 140 (match tree with Node { left = left, right = right } then 141 simTree left tree lambda0 mu0 m1 logAlpha sigma rho;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 142 simTree right tree lambda0 mu0 m2 logAlpha sigma rho 143 else ());' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 144 145 -- Returns nothing, as the current model is only used to compute the 146 -- normalizing constant 147 () A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3 SMC: State-Space Aircraft Localization Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 11 presents the aircraft model used for the experiment in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' An aircraft flies along a one-dimensional axis in discrete time steps, and the crew needs to estimate the aircraft’s current position using noisy satellite position data available for the ten most recent time steps (defined at line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A second model component—the aircraft’s altitude—further complicates the model as the crew cannot observe it (the altimeter is not functioning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The aircraft’s velocity and the precision of the satellite observations depend on the altitude, as dictated by the functions velocity (defined at line 13) and positionObsStDev (defined at line 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The velocity (in meters per second) increases linearly with increasing altitude (less air resistance) but is capped to the range [100, 500].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' On the other hand, the observation standard deviation (in meters) decreases linearly with increasing altitude (less interference between the satellites and the aircraft) but is never less than ten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lines 25 to 44 define the main function simulate iterating over the ten data items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The critical component illustrating the need for alignment is the weight 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 at line 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This weight encodes that the pilot adjusts the aircraft’s pitch when air traffic control signals altitude deviations more than 100 feet from the assigned altitude of 35 000 feet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Each time step where the actual altitude deviates more than 100 feet from the assigned altitude thus gives a penalty factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Unlike the weight at line 29, this weight is unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 34 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1 let data = [ 2 603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='57, 860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='42, 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='07, 1163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='53, 3 1540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='29, 1818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='10, 2045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='38, 2363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='49, 4 2590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='77, 2801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='91 5 ] 6 let holdingAltitude = 35 000 in 7 let altitudeRange = 100 in 8 let position = assume Uniform(0, 1000) in 9 let altitude = 10 assume N(holdingAltitude, 2002) in 11 let positionStDev = 50 in 12 let baseVelocity = 250 in 13 let velocity = λaltitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 14 let k = baseVelocity holdingAltitude in 15 min (500, max (100, (k · altitude))) 16 in 17 let basePositionObsStDev = 50 in 18 let positionObsStDev : = λaltitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 19 let m = 100 in 20 let k = − basePositionObsStDev holdingAltitude in 21 max (10, m + k · altitude) 22 in 23 let altitudeStDev = 100 in 24 let rec simulate = 25 λdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' λposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' λaltitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 26 match data with d :: ds then 27 let σ = 28 positionObsStDev altitude in 29 weight fN (position,σ2)(d) 30 if |altitude − holdingAltitude| 31 > altitudeRange then 32 weight 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 33 else ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 34 let position = 35 assume N( 36 position + velocity altitude, 37 positionStDev 2 38 ) in 39 let altitude = 40 assume N(altitude, altitudeStDev 2) 41 in 42 simulate ds position altitude 43 else position 44 in 45 simulate data position altitude Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 11: A state-space model for estimating an aircraft’s position given a set of noisy position estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The text contains further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The program uses the syntax (1), extended with sequences, pattern matching over sequences, and the pattern :: for sequence deconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The function fN(µ,σ2) is the PDF of the normal distribution at µ with variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The simulation also accounts for variations in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', wind resistance when updating the position at line 34 through a standard deviation of positionStDev meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Similarly, the altitude varies with a standard deviation of altitudeStDev feet when updating the altitude at line 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We generated the ten data points used for the experiment in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3 by running the model (ignoring line 32) and sampling from N(position, σ2) at line 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Listing 3 gives the Miking CorePPL source code used for the case study model in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 MCMC: Latent Dirichlet Allocation (LDA) Listing 4 gives the Miking CorePPL source code used for the case study model in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, we conduct an additional LDA experiment justi- fying the correctness of the aligned lightweight MCMC and lightweight MCMC implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The experiment uses a simplified generated data set with only two topics, a vocabulary of two words, and three documents with 10 words each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To generate the data, we use the true values θ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='95, θ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='05, and θ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 for the document topic distributions, and φ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='99 and φ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='01 for the word distribution within the two topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that the true proportions above are uniquely determined by the proportion of the first topic and first word, as there are only two topics and two words in the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The simplicity of the Automatic Alignment in Higher-Order PPLs 35 Listing 3: The source code for the experiment in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3 1 --------------------------------------------------- 2 -- A state-space model for aircraft localization -- 3 --------------------------------------------------- 4 5 include "math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 6 7 mexpr 8 9 -- Noisy satellite observations of position (accuracy is improved at higher 10 -- altitude) 11 let ysPos: [Float] = [ 12 603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5736741666899, 860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4207338929477, 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='0766100484578, 1163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5339974878366, 13 1540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2972028551385, 1818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1023092741882, 2045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='3888580253108, 14 2363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4902615131796, 2590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='773153142429, 2801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='9143537470927 15 ] in 16 17 let holdingAltitude = 35000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in 18 let altitudeRange = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in 19 let position: Float = assume (Uniform 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=') in 20 let altitude: Float = assume (Gaussian holdingAltitude 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=') in 21 22 let positionStDev = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in 23 24 let baseVelocity = 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in 25 let velocity: Float -> Float = lam altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 26 let k = divf baseVelocity holdingAltitude in 27 minf 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (maxf 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (mulf k altitude)) 28 in 29 30 let basePositionObsStDev = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in 31 let positionObsStDev: Float -> Float = lam altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 32 let m = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in 33 let k = negf (divf basePositionObsStDev holdingAltitude) in 34 maxf 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (addf m (mulf k altitude)) 35 in 36 37 let altitudeStDev = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in 38 39 recursive let simulate: Int -> Float -> Float -> Float = 40 lam t: Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' lam position: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' lam altitude: Float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 41 42 -- Observe position 43 let dataPos: Float = get ysPos t in 44 observe dataPos (Gaussian position (positionObsStDev altitude));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 45 let t = addi t 1 in 46 47 -- Penalize altitude divergence of more than ‘altitudeRange‘ feet from 48 -- holding altitude 49 (if gtf (absf (subf altitude holdingAltitude)) altitudeRange then 50 weight (log 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5) 51 else ());' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 52 53 -- Transition 54 let position: Float = 55 assume (Gaussian (addf position (velocity altitude)) positionStDev) in 56 let altitude: Float = assume (Gaussian altitude altitudeStDev) in 57 58 if eqi (length ysPos) t then position 59 else simulate t position altitude 60 in 61 62 simulate 0 position altitude 36 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 1 θ1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 1 θ2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 1 θ3 (a) Aligned lightweight MCMC posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 1 θ1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 1 θ2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 1 θ3 (b) Lightweight MCMC posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 12: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (a) and (b) plots aligned lightweight MCMC and lightweight MCMC posterior distributions for the three documents θ1, θ2, and θ3 in the simplified LDA data set in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The posteriors are the combined samples of 300 independent MCMC runs, each with 3 · 106 iterations and 10% burn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' model and rather extreme true values used to generate the data allows for easy visualization of the document topic posteriors and justification of their correct- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 12 presents the posterior topic distributions for the three documents for a very large number of MCMC iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' As expected, aligned lightweight MCMC and lightweight MCMC produce identical results agreeing with the true values for θ1, θ2, and θ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The bimodal posteriors for θ1 and θ2 are due to the interchangeability of topics in LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5 MCMC: Constant Rate Birth-Death (CRBD) Listing 1 gives the Miking CorePPL source code used for the case study model in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, Fig 13 shows the posterior distributions over lambda, justifying the use of the mean as a measure of accuracy as the posterior is clearly unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' B Alignment Analysis, Continued This section presents the full alignment constraint propagation algorithm (Sec- tion B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1) and proof of soundness of the alignment analysis (Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 Algorithm Algorithm 4 presents the full alignment algorithm that produces a solution to the constraints generated by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For reference, we now also give a more formal definition of constraints c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 37 Listing 4: The source code for the experiment in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 1 include "common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 2 include "string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 3 include "seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 4 include "ext/dist-ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 5 6 -- The data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc file contains the generated data 7 include "data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='mc" 8 9 mexpr 10 11 let alpha: [Float] = make numtopics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in 12 let beta: [Float] = make vocabsize 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' in 13 let phi = create numtopics (lam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' assume (Dirichlet beta)) in 14 let theta = create numdocs (lam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' assume (Dirichlet alpha)) in 15 repeati (lam w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 16 let word = get docs w in 17 let counts = assume (Multinomial word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='1 (get theta (get docids w))) in 18 iteri (lam z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' lam e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 19 weight (mulf (int2float e) 20 (bernoulliLogPmf (get (get phi z) word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='0) true)) 21 ) counts 22 ) (length docs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 23 24 -- Returns the joint posterior distribution over theta and phi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 25 (theta, phi) 26 27 -- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' or the marginal posterior over theta (simple data set) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 28 -- theta 29 30 -- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' or nothing if only comparing exeuction time (C3 data set) 31 -- () 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='8 1 (a) Aligned lightweight MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='8 1 (b) Lightweight MCMC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 13: One iteration of the CRBD experiment in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (a) shows posteriors for aligned lightweight MCMC (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' From left to right: 3 · 104 iter- ations, 3 · 105 iterations, and 3 · 106 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (b) shows the corresponding posteriors for lightweight MCMC (white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 38 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Algorithm 4 Alignment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' function analyzeAlign(t): TANF → ((X → P(A)) × P(X)) = 1 worklist: [X] := [] 2 data: X → P(A) := {(x, ∅) | x ∈ X} 3 unaligned: P(X) := ∅ 4 edges: X → P(R) := {(x, ∅) | x ∈ X} 5 for c ∈ generateConstraints(t): 6 initializeConstraint(c) 7 iter();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' return (data, unaligned) 8 9 function iter: () → () = match worklist with 10 | [] → () 11 | x :: worklist’ → 12 worklist := worklist’ 13 for c ∈ edges(x): 14 propagateConstraint(c) 15 iter () 16 17 function initializeConstraint(c): R → () = 18 match c with 19 | a ∈ Sx → addData(x, {a}) 20 | Sx ⊆ Sy → initializeConstraint′(x, c) 21 | a1 ∈ Sx ⇒ a2 ∈ Sy → 22 initializeConstraint′(x, c) 23 | ∀x∀y λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='y ∈ Slhs 24 ⇒ (Srhs ⊆ Sx) ∧ (Sy ⊆ Sapp) → 25 initializeConstraint′(lhs, c) 26 | ∀n (const n ∈ Slhs) ∧ (n > 1) 27 ⇒ const n − 1 ∈ Sapp → 28 initializeConstraint′(lhs, c) 29 | const _ ∈ Slhs 30 ⇒ (stoch ∈ rhs ⇒ stoch ∈ app) → 31 initializeConstraint′(lhs, c) 32 | unalignedx ⇒ unalignedy → 33 initializeConstraint′(x, c) 34 | stoch ∈ Sx ⇒ unalignedy → 35 initializeConstraint′(x, c) 36 | ∀x λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedx → 37 initializeConstraint′(lhs, c) 38 | unalignedres ⇒ 39 (∀x λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedx) → 40 initializeConstraint′(res, c) 41 | stoch ∈ Slhs ⇒ 42 (∀x λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedx) → 43 initializeConstraint′(lhs, c) 44 45 function initializeConstraint′(x,c) 46 : X → () = 47 edges(x) := edges(x) ∪ {c};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 48 propagateConstraint(c) 49 50 function addData(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A): X × P(A) → () = 51 if A ̸⊆ data(x) then 52 data(x) := data(x) ∪ A 53 worklist := x :: worklist 54 55 function addUnaligned(x): X → () = 56 if x ̸∈ unaligned then 57 unaligned := unaligned ∪{x} 58 worklist := x :: worklist 59 60 function propagateConstraint(c): R → () = 61 match c with 62 | a ∈ Sx → () 63 | Sx ⊆ Sy → addData(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' data(x)) 64 | a1 ∈ Sx ⇒ a2 ∈ Sy → 65 if a1 ∈ data(x) then addData(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='{a2}) 66 | ∀x∀y λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='y ∈ Slhs 67 ⇒ (Srhs ⊆ Sx) ∧ (Sy ⊆ Sapp) → 68 for λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='y ∈ data(lhs): 69 initializeConstraint(Srhs ⊆ Sx) 70 initializeConstraint(Sy ⊆ Sapp) 71 | ∀n (const n ∈ Slhs) ∧ (n > 1) 72 ⇒ const n − 1 ∈ Sapp → 73 for const n ∈ data(lhs): 74 if n > 1 then 75 addData(app,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' {const n − 1}) 76 | const _ ∈ Slhs 77 ⇒ (stoch ∈ rhs ⇒ stoch ∈ app) → 78 if ∃n const n ∈ Slhs then 79 initializeConstraint( 80 stoch ∈ rhs ⇒ stoch ∈ app 81 ) 82 | unalignedx ⇒ unalignedy → 83 if x ∈ unaligned then addUnaligned(y) 84 | stoch ∈ Sx ⇒ unalignedy → 85 if stoch ∈ data(x) then addUnaligned(y) 86 | ∀x λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedx → 87 for λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ data(lhs): addUnaligned(x) 88 | unalignedres ⇒ 89 (∀x λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedx) → 90 if res ∈ unaligned then 91 initializeConstraint( 92 ∀x λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedx 93 ) 94 | stoch ∈ Slhs ⇒ 95 (∀x λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedx) → 96 if stoch ∈ data(lhs) then 97 initializeConstraint( 98 ∀x λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedx 99 ) Automatic Alignment in Higher-Order PPLs 39 Definition 8 (Constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' c ::= a ∈ Sx | Sx ⊆ Sy | a ∈ Sx ⇒ a ∈ Sy | ∀x∀y λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='y ∈ Slhs ⇒ (Srhs ⊆ Sx) ∧ (Sy ⊆ Sapp) | ∀n (const n ∈ Slhs) ∧ (n > 1) ⇒ const n − 1 ∈ Sapp | const _ ∈ Slhs ⇒ (stoch ∈ Srhs ⇒ stoch ∈ Sapp) | unalignedx ⇒ unalignedy | stoch ∈ Sx ⇒ unalignedy | ∀x λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedx | unalignedres ⇒ (∀x λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedx) | stoch ∈ Slhs ⇒ (∀x λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Slhs ⇒ unalignedx) x, y, lhs, rhs, app, res ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (8) The main function analyzeAlign consists of two steps: initialization and itera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In the initialization step, generateConstraints provides constraints to the initializeConstraint function, which initializes the maps data and edges, and the set unaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The map data contains the sets of abstract values for all program variables and is initially empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' At termination, data(x) is a sound approximation of Sx for each x (Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The map edges associates a set of constraints with each variable in the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Specifically, we must propagate the constraints associated with a variable x after updating data(x) with new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Finally, the set unaligned tracks unaligned expressions and is ini- tially empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' At termination, unaligned contains the set of all unaligned variables identified by the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' This set is sound according to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The iteration step iter propagates constraints with propagateConstraint for all variables updated with new abstract values or unalignment since their last propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We store these updated variables in the sequence worklist, which, when empty, signals fixpoint and termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', the lambda ap- plication constraint at line 67 initializes new constraints dynamically during propagation, depending on which abstract lambdas flow to the left-hand side of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='2 Correctness Proof This section presents the correctness proof that is ultimately used to prove The- orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Throughout this section, t1 = t2 means that the terms t1 and t2 are alpha equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For constant comparisons c1 = c2, we assume the prior existence of an equality function over constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We first require a specific equality relation on values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Definition 9 (Value equality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' v1 V= v2 iff – v1 = ⟨λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t1, ρ1⟩, v2 = ⟨λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='t2, ρ2⟩, and t1 = t2, or – v1 = c1, v2 = c2, and c1 = c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 40 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note, in particular, that V= treats closures as equal even if their environments differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' As we will see, this property is critical in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Next, we formally define subterms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Definition 10 (Subterms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We say that t′ is a subterm of t iff (1) t′ = t, or (2) either t = λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' t1, t = t1 t2, t = let x = t1 in t2, t = if t1 then t2 else t3, t = assume t1, or t = weight t1, and t′ is a subterm of either t1, t2, or t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In the below, we assume a – fixed t ∈ TANF, – an assignment to Sx and unalignedx for x ∈ X from analyzeAlign(t), and – �At = {x | ¬unaligned x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We begin with a lemma concerning unaligned expressions in single evaluations of ⇓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lemma 2 (Unaligned evaluations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Let – t′ be a subterm of t, t′ ∈ TANF, and – ρ ⊢ t′ s⇓w l v with ρ such that, for each x ∈ X, (C1) ρ(x) = ⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty, ρy⟩ implies that (λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty) is a subterm of t, λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='name(ty) ∈ Sx, and that (C1) holds for ρy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Also, ρ(x) = c such that |c| > 1 implies const |c| ∈ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Then, (R1) if unalignedn for all n ∈ names(t′), then l| � At = [], and (R2) v = ⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty, ρy⟩ implies (λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty) is a subterm of t and λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='name(ty) ∈ Sname(t′), and that (C1) holds for ρy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, v = c such that |c| > 1 implies const |c| ∈ Sname(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We proceed by structural induction over ρ ⊢ t′ s⇓w l v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Case t′ = x: The derivation is ρ ⊢ x []⇓1 [] ρ(x) (Var) (R1) Immediate as l = [] = l| �At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R2) By definition, name(t′) = x and ρ(x) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result follows from (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 41 Case t′ = (let x = t1 in t2): The derivation is ρ ⊢ t1 s1⇓w1 l1 v′ ρ, x �→ v′ ⊢ t2 s2⇓w2 l2 v ρ ⊢ let x = t1 in t2 s1∥s2⇓w1·w2 l1∥[x]∥l2 v (Let) Note that unalignedn for all n ∈ names(t′) and the definition of �At implies [x]| � At = [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Also, l| � At = (l1 ∥ [x] ∥ l2)| � At = l1| � At ∥ [x]| � At ∥ l2| � At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To show (R1), we therefore only need l1| � At = l2| � At = [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Now, let ρ′ = ρ, x �→ ρ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To apply the induction hypothesis, we must establish (C1) for ρ′, denoted (C1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To prove (C1′), note that we only need to consider ρ′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For all other x′ ∈ X, ρ′(x′) = ρ(x′) and (C1′) follows directly as a result of (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We denote the induction hypothesis results (R1)–(R2) for ρ′ ⊢ t2 s2⇓w2 l2 v with (R1′)– (R2′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Next, we consider each case for t1 (according to t′ ANF in (3), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that (R2) follows directly from (R2′) as name(t2) = name(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Thus, we only need to consider (C1′) and (R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = y The derivation for t1 is ρ ⊢ y []⇓1 [] ρ(y) (Var) Clearly ρ′(x) = ρ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Also, Sy ⊆ Sx from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C1′) If ρ′(x) = ⟨λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='tz, ρz⟩, then λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='tz is a subterm of t by ρ′(x) = ρ(y) and (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Also, clearly (C1) holds for ρz by (C1) for ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='name(tz) ∈ Sy by (C1) and Sy ⊆ Sx implies λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='name(tz) ∈ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By a similar argument, if ρ′(x) = c such that |c| > 0, const |c| ∈ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R1′)–(R2′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R1) We clearly have l1 = [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result now follows immediately from (R1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = c The derivation for t1 is ρ ⊢ c []⇓1 [] c (Const) Clearly, ρ′(x) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C1′) If |c| > 0, we have const |c| ∈ Sx as a result of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R1′)–(R2′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R1) We clearly have l1 = [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result now follows immediately from (R1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 42 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty The derivation for t1 is ρ ⊢ λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty []⇓1 [] ⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty, ρ⟩ (Lam) Clearly, ρ′(x) = ⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty, ρ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C1′) First, it is clear that λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty is a subterm of t and that (C1) holds for ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lastly, Lemma 1 also gives λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='name(ty) ∈ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R1′)–(R2′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R1) We clearly have l1 = [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result now follows immediately from (R1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = y z The possible derivations are ρ ⊢ y []⇓1 [] ⟨λy′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty′, ρy′⟩ ρ ⊢ z []⇓1 [] ρ(z) ρy′, y′ �→ ρ(z) ⊢ ty′ s1⇓w1 l1 v′ ρ ⊢ y z s1⇓w1 l1 v′ (App) ρ ⊢ y []⇓1 [] c ρ ⊢ z []⇓1 [] ρ(z) |c| > 0 ρ ⊢ y z []⇓1 [] δ(c, ρ(z)) (Const-App) (C1′) We first consider the case (App).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Let ρ′′ = ρy′, y′ �→ ρ(z) and consider the derivation ρ′′ ⊢ ty′ s1⇓w1 l1 v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To apply the induction hypothesis, we must establish (C1) for ρ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' First, (C1) holds for ρy′ by (C1) for ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C1) therefore also holds for ρ′′ by a similar argument to ρ′ in the subcase t1 = y above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' From the induction hypothesis, we then get (R1′′)–(R2′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' From Lemma 1, we have Sname(ty′ ) ⊆ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Combined with (R2′′), the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Now, consider the case (Const-App).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' From Lemma 1, ∀n const n ∈ Sy ∧ n > 1 ⇒ const n − 1 ∈ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' From Definition 2, we also have |δ(c, ρ(z))| = |c| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R1′)–(R2′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R1) The (Const-App) case is immediate by l1 = [] and (R1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Therefore, assume the derivation is (App) and that unalignedn for all n ∈ names(t′) (in particular unalignedx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By Lemma 1, we have unalignedy′ and unalignedn′ for n′ ∈ names(ty′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By (R1′′), we then have l1| � At = [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' From (R1′), we also have l2| � At = [] and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = if y then tt else te The possible derivations are ρ ⊢ y []⇓1 [] true ρ ⊢ tt s1⇓w1 l1 vt ρ ⊢ if y then tt else te s1⇓w1 l1 vt (If-True) ρ ⊢ y []⇓1 [] false ρ ⊢ te s1⇓w1 l1 ve ρ ⊢ if y then tt else te s1⇓w1 l1 ve (If-False) Automatic Alignment in Higher-Order PPLs 43 Without loss of generality, we only consider (If-True).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that for the sub- derivation ρ ⊢ tt s1⇓w1 l1 vt, (R1)–(R2), denoted (R1t)–(R2t) below, holds im- mediately by the induction hypothesis as (C1) holds for ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C1′) Follows from (R2t) and name(tt) ⊆ Sx from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R1′)–(R2′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R1) Assume we have unalignedn for all n ∈ names(t′) (including unalignedx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Then, by Lemma 1, unalignedn′ for n′ ∈ names(tt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Therefore, l1| � At = [] by (R1t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' From (R1′), we also have l2| � At = [] and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = assume y The derivation is ρ ⊢ y []⇓1 [] d w = fd(c) ρ ⊢ assume y [c]⇓w [] c (Assume) (C1′) By Definition 3, |c| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R1′)–(R2′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R1) We clearly have l1 = [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result now follows immediately from (R1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = weight y The derivation is ρ ⊢ y []⇓1 [] w ρ ⊢ weight y []⇓w [] () (Weight) (C1′) We have w ∈ R and |w| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R1′)–(R2′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R1) We clearly have l1 = [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result now follows immediately from (R1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' ⊓⊔ With Lemma 2 established, we now give the main lemma used to prove Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lemma 3 (Aligned evaluations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Let – t′ ∈ t, t′ ∈ TANF, – ρ1 ⊢ t′ s1⇓w1 l1 v1, and – ρ2 ⊢ t′ s2⇓w2 l2 v2 with ρ1 and ρ2 such that, for each x ∈ X, (C2) for ρ ∈ {ρ1, ρ2}, (C1) holds, (C3) if ρ1(x) = ⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty, ρ′ 1⟩, ρ2(x) = ⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty, ρ′ 2⟩, and stoch ̸∈ Sx, then ρ′ 1 and ρ′ 2 fulfill (C2)–(C4), and (C4) If ρ1(x) ̸ V= ρ2(x), then stoch ∈ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 44 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Then, (R3) l1| � At = l2| � At, (R4) if v1 = ⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty, ρ′ 1⟩, v2 = ⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty, ρ′ 2⟩, and stoch ̸∈ Sname(t′), then ρ′ 1 and ρ′ 2 fulfill (C2)–(C4), and (R5) If v1 ̸ V= v2, then stoch ∈ Sname(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We proceed by simultaneous structural induction over ρ1 ⊢ t′ s1⇓w1 l1 v1 and ρ2 ⊢ t′ s2⇓w2 l2 v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Case t′ = x: The possible derivations are ρ1 ⊢ x []⇓1 [] ρ1(x) (Var) ρ2 ⊢ x []⇓1 [] ρ2(x) (Var) (R3) We have l1 = l2 = [] = l1| � At = l2| � At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R4) By name(t′) = x and (C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R5) By name(t′) = x and (C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Case t′ = (let x = t1 in t2): The possible derivations are ρ1 ⊢ t1 s11⇓w11 l11 v′ 1 ρ1, x �→ v′ 1 ⊢ t2 s12⇓w12 l12 v1 ρ1 ⊢ let x = t1 in t2 s11∥s12⇓w11·w12 l11∥[x]∥l12 v1 (Let) ρ2 ⊢ t1 s21⇓w21 l21 v′ 2 ρ2, x �→ v′ 2 ⊢ t2 s22⇓w22 l22 v2 ρ2 ⊢ let x = t1 in t2 s21∥s22⇓w21·w22 l21∥[x]∥l22 v2 (Let) Assume l11| � At = l21| � At and l12| � At = l22| � At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Then, l1| � At = (l11 ∥ [x] ∥ l12)| � At = l11| � At ∥ [x]| � At ∥ l12| � At = l21| � At ∥ [x]| � At ∥ l22| � At = (l21 ∥ [x] ∥ l22)| � At = l2| � At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' That is, for (R3), we only need l11| � At = l21| � At and l12| � At = l22| � At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Let ρ′ 1 = ρ1, x �→ v′ 1) and ρ′ 2 = ρ2, x �→ v′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To apply the induction hypothesis, we must establish (C2)–(C4) for ρ′ 1 and ρ′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To avoid confusion with the original as- sumptions (C2)–(C4) for ρ1 and ρ2, we use the notation (C2′)–(C4′) for the ρ′ 1 and ρ′ 2 conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To prove (C2′)–(C4′), note that we only need to consider ρ′ 1(x) and ρ′ 2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For all other x′ ∈ X, ρ′ 1(x′) = ρ1(x′) and ρ′ 2(x′) = ρ2(x′), and (C2′)–(C4′) follow directly as a result of (C2)–(C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We denote the induc- tion hypothesis results (R3)–(R5) for ρ′ 1 ⊢ t2 s12⇓w12 l12 v1 and ρ′ 2 ⊢ t2 s22⇓w22 l22 v2 with (R3′)–(R5′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Next, we consider each case for t1 (according to t′ ANF in (3), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that (R4) and (R5) follow directly from (R4′) and (R5′) as name(t2) = name(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Thus, we only need to consider (C2′)–(C4′) and (R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 45 Subcase t1 = y The derivations for t1 are ρ1 ⊢ y []⇓1 [] ρ1(y) (Var) ρ2 ⊢ y []⇓1 [] ρ2(y) (Var) We first establish (C2′)–(C4′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Clearly ρ′ 1(x) = ρ1(y) and ρ′ 2(x) = ρ2(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Also, Sy ⊆ Sx from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for both ρ′ 1(x) and ρ′ 2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C3′) Assume ρ′ 1(x) = ⟨λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='tz, ρ′′ 1⟩, ρ′ 2(x) = ⟨λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='tz, ρ′′ 2⟩, and stoch ̸∈ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By Sy ⊆ Sx, stoch ̸∈ Sy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Because ρ′ 1(x) = ρ1(y) and ρ′ 2(x) = ρ2(y), the result follows from (C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C4′) If ρ′ 1(x) ̸ V= ρ′ 2(x), then clearly ρ1(y) ̸ V= ρ2(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Hence, stoch ∈ Sy by (C4) and the result follows by Sy ⊆ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R3′)–(R5′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R3) The result follows from l11 = l21 = [] and (R3′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = c The derivations for t1 are ρ1 ⊢ c []⇓1 [] c (Const) ρ2 ⊢ c []⇓1 [] c (Const) We first establish (C2′)–(C4′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for ρ′ 1(x) = ρ′ 2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C3′) Follows directly as ρ′ 1(x) = ρ′ 2(x) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C4′) Follows directly because ρ′ 1(x) = ρ′ 2(x) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R3′)–(R5′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R3) The result follows from l11 = l21 = [] and (R3′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty The derivations are ρ1 ⊢ λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty []⇓1 [] ⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty, ρ1⟩ (Lam) ρ2 ⊢ λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty []⇓1 [] ⟨λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty, ρ2⟩ (Lam) We first establish (C2′)–(C4′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for both ρ′ 1(x) and ρ′ 2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 46 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C3′) Follows because ρ1 and ρ2 fulfills (C2)–(C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C4′) Follows because ρ′ 1(x) V= ρ′ 2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R3′)–(R5′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R3) The result follows from l11 = l21 = [] and (R3′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = y z The possible derivations are ρ1 ⊢ y []⇓1 [] ⟨λy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty1, ρy1⟩ ρ1 ⊢ z []⇓1 [] ρ1(z) ρy1, y1 �→ ρ1(z) ⊢ ty1 s11⇓w11 l11 v′ 1 ρ1 ⊢ y z s11⇓w11 l11 v′ 1 (App) ρ2 ⊢ y []⇓1 [] ⟨λy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty2, ρy2⟩ ρ2 ⊢ z []⇓1 [] ρ2(z) ρy2, y2 �→ ρ2(z) ⊢ ty2 s21⇓w21 l21 v′ 2 ρ2 ⊢ y z s21⇓w21 l21 v′ 2 (App) ρ1 ⊢ y []⇓1 [] c1 ρ1 ⊢ z []⇓1 [] ρ1(z) |c1| > 0 ρ1 ⊢ y z []⇓1 [] δ(c1, ρ1(z)) (Const-App) ρ2 ⊢ y []⇓1 [] c2 ρ2 ⊢ z []⇓1 [] ρ2(z) |c2| > 0 ρ2 ⊢ y z []⇓1 [] δ(c2, ρ2(z)) (Const-App) We first establish (C2′)–(C4′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for both ρ′ 1(x) and ρ′ 2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C3′) Assume stoch ̸∈ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For (Const-App), ρ′ 1(x) ∈ C and ρ′ 2(x) ∈ C, and the result follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Therefore, assume that both derivations are (App).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By Lemma 1, stoch ∈ Sy ⇒ stoch ∈ Sx, and consequently stoch ̸∈ Sy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By (C4), this leads to ρ1(y) = ⟨λy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty1, ρy1⟩ V= ρ2(y) = ⟨λy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty2, ρy2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' That is, λy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty1 = λy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty2 = λy′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ty′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By (C3), ρy1 and ρy2 fulfill (C2)– (C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Let ρ′′ 1 = ρy1, y′ �→ ρ1(z) and ρ′′ 2 = ρy2, y′ �→ ρ2(z) and consider the derivations ρ′′ 1 ⊢ ty′ s11⇓w11 l11 v′ 1 and ρ′′ 2 ⊢ ty′ s21⇓w21 l21 v′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' It is straightforward to check that ρ′′ 1 and ρ′′ 2 fulfill (C1), and we apply the induction hypothe- sis and get the results (R3′′)–(R5′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Now, by Lemma 1, Sname(ty′ ) ⊆ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Combined with (R4′′), the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C4′) Assume ρ′ 1(x) ̸ V= ρ′ 2(x), and consider first the case where ρ1(y) ̸ V= ρ2(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Then, by (C4), stoch ∈ Sy and by stoch ∈ Sy ⇒ stoch ∈ Sx from Lemma 1, stoch ∈ Sx and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Therefore, assume ρ1(y) V= ρ2(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Consequently, both derivations are either (Const-App) or (App).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If both derivations are (Const-App), c1 V= c2 V= c and ρ′ 1(x) ̸ V= ρ′ 2(x) implies ρ1(z) ̸ V= ρ2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By (C4), this implies stoch ∈ Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lemma 1 gives const _ ∈ Sy ⇒ (stoch ∈ Sz ⇒ stoch ∈ Sx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Clearly, const |c| ∈ Sy by (C2) and |c| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 47 It follows that stoch ∈ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If both derivations are instead (App), we repeat the argument for (C3′) and get (R3′′)–(R5′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Furthermore, we must have v′ 1 = ρ′ 1(x) ̸ V= ρ′ 2(x) = v′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By Lemma 1, Sname(ty′ ) ⊆ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result now follows from (R5′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R3′)–(R5′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R3) First, by (R3′), we have l12| � At = l22| � At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now show l11| � At = l21| � At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Assume that stoch ∈ Sy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Then, in all cases we have l11| � At = l21| � At = [] and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' To see this, note first that for the (Const-App) derivations, the result holds immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Therefore, assume both deriva- tions are (App).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Now, by Lemma 1, we have stoch ∈ Sy ⇒ (∀y′ λy′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='_ ∈ Sy ⇒ unaligned ′ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In other words, unalignedy1 and unalignedy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Again by Lemma 1, unalignedn1 for all n1 ∈ names(ty1) and unaligned n2 for all n2 ∈ names(ty2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Let ρ′′ 1 = ρy1, y1 �→ ρ1(z) and ρ′′ 2 = ρy2, y2 �→ ρ2(z) and consider the derivations ρ′′ 1 ⊢ ty1 s11⇓w11 l11 v′ 1 and ρ′′ 2 ⊢ ty2 s21⇓w21 l21 v′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' It is straightforward to check that ρ′′ 1 and ρ′′ 2 fulfill (C1) and double applications of Lemma 2 give the required result l11| � At = l21| � At = [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Now, assume that stoch ̸∈ Sy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Clearly, both derivations are either (Const- App) or (App).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The (Const-App) case is trivial, because l11| � At = l21| � At = [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Therefore, assume both derivations are (App).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By repeating the reasoning in (C3′), we get (R3′′)–(R5′′) by the induction hypothesis for the derivations ρ′′ 1 ⊢ ty′ s11⇓w11 l11 v′ 1 and ρ′′ 2 ⊢ ty′ s21⇓w21 l21 v′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In other words, l11| � At = l21| � At by (R3′′) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='Subcase t1 = if y then tt else te ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='The possible derivations are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ1 ⊢ y []⇓1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='[] true ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ1 ⊢ tt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='s11⇓w11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='l11 vt1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ1 ⊢ if y then tt else te ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='s11⇓w11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='l11 vt1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='(If-True) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ2 ⊢ y []⇓1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='[] true ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ2 ⊢ tt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='s21⇓w21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='l21 vt2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ2 ⊢ if y then tt else te ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='s21⇓w21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='l21 vt2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='(If-True) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ1 ⊢ y []⇓1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='[] false ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ1 ⊢ te ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='s11⇓w11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='l11 ve1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ1 ⊢ if y then tt else te ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='s11⇓w11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='l11 ve1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='(If-False) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ2 ⊢ y []⇓1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='[] false ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ2 ⊢ te ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='s21⇓w21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='l21 ve2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='ρ2 ⊢ if y then tt else te ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='s21⇓w21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='l21 ve2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='(If-False) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='We first establish (C2′)–(C4′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C2′) Holds in all four cases by repeating the corresponding argument for (C1′) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C3′) Assume stoch ̸∈ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By Lemma 1, clearly stoch ̸∈ Sy and both deriva- tions are either (If-True) or (If-False).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Without loss of generality, assume both derivations are (If-True).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The induction hypothesis directly applies to ρ1 ⊢ tt s11⇓w11 l11 vt1 and ρ2 ⊢ tt s21⇓w21 l21 vt2, and we get the result (R3t)– (R5t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By Lemma 1, name(tt) ⊆ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result now follows from (R4t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 48 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C4′) Assume first that stoch ∈ Sy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Then stoch ∈ Sx by Lemma 1, and the result is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Therefore, assume stoch ̸∈ Sy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Again, both derivations are either (If-True) or (If-False) and we assume, without loss of gener- ality, that both are (If-True).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The induction hypothesis directly applies to ρ1 ⊢ tt s11⇓w11 l11 vt1 and ρ2 ⊢ tt s21⇓w21 l21 vt2, and we get the result (R3t)– (R5t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By Lemma 1, name(tt) ⊆ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result now follows from (R5t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R3′)–(R5′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R3) First, by (R3′), we have l12| � At = l22| � At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now show l11| � At = l21| � At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If stoch ∈ Sy, then by Lemma 1, unaligned nt for all nt ∈ names(tt) and unalignedne for all ne ∈ names(te).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By repeating Lemma 2 twice, we get l11| � At = l21| � At = [] and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Assume stoch ̸∈ Sy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Again, both derivations are either (If-True) or (If- False) and we assume, without loss of generality, that both are (If-True).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The induction hypothesis directly applies to ρ1 ⊢ tt s11⇓w11 l11 vt1 and ρ2 ⊢ tt s21⇓w21 l21 vt2, and we get the result (R3t)–(R5t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' By (R3t), l11| � At = l21| � At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = assume y The derivations are ρ1 ⊢ y []⇓1 [] d1 w1 = fd1(c1) ρ1 ⊢ assume y [c1]⇓w1 [] c1 (Assume) ρ2 ⊢ y []⇓1 [] d2 w2 = fd2(c2) ρ2 ⊢ assume y [c2]⇓w2 [] c2 (Assume) We first establish (C2′)–(C4′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for both ρ′ 1(x) and ρ′ 2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C3′) Immediate as ρ′ 1(x) = c1 and ρ′ 2(x) = c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C4′) By Lemma 1, stoch ∈ Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R3′)–(R5′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R3) The result follows from l11 = l21 = [] and (R3′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Subcase t1 = weight y The derivations are ρ1 ⊢ y []⇓1 [] w1 ρ1 ⊢ weight y []⇓w1 [] () (Weight) ρ2 ⊢ y []⇓1 [] w2 ρ2 ⊢ weight y []⇓w2 [] () (Weight) We first establish (C2′)–(C4′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Automatic Alignment in Higher-Order PPLs 49 Algorithm 5 Unaligned SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The input is a program t ∈ TANF and the number of execution instances n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Initiate n execution instances {ei | i ∈ N, 1 ≤ i ≤ n} of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Execute all ei (for already terminated ei, do nothing) and suspend execution upon reaching a weight (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', let x = weight w in t) or when the execution terminates naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result is a new set of execution instances e′ i with weights w′ i (from w, or 1 if already terminated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If all e′ i = v′ i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', all executions have terminated and returned a value), terminate inference and return the set of samples v′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The samples approximate the probability distribution encoded by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Resample the e′ i according to their weights w′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The result is a new set of unweighted execution instances e′′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set ei ← e′′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Go to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C2′) By repeating the corresponding argument for (C1′) in Lemma 2 for both ρ′ 1(x) and ρ′ 2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C3′) Immediate as ρ′ 1(x) = ρ′ 2(x) = ().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (C4′) Immediate as ρ′ 1(x) V= ρ′ 2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now apply the induction hypothesis and get (R3′)–(R5′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (R3) The result follows from l11 = l21 = [] and (R3′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' ⊓⊔ C Unaligned SMC Algorithm 5 presents the unaligned SMC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' It is in many ways similar to Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' D Lightweight MCMC Algorithm 6 presents the lightweight MCMC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The algorithm is in many ways similar to Algorithm 3, but relies on databases represented with Di (random draws) and pi (probability densities/masses of the draws) to reuse random draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The Run function keeps track of the current stack trace t at all times and uses it to index the databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' E Metropolis–Hastings Acceptance Ratio This section derives the Metropolis–Hastings acceptance ratio used in Algo- rithm 3 and Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We assume basic familiarity with Bayesian statistics and the Metropolis–Hastings algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Bayes’ theorem on probability density/mass functions is usually written as p(x|y) = p(y|x)p(x) p(y) (10) 50 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Algorithm 6 Lightweight MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The input is a program t ∈ TANF, the num- ber of steps n, and the global step probability g > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set i ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Call Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set i ← i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If i = n, terminate inference and return the samples {vj | j ∈ N, 0 ≤ j < n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' They approximate the probability distribution encoded by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Uniformly draw a trace t′ from dom(Di−1) at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set global ← true with probability g, and global ← false otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set w′ −1 ← 1, and w′ ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Call Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Compute the Metropolis–Hastings acceptance ratio A = min � 1, wi wi−1 w′ w′ −1 |dom(Di−1)| |dom(Di)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (9) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' With probability A, accept vi and go to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Otherwise, set vi ← vi−1, wi ← wi−1, Di ← Di−1, and pi ← pi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Go to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' function run() = Let t represent the current stack trace throughout execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Run t and do the following: – Record the total weight wi accumulated from calls to weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – Record the final value vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – At terms let c = assume d in t, do the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' If t = t′, global = true, or if t ̸∈ dom(Di−1), sample a value x from d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Otherwise, reuse the sample x = Di−1(t) and set w′ −1 ← w′ −1 · pi−1(t) and w′ ← w′ · fd(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Set Di(t) ← x and pi(t) ← fd(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In the program, bind c to the value x and resume execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' where y is some fixed observed random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The standard Metropolis– Hastings ratio for a proposal distribution with probability density/mass q(x′|x) is then A(x, x′) = min � 1, p(x′|y) p(x|y) q(x|x′) q(x′|x) � = min � 1, p(y|x′)p(x′) p(y|x)p(x) q(x|x′) q(x′|x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (11) Assume a fixed program t ∈ T in the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' For such a program, Bayes’ theorem takes a generalized form ˆp(s) = L(s)p(s) Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (12) Here, we have replaced x with a trace s (a sequence of random values during evaluation of a probabilistic program) and removed the dependence on y entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We use the notation ˆp and p to differentiate between the posterior and prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The likelihood function is denoted L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Z is a normalizing constant that disappears in the Metropolis–Hastings ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' One can view (12) in the context of the semantics in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We (very infor- mally) have ˆp(s) = w up to normalization iff ∅ ⊢ t l⇓w s v for some l and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' L(s) is then the contribution to w from (Weight), and p(s) from (Assume).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' The PPL version of the Metropolis–Hastings ratio is A(s, s′) = min � 1, ˆp(s′) ˆp(s) q(s|s′) q(s′|s) � = min � 1, L(s′)p(s′) L(s)p(s) q(s|s′) q(s′|s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (13) Automatic Alignment in Higher-Order PPLs 51 The most trivial proposal, amounting to not reusing any draws, is q(s′|s) = p(s′) (14) This directly gives the ratio A(s, s′) = min � 1, L(s′) L(s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' (15) To derive the ratio for aligned lightweight MCMC and lightweight MCMC, we need to first capture the proposal q used in Algorithm 3 and Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We capture the reuse mechanisms (alignment and the stack trace database) in both algorithms through functions D1 : S × S → P(N) and D2 : S × S → P(N) such that |D1(s, s′)| = |D2(s, s′)|, D1(s, s′) = D2(s′, s), and D2(s, s′) = D1(s′, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Intuitively, D1(s, s′) gives the indices in s that match the indices D2(s, s′) in s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We now define the proposal q as q(s′, i|s) = [s′|A′ = s|A]p|A′C(s′)pi(i|s) (16) and make the following definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – A′ = D2(s, s′) \\ f(i, s′) and A = D1(s, s′) \\ f(i, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – The function f(i, s) transforms the index i in the context of s (explained further below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – The function pi(i|s) is the density for selecting an i given s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – The trace s|A is the restriction of s to A (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Definition 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – [· · · ] is the Iverson bracket (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=', evaluates to zero if the predicate · · · is false and to one if the predicate is true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' – We denote the contribution to p(s) from the indices A in s with p|A(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Importantly, p|A(s) · p|AC(s) = p(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Note that q also proposes an auxiliary variable i—the trace index that we choose to redraw in the proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Due to the auxiliary variable i, the acceptance ratio is now a function of three arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' A(s, s′, i) = min � 1, L(s′)p(s′) L(s)p(s) q(s, i|s′) q(s′, i|s) � = min � 1, L(s′)p(s′) L(s)p(s) [s|A = s′|A′] [s′|A′ = s|A] p|AC(s) p|A′C(s′) pi(i|s′) pi(i|s) � = min � 1, L(s′) L(s) p(s′) p|A′C(s′) p|AC(s) p(s) pi(i|s′) pi(i|s) � = min � 1, L(s′) L(s) p|A′(s′) p|A(s) pi(i|s′) pi(i|s) � (17) This acceptance ratio is equivalent to the ratio derived by van de Meent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' [45, Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' We first view (17) in the context of aligned lightweight MCMC in Algo- rithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Here, f(i, s) returns the i-th aligned index in s (not index i in s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' In 52 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Lundén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' aligned lightweight MCMC, we only select what to redraw among the aligned draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' As we know, the number of aligned draws is fixed across all possible executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' pi(i|s) is thus a constant, and (17) reduces to A(s, s′, i) = min � 1, L(s′) L(s) p|A′(s′) p|A(s) � (18) This is the ratio computed in step 5 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Next, we consider lightweight MCMC in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Here, we simply choose f(i, s) = i (the identity function), and select an element to redraw uniformly over the previous trace s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} +page_content=' Thus, pi(i|s) = 1/|s| and (17) reduces to A(s, s′, i) = min � 1, L(s′) L(s) p|A′(s′) p|A(s) |s| |s′| � (19) This is the ratio computed in step 4 of Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQf4C1z/content/2301.11664v1.pdf'} diff --git a/VNE2T4oBgHgl3EQfDAaO/content/tmp_files/2301.03620v1.pdf.txt b/VNE2T4oBgHgl3EQfDAaO/content/tmp_files/2301.03620v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7cdd4d8053fd2b62a0139cb2b0ecac653de21002 --- /dev/null +++ b/VNE2T4oBgHgl3EQfDAaO/content/tmp_files/2301.03620v1.pdf.txt @@ -0,0 +1,1474 @@ +Triggering Higgs vacuum decay +Alessandro Strumia +Dipartimento di Fisica dell’Universit`a di Pisa +Abstract +The Standard Model Higgs potential seems unstable at field values h > htop ∼ +1010 GeV. Vacuum decay can be triggered by N ∼ 4π/λ ∼ 1000 overlapped Higgs +bosons with energy +√ +λhtop. +However, this configuration is stimulated by ultra- +high energy collisions with a exp(−O(N)) suppression, comparable to spontaneous +vacuum decay: no ‘Higgspolosion’ enhancement arises. +This implies that ultra- +high energy cosmic ray collisions are safe, despite that their number (in production +sites) likely is tens of orders of magnitude higher than what usually estimated (in +space). We speculate on how vacuum decay could be induced classically, forming a +in-coming wave of N boosted Higgs bosons at futuristic ultra-high energy colliders, +and on how the resulting vacuum bubble could be controlled to extract energy. +Contents +1 +Introduction +2 +2 +Triggering Higgs vacuum decay +3 +2.1 +Simple estimates +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2.2 +Numerical computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +3 +Vacuum decay stimulated by few particles +5 +3.1 +Stimulated tunnelling in the thin-wall limit . . . . . . . . . . . . . . . . . . . . . +6 +3.2 +Stimulated tunnelling beyond the thin-wall limit . . . . . . . . . . . . . . . . . . +8 +3.3 +Vacuum decay via Higgsplosion? . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +4 +Highest-energy cosmic-ray collisions +11 +4.1 +Collisions of 2 cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +4.2 +Collisions of N cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +5 +Vacuum decay triggered by many particles +14 +5.1 +Collision scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +5.2 +Classical Higgs wave +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +5.3 +Can a vacuum bubble be controlled, extracting energy? . . . . . . . . . . . . . . +18 +6 +Conclusions +19 +arXiv:2301.03620v1 [hep-ph] 9 Jan 2023 + +1 +Introduction +The discovery of Higgs boson h opens the door to a new form of energy, the vacuum energy +in the Higgs potential V (h). +The Higgs currently sits at the Standard Model (SM) local +minimum of its potential. If this is the global minimum, no vacuum energy remains stored in +our Universe, and vacuum cannot be used as a source of energy. However, extrapolating the SM +Higgs potential V (h) to ultra-high field values h suggests that the potential might have a local +maximum at h = htop ∼ 1010 GeV becoming negative at h > e1/4htop [1,2]. This happens for +current best-fit values of the top quark mass, Higgs mass and strong coupling. Within current +uncertainties a much larger instability scale or even stability is possible. Clarifying if V (h) is +really unstable needs improved future measurements of the top mass, doable at a lepton collider +at the top threshold (such as an e−e+ collider in the LEP tunnel, or a µ−µ+ collider) [3]. +Assuming that the Higgs instability exists, we explore under which conditions it can result +in vacuum decay. The following possibilities have been considered in the literature: +• Spontaneous vacuum tunnelling [4] is exponentially slow [5,6,1,2,7]. +• The thermal energy in the early universe at temperatures T >∼ htop could have overcome +the potential barrier in V (h), but thermal effects also add a thermal barrier: as a result +vacuum decay remains exponentially slow at any T (see e.g. [8]). +• Inflationary fluctuations could have trigger the instability, but this depends on the un- +known inflation model and non-minimal Higgs coupling to gravity (see e.g. [2]). +• Small hypothetical primordial black holes with Hawking temperature T >∼ htop are a pos- +sible seed of vacuum decay, but the rate is again exponentially slow, similarly to thermal +vacuum decay (claims in the literature differ, I think this is the correct conclusion as +discussed in [9,10]). +• Vacuum decay seeded by particle collisions with ultra-high energy E >∼ htop has been +considered for some scalar potentials, finding that the rate remains partially exponentially +suppressed [11–21]. +In this paper we reconsider the latter issue. The exponential suppression persists because, as +discussed in section 2, the minimal configuration needed to seed vacuum decay contains N ≫ 1 +scalar quanta. In the case of SM Higgs vacuum decay, one needs N ∼ 4π/λ ∼ 1000 Higgs +boson quanta. Such a semi-classical state is generated out of collisions of a few quanta with +rate exponentially suppressed by a ‘one to many particles’ factor. In section 3 we revisit and +extend previous computations, focusing on the specific case of Higgs vacuum decay, that needs +going beyond the thin-wall approximation, and summarizing why cross sections for producing +N Higgs bosons remain small at large N (no ‘Higgsplosion’ happens). +In section 4 we reconsider collisions of two ultra-high-energy cosmic rays (CR), showing that +a significantly enhanced CR collision rate happened if cosmic rays are accelerated by relatively +compact astrophysical objects, from magnetars up to Active Galactic Nuclei. We will also +2 + +find that few-particle ultra-high energy CR collisions can have happened, while many-particle +collisions never occurred. As a result, CR collisions could not trigger Higgs vacuum decay. +The main simple lesson is that an appropriate many-particle classical process is needed to +trigger vacuum decay avoiding the exponential suppression that would make the rate negligible. +We then explore in section 5 if the critical configuration could be engineered classically via +futuristic collider schemes with ultra-high energy √s >∼ htop. We next discuss if the true vacuum +bubble can be artificially controlled and used to extract energy from the vacuum. +Section 6 presents our conclusions. +2 +Triggering Higgs vacuum decay +We consider an initial configuration of the Higgs field h(t, r) at t = 0 with field value h ∼ h0 +within a space region of size r <∼ r0. For simplicity we assume spherical symmetry, denote as r +the radius, and assume that the initial configuration is at rest. An example is +h(0, r) = +h0 +1 + r2/r2 +0 +, +˙h(0, r) = 0. +(1) +The classical Higgs field equation in flat space is ¨h − h′′ − 2h′/r = −V ′, where, as usual, +˙h = ∂h/∂t and h′ = ∂h/∂r. +2.1 +Simple estimates +The total energy of the configuration is +E = +� +dr 4πr2 +�h′2 +2 + V +� +∼ 4πr0h2 +0[1 + λr2 +0h2 +0] +(2) +having here simply approximated the potential energy as V = λh4/4. +In its subsequent evolution this field configuration triggers vacuum decay rather than dis- +solving if two conditions are met: +1) h0 >∼ htop is over the top of the SM potential barrier; +2) h0r0 +� +|λ| >∼ 1 such that the potential energy wins over the gradient energy in the classical +evolution. This estimate can be obtained by demanding that potential energy dominates +over the gradient energy in the total energy of eq. (2), or by considering the classical +equation of motion at the point r = 0, or in thin-wall approximation (eq. (13) later). +According to conditions 1) and 2) above, an initially static configuration h(0, r) evolves towards +the true vacuum if its energy is +E >∼ Emin ∼ 4πhtop +� +|λ| +. +(3) +3 + +The Fourier transform of h(0, r) is large at k <∼ 1/r0, showing that the scalar field configuration +contains +N ∼ E +k ∼ 4π(h0r0)2 >∼ Nmin ∼ Emin +k +∼ 4π +|λ| +(4) +Higgs quanta. +2.2 +Numerical computations +We validate and precise the above analytic approximations by numerically solving the classical +Higgs field equation in flat space. The SM Higgs potential can be accurately approximated +through an effective running coupling as +V (h) ≈ λ(h)h4 +4 , +λ(h) ≈ −b ln +h2 +e1/2h2 +top +with +b ≈ 0.15 +(4π)2 +(5) +for best-fit values of the SM parameters. Switching to the dimension-less field ˜h = h/htop and +coordinates ˜x = xhtop +√ +b, the Higgs action simplifies to +S = 1 +b +� +d4˜x +� +�1 +2 +� +∂˜h +∂˜xµ +�2 ++ +˜h4 +4 ln +˜h2 +e1/2 +� +� , +(6) +showing that the dynamics depends in a simple way on the values of b and htop. +We evolve the Higgs field starting from an initial field profile with vanishing time derivative. +The result for the profile in eq. (1) is shown in fig. 1a: the ‘drainage divide’ critical boundary +in the (h0, r0) plane that separates initial configuration that end up in the false vacuum or in +the true vacuum is well approximated by the analytic conditions 1) and 2). Fig. 1a also shows +iso-curves of the approximate number of Higgs quanta, computed from the Fourier transform +hk = +� +dr 4πr2 h(r)sin kr +kr +as +N = +� dk 4πk2 +(2π)3k +k2 +2 |hk|2. +(7) +The expression for N applies to an in-going or out-going Higgs wave with h ≪ htop such that +Higgs quanta are free with energy k. So eq. (7) can be applied to the field configuration at a +time such that the bubble is dissolving into the false vacuum, and field values h got much below +htop. For simplicity, in fig. 1 we will evaluate eq. (7) at t = 0 on the static configuration of +eq. (1) with h ∼ htop obtaining N = (πr0h0/2)2. But the true number of quanta is kN/2 ≈ N, +where the factor 1/2 accounts for the static initial configuration, and k for the potential energy, +with k ∼ 1 − 2 for critical bubbles. +Fig. 1a shows that increasing the energy k ∼ 1/r0 of each quantum does not reduce the +needed N below Nmin ≈ 1/b ≈ 1000, while N ≫ Nmin quanta are needed for a large bubble +made of quanta with energy lower than the optimal energy k ∼ +� +|λ|htop ∼ 0.1htop. A nearly +identical figure is obtained replacing eq. (1) with h(r) = h0e−r2/r2 +0, for which N = π(r0h0)2/2 +is mildly lower. +4 + +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +Radius r = r htop +b +Higgs field intensity h(r)/htop +Sphaleron +Figure 1: Left: We numerically evolve a spherical Higgs bubble h0/(1+r2/r2 +0) initially at rest. In the +red region the bubble expands triggering vacuum decay, while in the green region the bubble dissolves. +The red region is well approximated as h0 > htop and h0r0 > 1/ +� +|λ| (dashed curve). The hatching +covers the unphysical region where the bubble is so large that its energy is negative, E < 0. The blue +iso-contours show the number N of Higgs quanta. This confirms that expansion needs N >∼ 4π/|λ| +quanta naively computed from the static configuration at t = 0. The sphaleron is denoted as ‘Sph’. +Right: sphaleron solution, and its analytic approximation (dashed curve). +A special configuration is the sphaleron, the static spherical solution to the Higgs field +equation. Thereby it is the critical configuration with lowest energy +Emin = Esph ≈ 9.7htop +√ +b +≈ 500 Joule +htop +1010 GeV. +(8) +The sphaleron has h0 ≈ 4.1htop and its profile is plotted in fig. 1b. It can be approximated as +eq. (1) with r0 ≈ 0.4/htop +√ +b, corresponding to the point denoted as ‘Sph’ in fig. 1a. By letting +the sphaleron dissolve in time [22], one finds Nsph ≈ 2.2/b >∼ Nmin. Critical configurations with +higher energy E > Esph do not have N significantly lower than Nsph. This physically suggests +that vacuum decay stimulated by particle collisions will remain exponentially suppressed, as +discussed in the next section. +3 +Vacuum decay stimulated by few particles +We have seen that a Higgs field configuration that evolves into the true vacuum contains N > +Nmin ∼ 4π/|λ| ∼ 103 Higgs quanta. Then, one expects that the probability of generating such +5 + +1 +10-1 +10-2 +10-3 +30 +10 +E<0 +Sph +3 +E>0 +Bubble expands +10 +1 +Bubble evaporates +106 +100 +1000 +101 +102 +1 +103 +Bubble radius ro htopa configuration out of high-energy collisions with few particles is suppressed by an exponential +factor ∼ e−O(N) parametrically similar to the vacuum tunnelling rate ∼ e−Sbounce. +To see this, let us over-simplify the problem considering free Higgs particles, such that one +can define the number operator ˆNk = ˆa† +kˆak in terms of the creation operator ˆa† +k. As well known, +states with exactly N quanta have zero field value ⟨N|ˆh|N⟩ and large fluctuations. The good +semi-classical states with small field fluctuations and large N = ⟨α| ˆN|α⟩ = |α|2 are instead the +coherent states |α⟩ = e−|α|2/2 �∞ +n=0 αn|n⟩/ +√ +n!, eigenstates of ˆa|α⟩ = α|α⟩. The scalar product +between two coherent states is exponentially suppressed as |⟨α|β⟩|2 = e−|α−β|2 +. +A qualitatively similar suppression persists taking into account the interaction in the scalar +potential. The vacuum decay rate induced by particle collisions was computed in thin-wall +approximation by Voloshin [15] using the following WKB-like result from Landau [23]. Let us +consider (for simplicity) a quantum system described by an Hamiltonian H that depends on +one degree of freedom q and its conjugated momentum p. The matrix element of an operator +O among quasi-classical states with energies E1, E2 is not easily computed, because their wave +functions oscillate wildly, averaging to nearly zero. Suitable analytic continuations to complex +q in the direction of decreasing WKB exponentials show that the matrix element is suppressed +as [23] +⟨E2|O|E1⟩ ∼ exp +� +−Im +�� q∗ +q1 +p(q, E1)dq + +� q2 +q∗ +p(q, E2)dq +�� +(9) +where q1 and q2 can be chosen anywhere in the classically domain of q corresponding to the +states with energy E1 and E2, as p is real there. The operator O does not affect the exponen- +tial suppression. Finally, q∗ is the complex ‘transition point’ that minimises the exponential +suppression, so that p(q∗, E1) = p(q∗, E2). +The usual vacuum tunnelling rate is recovered setting E1 = E2 = 0, so that the two terms +in eq. (9) merge giving the usual WKB penetration factor of the classically forbidden potential +barrier region. Eq. (9) shows that transitions among different classical states with E1 ̸= E2 +are generically exponentially suppressed. This is relevant for vacuum decay induced by particle +collisions. +3.1 +Stimulated tunnelling in the thin-wall limit +Eq. (9) was used by Voloshin [24] to compute the stimulated vacuum decay rate in the thin- +wall limit. We here review and validate the computation, and in the next section we try going +beyond the thin-wall limit. A scalar field contains many degrees of freedom. Its tunnelling +reduces to a quantum mechanical problem with one degree of freedom in thin-wall spherical +approximation: the scalar profile is approximated as constant in two regions, +h(t, r) ≃ +� htrue +for r < R(t), +hfalse +for r > R(t). +(10) +The only degree of freedom is the radius q(t) = R(t) of the thin-wall spherical bubble with +surface density σ that separates the two vacua with vacuum energy difference ∆V = V (hfalse)− +6 + +V (htrue). The Lagrangian for R is obtained inserting in the field action a smooth field profile +that depends on (r − R(t))/δ, where δ0 is the wall thickness at rest, and δ = δ0 +� +1 − ˙R2 takes +into account its Lorentz contraction. +In the limit of small δ0, integrating over the volume +gives [25] +L = +� +d3x L = −4πR2σ +� +1 − ˙R2 + 4πR3 +3 +∆V. +(11) +The ‘momentum’ conjugated to R is +pR = ∂L +∂ ˙R += 4πR2σ ˙R +� +1 − ˙R2 = +�� +H + 4πR3 +3 +∆V +�2 +− (4πR2σ)2 +(12) +in terms of the Hamiltonian +H = pR ˙R − L = +� +p2 +R + (4πR2σ)2 − 4πR3 +3 +∆V. +(13) +A bubble initially at rest, pR = 0, expands if R > R0 = 2σ/∆V corresponding to the critical +energy Ecr = 16πσ3/3∆V 2. The estimate r0 ∼ 1/ +√ +λhtop of section 2 is recovered taking into +account that σ >∼ h0 +√ +∆V . +Quantising this degree of freedom R(t) allows for quantum tunnelling. The initial state is +the false vacuum with no bubble, corresponding to R = 0. The final state is a super-critical +bubble. The tunnelling amplitude is given by eq. (9), that can be conveniently rewritten as +Asuper = Asub(E1, E2)Atunnel(E2) +(14) +by factoring out Atunnel, the usual WKB amplitude for tunnelling from the sub-critical to the +super-critical bubble. Then, Asub is interpreted as the amplitude for producing a classically +allowed sub-critical bubble out of the initial state. +• The tunneling amplitude |Atunnel|2 ≈ e−Stunnel from a thin-wall sub-critical bubble to a +super-critical bubble is given by the usual WKB computation. For vanishing initial-state +energy one sets E2 = 0, and the classically forbidden region extends from R1 = 0 to +R2 = 3σ/∆V , recovering [25] +Stunnel = S0 = 2 +� R2 +R1 +|pR| dR = π2σ +2 R3 +2 = 27π2σ4 +2∆V 3 +|pR| = 4πR2µ +� +1 − R2 +R2 +2 +. +(15) +For E2 = E > 0 the integral gets restricted to the smaller classically forbidden portion of +the potential barrier, so Stunnel decreases reaching zero when E = Ecr. +• The amplitude Asub is obtained inserting in eq. (9) the transition point +R3 +∗ = −3(E1 + E2)/8π∆V +(16) +with complex R∗. +This non-trivial point exists because the Lorentz factor induces a +non-trivial Hamiltonian, and it is determined solely by the volume term in pR eq. (12), +7 + +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Energy E in units of Ecr = 16πσ3/3ΔV2 +Actions S/S0 +← vacuum decay S0 = 27π2σ4/2ΔV3 +Ssuper = Ssub + Stunnel +Stunnel +Ssub +Figure 2: +Exponential suppression factors of vacuum decay induced by particle collisions with energy +E, and computed in thin-wall approximation, with surface density σ and potential energy difference +∆V . An exponential suppression remains even at energies large enough that the tunnelling suppression +vanishes. +imposing (E1 + 4πR3∆V/3) = −(E2 + 4πR3∆V/3). Next one sets E2 = E and E1 = 0, +since the energy E initially is in particles, not in the bubble that approximates the scalar +potential energy. The exponential suppression does not depend on the specific operator +O ∝ R3 [24]. So |Asub|2 ≈ e−Ssub consists of two terms: from R = 0 up to R∗ with energy +E1 = 0, next from R∗ to Re R∗ with energy E2. +Fig. 2 shows our numerical result, that agrees with [24]: increasing E reduces the exponential +suppression Ssuper = Ssub + Stunnel, but only partially. The minimum is Ssuper ≈ 0.16S0 at +E = Ecr. +At E > Ecr Stunnel = 0 but the total action Ssuper = Ssub grows because the +computation assumes that all energy must go in the formation of an unnecessarily big classical +bubble. Vacuum decay stimulated by few-particles collisions remains exponentially suppressed, +and thereby negligible. +3.2 +Stimulated tunnelling beyond the thin-wall limit +The previous section rests on the simplifying thin-wall limit, that affects the result in a crucial +way: a non-trivial complex transition point exists thanks to the non-trivial relativistic form +of the thin-wall action, eq. (11). However the thin-wall approximation only applies when two +vacua are nearly degenerate implying a huge exponential suppression. The approximation does +not apply to SM vacuum decay, where the true vacuum can be much deeper than the false SM +vacuum. The literature contains the following two results about stimulated tunnelling beyond +the thin-wall approximation: +• Enqvist computed corrections to the thin-wall approximation at leading order in the +8 + +thickness of the wall [16], finding that stimulation with E > 0 provides a milder reduction +(compared to the exact thin-wall limit) of the exponential suppression of the spontaneous +tunnelling rate at E = 0. +• Beyond the thin-wall limit, the dynamics does not reduce to a simpler problem with one +degree of freedom. Kuznetsov and Tinyakov [26] employed a coherent-state approxima- +tion to compute thick-wall vacuum decay induced by particle collisions in a theory with +potential V (h) = m2h2/2 − |λ|h4/4, finding that the exponential suppression present at +E = 0 almost entirely persists, at least up to energies of order Esph ≈ 19m/|λ|. However +the Higgs has a different potential, with sphaleron energy computed in eq. (8). +We provide two arguments why particle collisions with energy E negligibly reduce the expo- +nential suppression of SM Higgs vacuum decay. +First, we try going beyond the thin-wall limit by introducing arbitrary reasonable simplifying +approximations consisting in assuming special forms for the scalar field profile. In particular, +we assume a profile h(t, r) = h0(t)/(1+r2/R2), quantise h0(t) and apply the Landau formalism +to complex h0 (rather than complex R). The SM Lagrangian with a constant quartic coupling +becomes +L(h0, ˙h0) = +� +d3x L = π2R3 +� ˙h2 +0 +2 − Veff +� +where +Veff(h0) = h2 +0 +4R2 + λ +32h4 +0. +(17) +The gradient energy produced a barrier term in Veff, so that the above approximation can +be applied to constant λ < 0. The momentum conjugated to h0 is ph = π2R3 ˙h0, giving the +Hamiltonian H = p2 +h/2π2R3 + π2R3Veff. Tunnelling with E = 0 gets approximated as +S0 = 2Im +� hFubini +0 +0 +ph dh = +√ +2SFubini, +SFubini = 8π2 +3|λ| +(18) +showing that our simplifying assumption is sub-optimal, giving an action S0 larger than the +action of the QFT Fubini bounce. Nevertheless, our simplifying assumption allows to extend +the computation to E > 0 via the Landau eq. (9). Unlike in the thin-wall limit, the action now +has a simple form, so that the only transition point is h∗ = ∞ along the real axis, implying that +Ssuper(E) = S0. In other words, the decrease in Stunnel is exactly compensated by the increase +in Ssub, the suppression needed to convert the particle energy E into a sub-critical bubble. +Our simplified analytic argument gives a result that qualitatively agrees with the numerical +computation [26], and is easily extended to the SM potential of eq. (5). One obtains the same +Veff as in eq. (17) with the quartic λ replaced by −b ln(e7/6h2 +0/16h2 +top) i.e. renormalized around h0. +Despite this change, the only transition point remains h∗ = ∞ along the real axis, so Ssuper(E) = +S0. This result persists assuming the more complicated ansatz h0(t)/[1 + (r2 − R2(t))/r2 +0]. +We next show that, even if stimulation can mildly reduces the exponential suppression, in +the SM case this would need an energy E ≫ htop. The reason is that the SM Higgs potential +9 + +Figure 3: Example of a maximally factorially-enhanced Feynman diagram for the production of N = 3k +Higgs bosons out of one virtual Higgs using quartic Higgs interactions. Here k = 6, giving N ∼ 1000. +of eq. (5) is approximatively scale invariant. +As a result, vacuum decay at E = 0 is well +approximated by Fubini instantons +h(t, r) = +h0 +1 + (r2 − t2)/r2 +0 +with +h0 = hFubini +0 += +� +8/|λ| +r0 +, +(19) +and generic r0. +The vacuum decay rate is exponentially suppressed by the Fubini action +SFubini ≈ 8π2/3|λ(hFubini +0 +)| where the quartic coupling is renormalized around hFubini +0 +. Since +λ(h) runs in the SM becoming more negative at h ≫ htop, vacuum decay is dominated by field +values hFubini +0 +much larger than htop. Stimulation by particle collisions with energy E ∼ htop +can only reduce the exponential suppression of configurations with hFubini +0 +∼ htop, which are +exponentially sub-dominant. +In section 3.3 we clarify an additional possible doubt. +3.3 +Vacuum decay via Higgsplosion? +Studies done before [27, 28, 24, 29, 30] and after [31–33] the Higgs discovery considered the +perturbative cross section σN for producing N Higgs bosons in collisions with √s >∼ NMh, +finding that it scales as N!(λ/4π)N ∼ (Nλ/4π)N for large N. If this behaviour holds up to N ∼ +4π/λ where perturbativity starts failing, the cross section σN would be large, a phenomenon +dubbed ‘Higgsplosion’. If this phenomenon would also hold for ultra-relativistic Higgs bosons, +then a Higgs bubble similar to what needed to trigger vacuum decay would be produced with +large cross section. +We here recall the standard ‘Higgsplosion’ argument, extending it to ultra-relativistic Higgs +bosons. For simplicity, we can focus on the Feynman diagram in fig. 3 where some unspecified +scattering (for example a pp collision) produced with amplitude A1 the first one Higgs boson +with virtuality s = E2. For simplicity, we can focus on the symmetric configuration where the +first Higgs splits into 3 Higgses with energy E/3 via a quartic interaction. Iterating k times, +one gets N = 3k Higgses with energy E1 = E/N. The diagram in fig. 3 provides the maximal +combinatorial enhancement. It contains NV = (N − 1)/2 quartic vertices and NP = (N − 3)/2 +Higgs propagators. The scattering amplitude is +AN ∼ A1 N!λNV Π +(20) +10 + +where the product of Higgs propagators Π does not have a significant N dependence, being +dominated by the most numerous propagators with smaller virtuality E1: +Π = +k−1 +� +ℓ=1 +1 +[s/32ℓ]3ℓ = +N 3 +(E2 +1/3)NP . +(21) +Taking into account the N identical particles in the final state, the cross section is +σN ∼ |AN|2 +s +Φ(N) +N! ∼ σ1 +� Nλ +3(4π)2 +�N−1 +Rnr. +(22) +having written the volume of the N-body phase space as Φ(N) = Φ(N) +rel Rnr where +Φ(N) +rel = 1 +8π +(E/4π)2N−4 +(N − 2)!(N − 1)! ≃ E2N−4 +1 +(4π)2 +N 2 +� e +4π +�2N +, +Rnr ∼ min(1, ϵ)3N/2. +(23) +Here Φ(N) +rel is the ultra-relativistic phase space, and Rnr is the kinematical suppression that arises +when N ∼ E/Mh is so large that the kinetic energy per particle ϵ = K/Mh = E/NMh − 1 is +non-relativistic. +One first simple argument to be suspicious about ‘Higgspolosion’ considers a similar toy +problem in d = 0 space-time dimensions [34], where the path-integral that gives the amplitude +(up to negligible disconnected diagrams) reduces to the ordinary integral +A d=0 +N += +1 +√ +2π +� ∞ +−∞ +dh hN exp +� +−h2 +2 − λh4 +4 +� +. +(24) +Since propagators are Π = 1 in d = 0, the ‘amplitude’ A d=0 +N +counts the number of diagrams. +Its perturbative expansion gives the same N! enhancement as in d = 3+ 1, and breaks down at +λN >∼ 1. The non-perturbative eq. (24) shows that the toy amplitude A d=0 +N +remains exponen- +tially suppressed: it can be approximated by moving hN into the exponential, finding that the +‘saddle point’ that minimises the effective action has h ≈ (N/λ)1/4, away from the perturbative +expansion point h = 0. +This argument clarifies the physics, suggesting that, in d = 3+1, a large number N ≫ 4π/λ +of Higgs quanta forms a classical configuration that invalidates the perturbative expansion +around the vacuum (see also [35]). However, the argument does not tell if the σN cross section +gets large in the physical Higgs problem at N ∼ 4π/λ. This is the relevant regime for stimulated +vacuum decay. +The first computation that covers this critical regime in d = 3 + 1 was performed in [36] +using a method developed in [37–39], finding an exponentially suppressed cross section. +4 +Highest-energy cosmic-ray collisions +In section 3 we found that vacuum decay stimulated by few-particle collisions remains expo- +nentially suppressed by a factor of order SFubini ∼ 1000. We here estimate that the number of +11 + +cosmic-ray collisions with energy √s comparable to the SM vacuum instability scale is below +e160 even under most optimistic assumptions. So we will conclude that cosmic rays would not +have triggered SM vacuum decay. +Auger (and the smaller TA experiment in the northern hemisphere) detected cosmic rays +up to to energy E ≈ 2 1011 GeV with flux roughly given by [42–44] +dΦ +d ln E ≈ EeV2 +E2 +100 +km2 yr sr +(25) +but could not clarify their composition, finding something intermediate between proton and +nuclei. +Furthermore, they could not identify the production sites of ultra-high-energy CR +(hints are claimed in [40, 41]). Our numerical computations will use a precise CR flux, raher +than eq. (25). +4.1 +Collisions of 2 cosmic rays +The CR collision with highest energy ever occurred has been estimated to be √s ∼ 1011 GeV [45]. +Indeed, the CR number density n(E) = 4π +� ∞ +E dE dΦ/dE implies that the number of CR colli- +sions with energy s >∼ E2 is +N2(s) ∼ TUR3 +Uσ2n2 ∼ (E/EeV)6 +(26) +having integrated over the universe time and volume TU ∼ RU ∼ 10 Gyr and adopted σ2(s) = +1/s as a reference cross section among two CR [45] (this is roughly appropriate if CR have a +non-negligible proton component). This estimate was employed by collider safety reports such +as [46–48]. A more precise expression is +dN2 +ds = B2 +� +d4x +� +1 +−1 +dc +2 (1 − c) +� +dr +2ry +dn +dE +�� s +ry +� dn +dE +��rs +y +� +σ(s) +(27) +where +� +d4x ≈ 16π/1485H4 +0 is the volume of our past light-cone (neglecting the late-time +acceleration), r = E1/E2 is the ratio between the collision energies E1 and E2, c = cos θ is the +collision angle, and y = +� +2(1 + c). Apart from averaging over the angle, we introduced a ‘boost +factor’ B2 that accounts for CR inhomogeneities. Since CR are likely produced in astrophysical +sources, this enhances the CR collision rate in a significant way. Let us assume that CR have +a higher density around Ns production sources with size Rs ≪ RU and duration Ts, where CR +stay for a time τs. Then the CR density around one source is ns ∼ n(RU/Rs)3(TU/Ts)/Ns, and +the boost factor is +B2 ≈ ⟨n2⟩ +⟨n⟩2 ∼ 1 +Ns +τsTU +T 2 +s +R3 +U +R3 +s +(28) +where ⟨⟩ denotes the average over space and time. The values of Rs, Ts, τs are unknown, as +Auger and TA could not identify the production sites of ultra-high-energy CR (see [40,41] for +hints), and the plausible possibilities significantly differ. In particular, smaller Rs increases B2. +If multiple CR sources contribute comparably, the largest enhancement applies to the boost +factor. Plausible production sites are [42,43]: +12 + +SM instability scale? +104 +106 +108 +1010 +1012 +1014 +11 +1010 +1020 +1030 +1040 +1050 +Collision energy +s in GeV +Number of collisions assuming σ ≈ 1/s +Collisions of homogeneous Cosmic Rays +CR collisions in sources, B2 ≈ 1020 +CR collisions +in sources +without +GZK cut-off +LHC +Figure 4: +Number of collisions of two cosmic-rays: a) in the minimal uniform assumption (black +curve); b) including a boost due to collisions in small CR acceleration sites, B2 ∼ 1020 (blue curve, +B2 up to 1059 are discussed in the text); c) assuming no GZK cut-off in CR acceleration sites. +• The smallest sources are pulsars of magnetar type, neutron stars rotating with angu- +lar velocity ωs formed from supernovæ collapses that have the largest magnetic fields +Bs <∼ 1011 T, thereby allowing acceleration up to E <∼ eBsRs(ωsRs)2 [40] in a small accel- +eration region with size comparable to the Schwarzschild radius, Rs ∼ 2GM ∼ 10 km for +M ∼ (1.4 − 10)Msun. About Ns ∼ 1019 such objects are estimated to exist in our past +light-cone, and their magnetic field decays in Ts ∼ 104 yr. Assuming the minimal τs ∼ Rs, +this leads to B2 ∼ 1038, or even to B2 ∼ 1059 if a significant part of CR acceleration +happens just after the collapse, in Ts ∼ 10 s. Loosely similar estimates apply if CR are +accelerated during γ-ray bursts from various kinds of supernovæ [49]. +• The larger sources are Active Galactic Nuclei [50], objects with rotating magnetospheres +around super-massive galactic black holes of mass M ∼ 108Msun and duration Ts ∼ +105 yr [51]. +Given that our horizon contains about 1012 galaxies, we estimate Ns ∼ +1011 super-massive galactic black holes. This gives B2 ∼ 1028 if acceleration happens in +the inner region around the Schwarzschild radius Rs ∼ 2GM ∼ 103 s, or B2 ∼ 106 if +acceleration happens in a galactic-size Mpc region. Much larger B2 ∼ 1035 could arise if +CR acceleration significantly happens in brief events, while the black hole accretes nearby +stars by tidal disruption [43]. +These objects tend to have a geometry favourable to high-energy collisions, as incoming accel- +erated particles are channelled towards out-going jets at magnetic poles. +13 + +Assuming a negligible boost, B2 ∼ 1, fig. 4 shows that the CR maximal collision energy +is around the cut-off observed by Auger and TA, and possibly due to the GZK effect (the +pγCMB → ∆+ process opens at high energy and leads to absorption on distances larger than +∼ 100 Mpc). This collision energy is comparable to the uncertain SM Higgs instability scale. +Fig. 4 shows that a ‘modest’ boost factor B2 ∼ 1020 brings the maximal collision energy above +the Higgs instability scale. +Finally, CR around production sources could reach higher energies not limited by the GZK +effect. In such a case, significantly larger collision energies happened (red curve in fig. 4) if +around production sites the CR energy spectrum is a power law with no GZK cut-off. Some +other cut-off is however expected from the acceleration mechanism itself, and maybe this (rather +than the GZK effect) generated the cut-off observed in CR. +4.2 +Collisions of N cosmic rays +Collisions among N ≫ 1 initial-state particles are relevant for vacuum decay. +Starting from N = 3, the number of 3-collisions among cosmic rays with energy E is +N3 = B3 +� +d4x n3σ3 where we can estimate the 3-particle ‘cross section’ as σ3 ∼ 1/E5 and the +boost factor as +B3 ≈ ⟨n3⟩ +⟨n⟩3 ∼ B2B1 +B1 = ns +n ∼ TUR3 +U +NsTsR3 +s +. +(29) +The plausible CR accelerations sites discussed in the previous section lead to the following +extreme cases. At one extremum, CR accelerated by AGN on Mpc scales have a mild B1 ∼ 105 +resulting in a negligible N3 ≪ 1 at GZK energies comparable to the SM instability scale. +At the opposite extremum, CR accelerated by small magnetars on short time-scales can have +B1 ∼ 1060, implying N3 ≫ 1 at GZK energies. +At larger N, the boost factors grow as BN/BN−1 ∼ B1 resulting in the number of N- +collisions NN/NN−1 ∼ B1n/E3 ∼ (E0/E)5. The energy E0 is low, E0 ∼ 10 GeV, even in the +most optimistic case of small magnetars. This means that collisions of a large number N ∼ 1000 +of cosmic rays never occurred at energies comparable to the SM instability scale. +5 +Vacuum decay triggered by many particles +As discussed in the previous sections, vacuum decay must be triggered by an initial state that +contains a large number N of particles to avoid exponentially small rates. +One might think that vacuum decay can be triggered by colliding N beams into one center. +However, this N-collision would produce all SM particles (not only the Higgs), approximatively +giving a thermal-like environment that ‘burns’ the Higgs instability, as the potential V (h) +would be contain an extra stabilizing thermal-like Higgs mass trem M th +h = κT, similarly to what +happens at finite temperature [8]. Here κ2 ≈ (g2 +Y +2g2 +2)/16+y2 +t /4+· · · ≈ 0.352 is a combination +of SM couplings [8]. As SM couplings are perturbative, collisions between many particles occur +with negligible rate in the finite temperature plasma that filled the early universe. This is a +14 + +ℓ+ +ℓ- +θ +h +ℓ+ +ℓ- +θ +h +ℓ+ +ℓ- +θ +h +ℓ+ +ℓ- +θ +h +ℓ+ +ℓ- +θ +h +ℓ+ +ℓ- +θ +h +ℓ+ +ℓ- +θ +h +Figure 5: ‘Star’ collision scheme with N beams of ℓ− (blue) and ℓ+ (red) with equal energy collide at +small angle θ producing on-shell ℓ−ℓ+ → h. The N boosted Higgs bosons (in green) next collide at the +center, inducing Higgs vacuum decay for large enough N. We here plotted for N = 6. +special system where particle collisions lead to a vacuum decay rate exponential suppressed by +a factor Sthermal ≈ 6.015πκ/|λ| that can be computed via Euclidean methods [8]. +The above discussion indicates that triggering vacuum decay needs a cleaner initial state +of N Higgs quanta, not accompanied by a much larger number of SM particles, as they would +effectively contribute to κ. +5.1 +Collision scheme +A lepton collider with energies Eℓ± and collision angle θ can in principle produce a clean Higgs +source with controllable energy Eh = Eℓ− + Eℓ+ via resonant ℓ−ℓ+ → h production +M 2 +h = s = 2(Eℓ−Eℓ+ + m2 +ℓ − pℓ−pℓ+ cos θ) +(30) +The cross section is σpeak = 4π BR(h → ℓ+ℓ−)/M 2 +h. +The SM predicts BR(h → ℓ+ℓ−) ≈ +0.22 10−3. Unlike in an asymmetric e−e+ → φ factory, the produced h is highly boosted. +By using N such colliders one could engineer N Higgs bosons simultaneously hitting a +central interaction point, while SM particles produced by other processes with comparable +cross sections σ ∼ g4/4πM 2 +h are not focused into this center by the resonant kinematics. Fig. 5 +illustrates an example with Eℓ+ = Eℓ− and N = 6. The distance from the production point to +the interaction point must be smaller than the life-time of the boosted Higgs bosons, +τh +Eh +Mh += 0.37 µm +Eh +109 GeV. +(31) +15 + +Figure 6: +Classical evolution of an in-coming spherical Higgs wave containing about 1000 ultra- +relativistic Higgs quanta with individual energy k peaked around htop and total energy E. a) Forming +a slightly sub-critical bubble that dissolves, despite reaching h ≫ htop; b) Forming a slightly super- +critical bubble that ignites vacuum decay. +Needless to say, the needed energies are highly futuristic. A circular collider would need an as- +tronomical radius R = Eℓ/eB = 1 sec (Eℓ/ 1 +2109 GeV)(5.5 T/B) just to circulate in the magnetic +field B. The fraction of ℓ energy lost per turn is ϵ = 2e2E3 +ℓ /3m4 +ℓR: below unity for a muon +beam with energy Eµ = 1 +2109 GeV if R >∼ 11 hr, comparable to the boosted muon lifetime. +As a special case of possible interest, a ℓ+ beam hitting on ℓ− at rest needs energy Eℓ+ ≃ +M 2 +h/2mℓ to produce the Higgs resonantly. The Higgs boson energy Eh ≃ Eℓ+ equals Eh = +74 TeV for muon µ+µ− collisions, and Eh = 1.5 107 GeV for e−e+ collisions. The electron option +could roughly produce the desired energy Eh, and the boosted Higgs life-time would be 0.6 nm, +comparable to the atomic size. +Triggering vacuum decay needs machine optimisations different from those considered for +muon colliders, that aim at a high time-averaged luminosity L, that grows quadratically with +its beam energy (values up to √s ∼ 10 TeV are currently discussed). The geometry considered +here reverses the transverse and longitudinal beam size. One Higgs collision event is enough if +the N Higgs boson are synchronised in space and time up to the quantum uncertainty of order +1/Eh. Otherwise, if this level of synchronisation cannot be achieved, a high enough rate of +Higgs quanta is needed. For example, an instantaneous event peak rate Lpeakσ of order ∼ Eh +is needed to have consecutive Higgs quanta. +5.2 +Classical Higgs wave +While more practical configurations can be considered (such as two beams with N/2 Higgs +boson each), our goal here is to explore if triggering vacuum decay is theoretically possible. +16 + +Figure 7: As in fig. 6, but using about 2500 Higgs quanta with lower energy k peaked around 0.01htop, +so that a bubble with larger radius ∼ 1/k and lower field value h ≈ htop ignites vacuum decay. +We thereby focus on the system of N ≫ 1 Higgs bosons, that can be approximated as a +(theoretically simple) classical inward-going spherical Higgs wave. Triggering vacuum decay +via classical evolution would avoid the exponential suppression of the quantum rate. +We recall the classical Higgs equation in flat space, ¨h − h′′ − 2h′/r = −V ′ for a spherical +wave h(t, r). We denote as r = r0 the radius at the points in fig. 5 where the Higgs bosons +are produced at t = 0. In terms of u(t, r) = r h(t, r)/r0 the wave equation becomes ¨u − u′′ = +−rV ′/r0. Around the production point r ≈ r0 we can initially neglect the Higgs potential +V (h), because the Higgs field value is well below htop and because the Higgs quanta are ultra- +relativistic. So the time evolution of the inward wave is initially approximatively solved as +h(t, r) ≃ u0(r + t)r0/r. +(32) +Assuming as initial profile u0 a short wave-packet with length 1/k peaked at r ≈ r0, the +energy E = +� +4πr2 dr[˙h2/2 + h′2/2 + V ] of the full system is roughly given by E ∼ 4πr2 +0kh2 +0, +corresponding to N ≈ E/k = 4πr2 +0h2 +0 Higgs quanta. Here h0 = h(0, r0), where t = 0 is the +ℓ−ℓ+ → h production time. +In fig. 6 and 7 we show the numerical solution to the classical evolution equations, choosing +an incoming wave-packet with arbitrary profile of the form +u0(r) = h0 sin[k(r − r0)]e−k(r−r0)2/2 +(33) +dependent on two free parameters: k (that controls the typical energy of one Higgs quantum), +and h0 (that controls the intensity of the wave). +Fig. 6 shows the evolution using quanta with k = htop i.e. energies around htop. In the left +panel we see how a spherical in-ward wave forms a bubble with h > htop that however does not +17 + +:trigger vacuum decay and dissolves into an out-ward wave. Increasing the intensity of the wave +crosses the critical value above which the bubble ignites vacuum decay. This case is shown in +fig. 6b, and corresponds to roughly the minimal number N ∼ 1000 of Higgs quanta. +Fig. 7 shows the similar result using quanta with lower energy k peaked around 0.01htop. +This only allows to form bubbles with large radius r0 ∼ 1/k ∼ 100htop, that ignite vacuum +decay as soon as the Higgs field inside is slightly above the top of the SM potential barrier at +htop. Compared to the previous case of fig. 6, the total energy of the configuration is lower, +E ∼ 500htop, but a larger number of Higgs quanta N ∼ 2600 is needed. While in principle +colliders with k ∼ TeV can trigger vacuum decay, in practice this would need a huge number +N ∼ (htop/k)3 of Higgs bosons that would decay too fast, with low boost factor. +To avoid hitting a singularity at h = ∞, our numerical computations introduced an extra +non-renormalizable term in the Higgs potential that creates a deep minimum at large field +value. If this is absent and/or deep, gravity becomes relevant as the Higgs falls towards the +true vacuum. As shown in [52, 53], gravity cannot stop the explosion: a black hole can form +inside the bubble, that anyhow expands at the speed of light. +5.3 +Can a vacuum bubble be controlled, extracting energy? +The technique proposed to ignite a vacuum bubble is loosely similar to the colliding beam +technique that aims at nuclear fusion. A key difference is that vacuum energy, unlike energy +stored in atoms or nuclei, is not limited by the amount of matter. Does this imply that vacuum +decay only produces an unstoppable explosion, so that vacuum energy cannot be extracted and +used as an energy source? +In line of principle, the expansion of a vacuum bubble could be slowered or blocked by hitting +all its surface with beams of SM particles intense enough to balance the vacuum pressure ∆V . +This is possible because most SM particles get ultra-heavy inside. For theoretical simplicity +we here focus on the possibility of surrounding the bubble with a thermal bath at temperature +T. The system would behave as an ultra-hot star, stabilised only by artificially tuning the +temperature T to be comparable to the energy difference ∆V 1/4 between our vacuum and the +unknown vacuum inside the bubble. +The value of ∆V is currently unknown. +In line of principle, ∆V could be determined +(without creating the bubble) by measuring safe few-particle collisions at ultra-high energy, +and interpreting the observations in Quantum Field Theory. Three extreme possibilities are: +1) ∆V ∼ M 4 +Pl. In this case the bubble would be uncontrollable but scientifically interesting: +the true vacuum reaches the Planck scale, probing the multiverse. +2) ∆V ∼ λh4 +top/4. +3) ∆V ≪ h4 +top. In this case the bubble would be more controllable, but this is a tuned +possibility, perhaps motivated by the vague idea of Multi-Criticality [54]. +18 + +Let’s assume (for simplicity, and perhaps because it’s needed) that gravity remains weak, +namely that the bubble remains smaller than its Schwarzschild or AdS radius, Rs <∼ MPl/∆V 1/2, +corresponding to Buchdahl mass |M| ∼ ∆V R3 +s ∼ M 3 +Pl/∆V 1/2. As a numerical example, in +case 2) the vacuum energy density is ∼ 37 orders of magnitude larger than nuclear energy, +Rs <∼ 10−16 m, releasing |M| ∼ 1028 J as energy with power W ∼ R2 +sT 4. +The stabilised bubble would have negative mass, but it would would soon become an im- +practical source of energy: after a time of order Rs preventing the bubble explosion while +keeping the bubble in the weak-gravity regime costs more energy than what it released. The +bubble could be closed by ‘burning’ it with more intense beams, but this would cost at least all +energy it released, because of energy conservation. To extract vacuum energy, one needs to go +in the strong-gravity regime: one could ‘dispose’ the bubble behind a pre-existing black hole +horizon before the vacuum bubble explodes, or perhaps add extra energy such that the bubble +gets screened behind its own black hole horizon. +6 +Conclusions +We reconsidered vacuum decay stimulated by particle collisions, focusing on the possible in- +stability of the SM Higgs potential extrapolated up to ultra-high field values htop ∼ 1010 GeV. +This instability is suggested by current values of the top mass and strong coupling, but more +accurate measurements of these quantities are needed to establish if this instability would really +exist, and to infer a precise value of htop. +In section 2 we found that Higgs vacuum decay can be triggered by an initial Higgs field +configuration (for simplicity spherical and static) with energy E >∼ 4πhtop/ +� +|λ| ∼ 500 Joule that +contains a large number N >∼ Nmin ∼ 4π/|λ| ∼ 1000 of Higgs bosons. Fig. 1 shows the precise +threshold on the size and intensity of the initial Higgs configuration. The critical configuration +with minimal energy, the sphaleron, contains a number of Higgs quanta mildly higher than the +critical configuration with minimal number of quanta. Since N ≫ 1 these are semi-classical +configurations, and the quantum amplitude for forming a critical bubble out of collisions of +few particles is exponentially suppressed by exp(−O(N)), as typical for quantum transitions +between classically different states. +This exponential suppression was more precisely computed in thin-wall approximation by +Voloshin [15]. In section 3 we verified this result: our result in fig. 2 confirms that particle +collisions lift the exponential suppression only partially, because the reduced exponential sup- +pression of tunnelling gets compensated by the exponential suppression for forming the needed +semi-classical configuration. +However, the thin-wall approximation is not applicable to SM Higgs vacuum decay. Going +beyond this limit, in section 3.2 we argued that the lifting of its exponential suppression is +negligible. We provided arguments based on the approximate scale invariance of the Higgs +potential, and assuming special field configurations that allow a simple computation. +Furthermore, in section 3.3 we addressed a related issue: the cross section σN for producing +N Higgs bosons grows factorially with N, and some authors argue that σN might become +19 + +large at N >∼ 4π/λ, a possibility dubbed ‘Higgsplosion’. This number of quanta (extended to +relativistic Higgs bosons) would also form the semi-classical Higgs configuration that stimulates +SM Higgs vacuum decay. We thereby critically considered the issue, reviewing recent works +that find that σN remains exponentially suppressed. +Having clarified that stimulated vacuum decay remains exponentially suppressed, in sec- +tion 4 we compared its rate with the rate of ultra-high energy collisions of cosmic rays. By +considering collisions happened in relatively compact and possibly short-lived astrophysical +production sites of ultra-high energy cosmic rays, we estimated that the number of collisions +with √s ∼ htop can be tens (up to 60) orders of magnitude larger than the minimal number of +collisions usually estimated in outer space, as illustrated in fig. 4. The CR collision rate can be +so high that ultra-high energy collisions among N > 2 cosmic rays (but not N ≫ 2) occurred. +Furthermore, the √s of cosmic-ray collisions around their production sites could extend beyond +the GZK cut-off. Nevertheless, we find that these enhancements cannot compensate for the +exponential suppression of Higgs vacuum decay stimulated by particle collisions. +Finally, in section 5 we discussed how the exponential suppression can be bypassed classi- +cally by using futuristic ultra-high energy colliders to artificially engineer an in-going wave of +N >∼ 1000 highly boosted Higgs bosons. We proposed a collision scheme, illustrated in fig. 5, +that exploits ℓ−ℓ+ → h on-shell production to form an inward-going Higgs wave. We simu- +lated the classical evolution of the Higgs wave, finding the critical threshold above which it +triggers vacuum decay, rather than dissolving. Results are shown in fig. 6 and 7 for sub- and +super-critical waves, and for two different energies of the Higgs bosons. In section 5.3 we wildly +speculate about a basic issue: is it possible (at least in theory) to control a vacuum bubble +slowering its expansion and using it as an energy source? We suggest a technique based on +pressing beams and on disposing the dangerous negative-mass remnant behind a black-hole +horizon. +References +[1] D. Buttazzo, G. Degrassi, P.P. Giardino, G.F. +Giudice, F. Sala, A. Salvio, A. Strumia, ‘Inves- +tigating the near-criticality of the Higgs bo- +son’, JHEP 12 (2013) 089 [arXiv:1307.3536]. +[2] J.R. Espinosa, +G.F. Giudice, +E. Morgante, +A. +Riotto, +L. +Senatore, +A. +Strumia, +N. +Tetradis, ‘The cosmological Higgstory of the +vacuum +instability’, +JHEP 09 (2015) 174 +[arXiv:1505.04825]. +[3] R. Franceschini, A. Strumia, A. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' However, this configuration is stimulated by ultra- high energy collisions with a exp(−O(N)) suppression, comparable to spontaneous vacuum decay: no ‘Higgspolosion’ enhancement arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This implies that ultra- high energy cosmic ray collisions are safe, despite that their number (in production sites) likely is tens of orders of magnitude higher than what usually estimated (in space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We speculate on how vacuum decay could be induced classically, forming a in-coming wave of N boosted Higgs bosons at futuristic ultra-high energy colliders, and on how the resulting vacuum bubble could be controlled to extract energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Triggering Higgs vacuum decay 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='1 Simple estimates .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='3 Can a vacuum bubble be controlled, extracting energy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 18 6 Conclusions 19 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='03620v1 [hep-ph] 9 Jan 2023 1 Introduction The discovery of Higgs boson h opens the door to a new form of energy, the vacuum energy in the Higgs potential V (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The Higgs currently sits at the Standard Model (SM) local minimum of its potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' If this is the global minimum, no vacuum energy remains stored in our Universe, and vacuum cannot be used as a source of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' However, extrapolating the SM Higgs potential V (h) to ultra-high field values h suggests that the potential might have a local maximum at h = htop ∼ 1010 GeV becoming negative at h > e1/4htop [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This happens for current best-fit values of the top quark mass, Higgs mass and strong coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Within current uncertainties a much larger instability scale or even stability is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Clarifying if V (h) is really unstable needs improved future measurements of the top mass, doable at a lepton collider at the top threshold (such as an e−e+ collider in the LEP tunnel, or a µ−µ+ collider) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Assuming that the Higgs instability exists, we explore under which conditions it can result in vacuum decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The following possibilities have been considered in the literature: Spontaneous vacuum tunnelling [4] is exponentially slow [5,6,1,2,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The thermal energy in the early universe at temperatures T >∼ htop could have overcome the potential barrier in V (h), but thermal effects also add a thermal barrier: as a result vacuum decay remains exponentially slow at any T (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Inflationary fluctuations could have trigger the instability, but this depends on the un- known inflation model and non-minimal Higgs coupling to gravity (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Small hypothetical primordial black holes with Hawking temperature T >∼ htop are a pos- sible seed of vacuum decay, but the rate is again exponentially slow, similarly to thermal vacuum decay (claims in the literature differ, I think this is the correct conclusion as discussed in [9,10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Vacuum decay seeded by particle collisions with ultra-high energy E >∼ htop has been considered for some scalar potentials, finding that the rate remains partially exponentially suppressed [11–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In this paper we reconsider the latter issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The exponential suppression persists because, as discussed in section 2, the minimal configuration needed to seed vacuum decay contains N ≫ 1 scalar quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In the case of SM Higgs vacuum decay, one needs N ∼ 4π/λ ∼ 1000 Higgs boson quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Such a semi-classical state is generated out of collisions of a few quanta with rate exponentially suppressed by a ‘one to many particles’ factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In section 3 we revisit and extend previous computations, focusing on the specific case of Higgs vacuum decay, that needs going beyond the thin-wall approximation, and summarizing why cross sections for producing N Higgs bosons remain small at large N (no ‘Higgsplosion’ happens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In section 4 we reconsider collisions of two ultra-high-energy cosmic rays (CR), showing that a significantly enhanced CR collision rate happened if cosmic rays are accelerated by relatively compact astrophysical objects, from magnetars up to Active Galactic Nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We will also 2 find that few-particle ultra-high energy CR collisions can have happened, while many-particle collisions never occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' As a result, CR collisions could not trigger Higgs vacuum decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The main simple lesson is that an appropriate many-particle classical process is needed to trigger vacuum decay avoiding the exponential suppression that would make the rate negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We then explore in section 5 if the critical configuration could be engineered classically via futuristic collider schemes with ultra-high energy √s >∼ htop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We next discuss if the true vacuum bubble can be artificially controlled and used to extract energy from the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Section 6 presents our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 2 Triggering Higgs vacuum decay We consider an initial configuration of the Higgs field h(t, r) at t = 0 with field value h ∼ h0 within a space region of size r <∼ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' For simplicity we assume spherical symmetry, denote as r the radius, and assume that the initial configuration is at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' An example is h(0, r) = h0 1 + r2/r2 0 , ˙h(0, r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (1) The classical Higgs field equation in flat space is ¨h − h′′ − 2h′/r = −V ′, where, as usual, ˙h = ∂h/∂t and h′ = ∂h/∂r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='1 Simple estimates The total energy of the configuration is E = � dr 4πr2 �h′2 2 + V � ∼ 4πr0h2 0[1 + λr2 0h2 0] (2) having here simply approximated the potential energy as V = λh4/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In its subsequent evolution this field configuration triggers vacuum decay rather than dis- solving if two conditions are met: 1) h0 >∼ htop is over the top of the SM potential barrier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 2) h0r0 � |λ| >∼ 1 such that the potential energy wins over the gradient energy in the classical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This estimate can be obtained by demanding that potential energy dominates over the gradient energy in the total energy of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (2), or by considering the classical equation of motion at the point r = 0, or in thin-wall approximation (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (13) later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' According to conditions 1) and 2) above, an initially static configuration h(0, r) evolves towards the true vacuum if its energy is E >∼ Emin ∼ 4πhtop � |λ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (3) 3 The Fourier transform of h(0, r) is large at k <∼ 1/r0, showing that the scalar field configuration contains N ∼ E k ∼ 4π(h0r0)2 >∼ Nmin ∼ Emin k ∼ 4π |λ| (4) Higgs quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='2 Numerical computations We validate and precise the above analytic approximations by numerically solving the classical Higgs field equation in flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The SM Higgs potential can be accurately approximated through an effective running coupling as V (h) ≈ λ(h)h4 4 , λ(h) ≈ −b ln h2 e1/2h2 top with b ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='15 (4π)2 (5) for best-fit values of the SM parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Switching to the dimension-less field ˜h = h/htop and coordinates ˜x = xhtop √ b, the Higgs action simplifies to S = 1 b � d4˜x � �1 2 � ∂˜h ∂˜xµ �2 + ˜h4 4 ln ˜h2 e1/2 � � , (6) showing that the dynamics depends in a simple way on the values of b and htop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We evolve the Higgs field starting from an initial field profile with vanishing time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The result for the profile in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (1) is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 1a: the ‘drainage divide’ critical boundary in the (h0, r0) plane that separates initial configuration that end up in the false vacuum or in the true vacuum is well approximated by the analytic conditions 1) and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 1a also shows iso-curves of the approximate number of Higgs quanta, computed from the Fourier transform hk = � dr 4πr2 h(r)sin kr kr as N = � dk 4πk2 (2π)3k k2 2 |hk|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (7) The expression for N applies to an in-going or out-going Higgs wave with h ≪ htop such that Higgs quanta are free with energy k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' So eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (7) can be applied to the field configuration at a time such that the bubble is dissolving into the false vacuum, and field values h got much below htop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' For simplicity, in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 1 we will evaluate eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (7) at t = 0 on the static configuration of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (1) with h ∼ htop obtaining N = (πr0h0/2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' But the true number of quanta is kN/2 ≈ N, where the factor 1/2 accounts for the static initial configuration, and k for the potential energy, with k ∼ 1 − 2 for critical bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 1a shows that increasing the energy k ∼ 1/r0 of each quantum does not reduce the needed N below Nmin ≈ 1/b ≈ 1000, while N ≫ Nmin quanta are needed for a large bubble made of quanta with energy lower than the optimal energy k ∼ � |λ|htop ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='1htop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' A nearly identical figure is obtained replacing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (1) with h(r) = h0e−r2/r2 0, for which N = π(r0h0)2/2 is mildly lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 4 0 1 2 3 4 5 0 1 2 3 4 Radius r\uf02d = r htop b Higgs field intensity h(r)/htop Sphaleron Figure 1: Left: We numerically evolve a spherical Higgs bubble h0/(1+r2/r2 0) initially at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In the red region the bubble expands triggering vacuum decay, while in the green region the bubble dissolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The red region is well approximated as h0 > htop and h0r0 > 1/ � |λ| (dashed curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The hatching covers the unphysical region where the bubble is so large that its energy is negative, E < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The blue iso-contours show the number N of Higgs quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This confirms that expansion needs N >∼ 4π/|λ| quanta naively computed from the static configuration at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The sphaleron is denoted as ‘Sph’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Right: sphaleron solution, and its analytic approximation (dashed curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' A special configuration is the sphaleron, the static spherical solution to the Higgs field equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Thereby it is the critical configuration with lowest energy Emin = Esph ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='7htop √ b ≈ 500 Joule htop 1010 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (8) The sphaleron has h0 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='1htop and its profile is plotted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' It can be approximated as eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (1) with r0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='4/htop √ b, corresponding to the point denoted as ‘Sph’ in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' By letting the sphaleron dissolve in time [22], one finds Nsph ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='2/b >∼ Nmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Critical configurations with higher energy E > Esph do not have N significantly lower than Nsph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This physically suggests that vacuum decay stimulated by particle collisions will remain exponentially suppressed, as discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 3 Vacuum decay stimulated by few particles We have seen that a Higgs field configuration that evolves into the true vacuum contains N > Nmin ∼ 4π/|λ| ∼ 103 Higgs quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Then, one expects that the probability of generating such 5 1 10-1 10-2 10-3 30 10 E<0 Sph 3 E>0 Bubble expands 10 1 Bubble evaporates 106 100 1000 101 102 1 103 Bubble radius ro htopa configuration out of high-energy collisions with few particles is suppressed by an exponential factor ∼ e−O(N) parametrically similar to the vacuum tunnelling rate ∼ e−Sbounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' To see this, let us over-simplify the problem considering free Higgs particles, such that one can define the number operator ˆNk = ˆa† kˆak in terms of the creation operator ˆa† k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' As well known, states with exactly N quanta have zero field value ⟨N|ˆh|N⟩ and large fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The good semi-classical states with small field fluctuations and large N = ⟨α| ˆN|α⟩ = |α|2 are instead the coherent states |α⟩ = e−|α|2/2 �∞ n=0 αn|n⟩/ √ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=', eigenstates of ˆa|α⟩ = α|α⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The scalar product between two coherent states is exponentially suppressed as |⟨α|β⟩|2 = e−|α−β|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' A qualitatively similar suppression persists taking into account the interaction in the scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The vacuum decay rate induced by particle collisions was computed in thin-wall approximation by Voloshin [15] using the following WKB-like result from Landau [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Let us consider (for simplicity) a quantum system described by an Hamiltonian H that depends on one degree of freedom q and its conjugated momentum p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The matrix element of an operator O among quasi-classical states with energies E1, E2 is not easily computed, because their wave functions oscillate wildly, averaging to nearly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Suitable analytic continuations to complex q in the direction of decreasing WKB exponentials show that the matrix element is suppressed as [23] ⟨E2|O|E1⟩ ∼ exp � −Im �� q∗ q1 p(q, E1)dq + � q2 q∗ p(q, E2)dq �� (9) where q1 and q2 can be chosen anywhere in the classically domain of q corresponding to the states with energy E1 and E2, as p is real there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The operator O does not affect the exponen- tial suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Finally, q∗ is the complex ‘transition point’ that minimises the exponential suppression, so that p(q∗, E1) = p(q∗, E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The usual vacuum tunnelling rate is recovered setting E1 = E2 = 0, so that the two terms in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (9) merge giving the usual WKB penetration factor of the classically forbidden potential barrier region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (9) shows that transitions among different classical states with E1 ̸= E2 are generically exponentially suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This is relevant for vacuum decay induced by particle collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='1 Stimulated tunnelling in the thin-wall limit Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (9) was used by Voloshin [24] to compute the stimulated vacuum decay rate in the thin- wall limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We here review and validate the computation, and in the next section we try going beyond the thin-wall limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' A scalar field contains many degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Its tunnelling reduces to a quantum mechanical problem with one degree of freedom in thin-wall spherical approximation: the scalar profile is approximated as constant in two regions, h(t, r) ≃ � htrue for r < R(t), hfalse for r > R(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (10) The only degree of freedom is the radius q(t) = R(t) of the thin-wall spherical bubble with surface density σ that separates the two vacua with vacuum energy difference ∆V = V (hfalse)− 6 V (htrue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The Lagrangian for R is obtained inserting in the field action a smooth field profile that depends on (r − R(t))/δ, where δ0 is the wall thickness at rest, and δ = δ0 � 1 − ˙R2 takes into account its Lorentz contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In the limit of small δ0, integrating over the volume gives [25] L = � d3x L = −4πR2σ � 1 − ˙R2 + 4πR3 3 ∆V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (11) The ‘momentum’ conjugated to R is pR = ∂L ∂ ˙R = 4πR2σ ˙R � 1 − ˙R2 = �� H + 4πR3 3 ∆V �2 − (4πR2σ)2 (12) in terms of the Hamiltonian H = pR ˙R − L = � p2 R + (4πR2σ)2 − 4πR3 3 ∆V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (13) A bubble initially at rest, pR = 0, expands if R > R0 = 2σ/∆V corresponding to the critical energy Ecr = 16πσ3/3∆V 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The estimate r0 ∼ 1/ √ λhtop of section 2 is recovered taking into account that σ >∼ h0 √ ∆V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Quantising this degree of freedom R(t) allows for quantum tunnelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The initial state is the false vacuum with no bubble, corresponding to R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The final state is a super-critical bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The tunnelling amplitude is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (9), that can be conveniently rewritten as Asuper = Asub(E1, E2)Atunnel(E2) (14) by factoring out Atunnel, the usual WKB amplitude for tunnelling from the sub-critical to the super-critical bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Then, Asub is interpreted as the amplitude for producing a classically allowed sub-critical bubble out of the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The tunneling amplitude |Atunnel|2 ≈ e−Stunnel from a thin-wall sub-critical bubble to a super-critical bubble is given by the usual WKB computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' For vanishing initial-state energy one sets E2 = 0, and the classically forbidden region extends from R1 = 0 to R2 = 3σ/∆V , recovering [25] Stunnel = S0 = 2 � R2 R1 |pR| dR = π2σ 2 R3 2 = 27π2σ4 2∆V 3 |pR| = 4πR2µ � 1 − R2 R2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (15) For E2 = E > 0 the integral gets restricted to the smaller classically forbidden portion of the potential barrier, so Stunnel decreases reaching zero when E = Ecr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The amplitude Asub is obtained inserting in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (9) the transition point R3 ∗ = −3(E1 + E2)/8π∆V (16) with complex R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This non-trivial point exists because the Lorentz factor induces a non-trivial Hamiltonian, and it is determined solely by the volume term in pR eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (12), 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='0 Energy E in units of Ecr = 16πσ3/3ΔV2 Actions S/S0 ← vacuum decay S0 = 27π2σ4/2ΔV3 Ssuper = Ssub + Stunnel Stunnel Ssub Figure 2: Exponential suppression factors of vacuum decay induced by particle collisions with energy E, and computed in thin-wall approximation, with surface density σ and potential energy difference ∆V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' An exponential suppression remains even at energies large enough that the tunnelling suppression vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' imposing (E1 + 4πR3∆V/3) = −(E2 + 4πR3∆V/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Next one sets E2 = E and E1 = 0, since the energy E initially is in particles, not in the bubble that approximates the scalar potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The exponential suppression does not depend on the specific operator O ∝ R3 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' So |Asub|2 ≈ e−Ssub consists of two terms: from R = 0 up to R∗ with energy E1 = 0, next from R∗ to Re R∗ with energy E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 2 shows our numerical result, that agrees with [24]: increasing E reduces the exponential suppression Ssuper = Ssub + Stunnel, but only partially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The minimum is Ssuper ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='16S0 at E = Ecr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' At E > Ecr Stunnel = 0 but the total action Ssuper = Ssub grows because the computation assumes that all energy must go in the formation of an unnecessarily big classical bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Vacuum decay stimulated by few-particles collisions remains exponentially suppressed, and thereby negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='2 Stimulated tunnelling beyond the thin-wall limit The previous section rests on the simplifying thin-wall limit, that affects the result in a crucial way: a non-trivial complex transition point exists thanks to the non-trivial relativistic form of the thin-wall action, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' However the thin-wall approximation only applies when two vacua are nearly degenerate implying a huge exponential suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The approximation does not apply to SM vacuum decay, where the true vacuum can be much deeper than the false SM vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The literature contains the following two results about stimulated tunnelling beyond the thin-wall approximation: Enqvist computed corrections to the thin-wall approximation at leading order in the 8 thickness of the wall [16], finding that stimulation with E > 0 provides a milder reduction (compared to the exact thin-wall limit) of the exponential suppression of the spontaneous tunnelling rate at E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Beyond the thin-wall limit, the dynamics does not reduce to a simpler problem with one degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Kuznetsov and Tinyakov [26] employed a coherent-state approxima- tion to compute thick-wall vacuum decay induced by particle collisions in a theory with potential V (h) = m2h2/2 − |λ|h4/4, finding that the exponential suppression present at E = 0 almost entirely persists, at least up to energies of order Esph ≈ 19m/|λ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' However the Higgs has a different potential, with sphaleron energy computed in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We provide two arguments why particle collisions with energy E negligibly reduce the expo- nential suppression of SM Higgs vacuum decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' First, we try going beyond the thin-wall limit by introducing arbitrary reasonable simplifying approximations consisting in assuming special forms for the scalar field profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In particular, we assume a profile h(t, r) = h0(t)/(1+r2/R2), quantise h0(t) and apply the Landau formalism to complex h0 (rather than complex R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The SM Lagrangian with a constant quartic coupling becomes L(h0, ˙h0) = � d3x L = π2R3 � ˙h2 0 2 − Veff � where Veff(h0) = h2 0 4R2 + λ 32h4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (17) The gradient energy produced a barrier term in Veff, so that the above approximation can be applied to constant λ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The momentum conjugated to h0 is ph = π2R3 ˙h0, giving the Hamiltonian H = p2 h/2π2R3 + π2R3Veff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Tunnelling with E = 0 gets approximated as S0 = 2Im � hFubini 0 0 ph dh = √ 2SFubini, SFubini = 8π2 3|λ| (18) showing that our simplifying assumption is sub-optimal, giving an action S0 larger than the action of the QFT Fubini bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Nevertheless, our simplifying assumption allows to extend the computation to E > 0 via the Landau eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Unlike in the thin-wall limit, the action now has a simple form, so that the only transition point is h∗ = ∞ along the real axis, implying that Ssuper(E) = S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In other words, the decrease in Stunnel is exactly compensated by the increase in Ssub, the suppression needed to convert the particle energy E into a sub-critical bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Our simplified analytic argument gives a result that qualitatively agrees with the numerical computation [26], and is easily extended to the SM potential of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' One obtains the same Veff as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (17) with the quartic λ replaced by −b ln(e7/6h2 0/16h2 top) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' renormalized around h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Despite this change, the only transition point remains h∗ = ∞ along the real axis, so Ssuper(E) = S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This result persists assuming the more complicated ansatz h0(t)/[1 + (r2 − R2(t))/r2 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We next show that, even if stimulation can mildly reduces the exponential suppression, in the SM case this would need an energy E ≫ htop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The reason is that the SM Higgs potential 9 Figure 3: Example of a maximally factorially-enhanced Feynman diagram for the production of N = 3k Higgs bosons out of one virtual Higgs using quartic Higgs interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Here k = 6, giving N ∼ 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (5) is approximatively scale invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' As a result, vacuum decay at E = 0 is well approximated by Fubini instantons h(t, r) = h0 1 + (r2 − t2)/r2 0 with h0 = hFubini 0 = � 8/|λ| r0 , (19) and generic r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The vacuum decay rate is exponentially suppressed by the Fubini action SFubini ≈ 8π2/3|λ(hFubini 0 )| where the quartic coupling is renormalized around hFubini 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Since λ(h) runs in the SM becoming more negative at h ≫ htop, vacuum decay is dominated by field values hFubini 0 much larger than htop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Stimulation by particle collisions with energy E ∼ htop can only reduce the exponential suppression of configurations with hFubini 0 ∼ htop, which are exponentially sub-dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='3 we clarify an additional possible doubt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='3 Vacuum decay via Higgsplosion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Studies done before [27, 28, 24, 29, 30] and after [31–33] the Higgs discovery considered the perturbative cross section σN for producing N Higgs bosons in collisions with √s >∼ NMh, finding that it scales as N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (λ/4π)N ∼ (Nλ/4π)N for large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' If this behaviour holds up to N ∼ 4π/λ where perturbativity starts failing, the cross section σN would be large, a phenomenon dubbed ‘Higgsplosion’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' If this phenomenon would also hold for ultra-relativistic Higgs bosons, then a Higgs bubble similar to what needed to trigger vacuum decay would be produced with large cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We here recall the standard ‘Higgsplosion’ argument, extending it to ultra-relativistic Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' For simplicity, we can focus on the Feynman diagram in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 3 where some unspecified scattering (for example a pp collision) produced with amplitude A1 the first one Higgs boson with virtuality s = E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' For simplicity, we can focus on the symmetric configuration where the first Higgs splits into 3 Higgses with energy E/3 via a quartic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Iterating k times, one gets N = 3k Higgses with energy E1 = E/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The diagram in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 3 provides the maximal combinatorial enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' It contains NV = (N − 1)/2 quartic vertices and NP = (N − 3)/2 Higgs propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The scattering amplitude is AN ∼ A1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='λNV Π (20) 10 where the product of Higgs propagators Π does not have a significant N dependence, being dominated by the most numerous propagators with smaller virtuality E1: Π = k−1 � ℓ=1 1 [s/32ℓ]3ℓ = N 3 (E2 1/3)NP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (21) Taking into account the N identical particles in the final state, the cross section is σN ∼ |AN|2 s Φ(N) N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' ∼ σ1 � Nλ 3(4π)2 �N−1 Rnr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (22) having written the volume of the N-body phase space as Φ(N) = Φ(N) rel Rnr where Φ(N) rel = 1 8π (E/4π)2N−4 (N − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' ≃ E2N−4 1 (4π)2 N 2 � e 4π �2N , Rnr ∼ min(1, ϵ)3N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (23) Here Φ(N) rel is the ultra-relativistic phase space, and Rnr is the kinematical suppression that arises when N ∼ E/Mh is so large that the kinetic energy per particle ϵ = K/Mh = E/NMh − 1 is non-relativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' One first simple argument to be suspicious about ‘Higgspolosion’ considers a similar toy problem in d = 0 space-time dimensions [34], where the path-integral that gives the amplitude (up to negligible disconnected diagrams) reduces to the ordinary integral A d=0 N = 1 √ 2π � ∞ −∞ dh hN exp � −h2 2 − λh4 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (24) Since propagators are Π = 1 in d = 0, the ‘amplitude’ A d=0 N counts the number of diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Its perturbative expansion gives the same N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' enhancement as in d = 3+ 1, and breaks down at λN >∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The non-perturbative eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (24) shows that the toy amplitude A d=0 N remains exponen- tially suppressed: it can be approximated by moving hN into the exponential, finding that the ‘saddle point’ that minimises the effective action has h ≈ (N/λ)1/4, away from the perturbative expansion point h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This argument clarifies the physics, suggesting that, in d = 3+1, a large number N ≫ 4π/λ of Higgs quanta forms a classical configuration that invalidates the perturbative expansion around the vacuum (see also [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' However, the argument does not tell if the σN cross section gets large in the physical Higgs problem at N ∼ 4π/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This is the relevant regime for stimulated vacuum decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The first computation that covers this critical regime in d = 3 + 1 was performed in [36] using a method developed in [37–39], finding an exponentially suppressed cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 4 Highest-energy cosmic-ray collisions In section 3 we found that vacuum decay stimulated by few-particle collisions remains expo- nentially suppressed by a factor of order SFubini ∼ 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We here estimate that the number of 11 cosmic-ray collisions with energy √s comparable to the SM vacuum instability scale is below e160 even under most optimistic assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' So we will conclude that cosmic rays would not have triggered SM vacuum decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Auger (and the smaller TA experiment in the northern hemisphere) detected cosmic rays up to to energy E ≈ 2 1011 GeV with flux roughly given by [42–44] dΦ d ln E ≈ EeV2 E2 100 km2 yr sr (25) but could not clarify their composition, finding something intermediate between proton and nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Furthermore, they could not identify the production sites of ultra-high-energy CR (hints are claimed in [40, 41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Our numerical computations will use a precise CR flux, raher than eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='1 Collisions of 2 cosmic rays The CR collision with highest energy ever occurred has been estimated to be √s ∼ 1011 GeV [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Indeed, the CR number density n(E) = 4π � ∞ E dE dΦ/dE implies that the number of CR colli- sions with energy s >∼ E2 is N2(s) ∼ TUR3 Uσ2n2 ∼ (E/EeV)6 (26) having integrated over the universe time and volume TU ∼ RU ∼ 10 Gyr and adopted σ2(s) = 1/s as a reference cross section among two CR [45] (this is roughly appropriate if CR have a non-negligible proton component).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This estimate was employed by collider safety reports such as [46–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' A more precise expression is dN2 ds = B2 � d4x � +1 −1 dc 2 (1 − c) � dr 2ry dn dE �� s ry � dn dE ��rs y � σ(s) (27) where � d4x ≈ 16π/1485H4 0 is the volume of our past light-cone (neglecting the late-time acceleration), r = E1/E2 is the ratio between the collision energies E1 and E2, c = cos θ is the collision angle, and y = � 2(1 + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Apart from averaging over the angle, we introduced a ‘boost factor’ B2 that accounts for CR inhomogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Since CR are likely produced in astrophysical sources, this enhances the CR collision rate in a significant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Let us assume that CR have a higher density around Ns production sources with size Rs ≪ RU and duration Ts, where CR stay for a time τs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Then the CR density around one source is ns ∼ n(RU/Rs)3(TU/Ts)/Ns, and the boost factor is B2 ≈ ⟨n2⟩ ⟨n⟩2 ∼ 1 Ns τsTU T 2 s R3 U R3 s (28) where ⟨⟩ denotes the average over space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The values of Rs, Ts, τs are unknown, as Auger and TA could not identify the production sites of ultra-high-energy CR (see [40,41] for hints), and the plausible possibilities significantly differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In particular, smaller Rs increases B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' If multiple CR sources contribute comparably, the largest enhancement applies to the boost factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Plausible production sites are [42,43]: 12 SM instability scale?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 104 106 108 1010 1012 1014 11 1010 1020 1030 1040 1050 Collision energy s in GeV Number of collisions assuming σ ≈ 1/s Collisions of homogeneous Cosmic Rays CR collisions in sources, B2 ≈ 1020 CR collisions in sources without GZK cut-off LHC Figure 4: Number of collisions of two cosmic-rays: a) in the minimal uniform assumption (black curve);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' b) including a boost due to collisions in small CR acceleration sites, B2 ∼ 1020 (blue curve, B2 up to 1059 are discussed in the text);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' c) assuming no GZK cut-off in CR acceleration sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The smallest sources are pulsars of magnetar type, neutron stars rotating with angu- lar velocity ωs formed from supernovæ collapses that have the largest magnetic fields Bs <∼ 1011 T, thereby allowing acceleration up to E <∼ eBsRs(ωsRs)2 [40] in a small accel- eration region with size comparable to the Schwarzschild radius, Rs ∼ 2GM ∼ 10 km for M ∼ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='4 − 10)Msun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' About Ns ∼ 1019 such objects are estimated to exist in our past light-cone, and their magnetic field decays in Ts ∼ 104 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Assuming the minimal τs ∼ Rs, this leads to B2 ∼ 1038, or even to B2 ∼ 1059 if a significant part of CR acceleration happens just after the collapse, in Ts ∼ 10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Loosely similar estimates apply if CR are accelerated during γ-ray bursts from various kinds of supernovæ [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The larger sources are Active Galactic Nuclei [50], objects with rotating magnetospheres around super-massive galactic black holes of mass M ∼ 108Msun and duration Ts ∼ 105 yr [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Given that our horizon contains about 1012 galaxies, we estimate Ns ∼ 1011 super-massive galactic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This gives B2 ∼ 1028 if acceleration happens in the inner region around the Schwarzschild radius Rs ∼ 2GM ∼ 103 s, or B2 ∼ 106 if acceleration happens in a galactic-size Mpc region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Much larger B2 ∼ 1035 could arise if CR acceleration significantly happens in brief events, while the black hole accretes nearby stars by tidal disruption [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' These objects tend to have a geometry favourable to high-energy collisions, as incoming accel- erated particles are channelled towards out-going jets at magnetic poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 13 Assuming a negligible boost, B2 ∼ 1, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 4 shows that the CR maximal collision energy is around the cut-off observed by Auger and TA, and possibly due to the GZK effect (the pγCMB → ∆+ process opens at high energy and leads to absorption on distances larger than ∼ 100 Mpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This collision energy is comparable to the uncertain SM Higgs instability scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 4 shows that a ‘modest’ boost factor B2 ∼ 1020 brings the maximal collision energy above the Higgs instability scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Finally, CR around production sources could reach higher energies not limited by the GZK effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In such a case, significantly larger collision energies happened (red curve in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 4) if around production sites the CR energy spectrum is a power law with no GZK cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Some other cut-off is however expected from the acceleration mechanism itself, and maybe this (rather than the GZK effect) generated the cut-off observed in CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='2 Collisions of N cosmic rays Collisions among N ≫ 1 initial-state particles are relevant for vacuum decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Starting from N = 3, the number of 3-collisions among cosmic rays with energy E is N3 = B3 � d4x n3σ3 where we can estimate the 3-particle ‘cross section’ as σ3 ∼ 1/E5 and the boost factor as B3 ≈ ⟨n3⟩ ⟨n⟩3 ∼ B2B1 B1 = ns n ∼ TUR3 U NsTsR3 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (29) The plausible CR accelerations sites discussed in the previous section lead to the following extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' At one extremum, CR accelerated by AGN on Mpc scales have a mild B1 ∼ 105 resulting in a negligible N3 ≪ 1 at GZK energies comparable to the SM instability scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' At the opposite extremum, CR accelerated by small magnetars on short time-scales can have B1 ∼ 1060, implying N3 ≫ 1 at GZK energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' At larger N, the boost factors grow as BN/BN−1 ∼ B1 resulting in the number of N- collisions NN/NN−1 ∼ B1n/E3 ∼ (E0/E)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The energy E0 is low, E0 ∼ 10 GeV, even in the most optimistic case of small magnetars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This means that collisions of a large number N ∼ 1000 of cosmic rays never occurred at energies comparable to the SM instability scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 5 Vacuum decay triggered by many particles As discussed in the previous sections, vacuum decay must be triggered by an initial state that contains a large number N of particles to avoid exponentially small rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' One might think that vacuum decay can be triggered by colliding N beams into one center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' However, this N-collision would produce all SM particles (not only the Higgs), approximatively giving a thermal-like environment that ‘burns’ the Higgs instability, as the potential V (h) would be contain an extra stabilizing thermal-like Higgs mass trem M th h = κT, similarly to what happens at finite temperature [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Here κ2 ≈ (g2 Y +2g2 2)/16+y2 t /4+· · · ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='352 is a combination of SM couplings [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' As SM couplings are perturbative, collisions between many particles occur with negligible rate in the finite temperature plasma that filled the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This is a 14 ℓ+ ℓ- θ h ℓ+ ℓ- θ h ℓ+ ℓ- θ h ℓ+ ℓ- θ h ℓ+ ℓ- θ h ℓ+ ℓ- θ h ℓ+ ℓ- θ h Figure 5: ‘Star’ collision scheme with N beams of ℓ− (blue) and ℓ+ (red) with equal energy collide at small angle θ producing on-shell ℓ−ℓ+ → h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The N boosted Higgs bosons (in green) next collide at the center, inducing Higgs vacuum decay for large enough N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We here plotted for N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' special system where particle collisions lead to a vacuum decay rate exponential suppressed by a factor Sthermal ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='015πκ/|λ| that can be computed via Euclidean methods [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The above discussion indicates that triggering vacuum decay needs a cleaner initial state of N Higgs quanta, not accompanied by a much larger number of SM particles, as they would effectively contribute to κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='1 Collision scheme A lepton collider with energies Eℓ± and collision angle θ can in principle produce a clean Higgs source with controllable energy Eh = Eℓ− + Eℓ+ via resonant ℓ−ℓ+ → h production M 2 h = s = 2(Eℓ−Eℓ+ + m2 ℓ − pℓ−pℓ+ cos θ) (30) The cross section is σpeak = 4π BR(h → ℓ+ℓ−)/M 2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The SM predicts BR(h → ℓ+ℓ−) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='22 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Unlike in an asymmetric e−e+ → φ factory, the produced h is highly boosted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' By using N such colliders one could engineer N Higgs bosons simultaneously hitting a central interaction point, while SM particles produced by other processes with comparable cross sections σ ∼ g4/4πM 2 h are not focused into this center by the resonant kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 5 illustrates an example with Eℓ+ = Eℓ− and N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The distance from the production point to the interaction point must be smaller than the life-time of the boosted Higgs bosons, τh Eh Mh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='37 µm Eh 109 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (31) 15 Figure 6: Classical evolution of an in-coming spherical Higgs wave containing about 1000 ultra- relativistic Higgs quanta with individual energy k peaked around htop and total energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' a) Forming a slightly sub-critical bubble that dissolves, despite reaching h ≫ htop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' b) Forming a slightly super- critical bubble that ignites vacuum decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Needless to say, the needed energies are highly futuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' A circular collider would need an as- tronomical radius R = Eℓ/eB = 1 sec (Eℓ/ 1 2109 GeV)(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='5 T/B) just to circulate in the magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The fraction of ℓ energy lost per turn is ϵ = 2e2E3 ℓ /3m4 ℓR: below unity for a muon beam with energy Eµ = 1 2109 GeV if R >∼ 11 hr, comparable to the boosted muon lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' As a special case of possible interest, a ℓ+ beam hitting on ℓ− at rest needs energy Eℓ+ ≃ M 2 h/2mℓ to produce the Higgs resonantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The Higgs boson energy Eh ≃ Eℓ+ equals Eh = 74 TeV for muon µ+µ− collisions, and Eh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='5 107 GeV for e−e+ collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The electron option could roughly produce the desired energy Eh, and the boosted Higgs life-time would be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='6 nm, comparable to the atomic size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Triggering vacuum decay needs machine optimisations different from those considered for muon colliders, that aim at a high time-averaged luminosity L, that grows quadratically with its beam energy (values up to √s ∼ 10 TeV are currently discussed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The geometry considered here reverses the transverse and longitudinal beam size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' One Higgs collision event is enough if the N Higgs boson are synchronised in space and time up to the quantum uncertainty of order 1/Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Otherwise, if this level of synchronisation cannot be achieved, a high enough rate of Higgs quanta is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' For example, an instantaneous event peak rate Lpeakσ of order ∼ Eh is needed to have consecutive Higgs quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='2 Classical Higgs wave While more practical configurations can be considered (such as two beams with N/2 Higgs boson each), our goal here is to explore if triggering vacuum decay is theoretically possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 16 Figure 7: As in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 6, but using about 2500 Higgs quanta with lower energy k peaked around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='01htop, so that a bubble with larger radius ∼ 1/k and lower field value h ≈ htop ignites vacuum decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We thereby focus on the system of N ≫ 1 Higgs bosons, that can be approximated as a (theoretically simple) classical inward-going spherical Higgs wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Triggering vacuum decay via classical evolution would avoid the exponential suppression of the quantum rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We recall the classical Higgs equation in flat space, ¨h − h′′ − 2h′/r = −V ′ for a spherical wave h(t, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We denote as r = r0 the radius at the points in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 5 where the Higgs bosons are produced at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In terms of u(t, r) = r h(t, r)/r0 the wave equation becomes ¨u − u′′ = −rV ′/r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Around the production point r ≈ r0 we can initially neglect the Higgs potential V (h), because the Higgs field value is well below htop and because the Higgs quanta are ultra- relativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' So the time evolution of the inward wave is initially approximatively solved as h(t, r) ≃ u0(r + t)r0/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' (32) Assuming as initial profile u0 a short wave-packet with length 1/k peaked at r ≈ r0, the energy E = � 4πr2 dr[˙h2/2 + h′2/2 + V ] of the full system is roughly given by E ∼ 4πr2 0kh2 0, corresponding to N ≈ E/k = 4πr2 0h2 0 Higgs quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Here h0 = h(0, r0), where t = 0 is the ℓ−ℓ+ → h production time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 6 and 7 we show the numerical solution to the classical evolution equations, choosing an incoming wave-packet with arbitrary profile of the form u0(r) = h0 sin[k(r − r0)]e−k(r−r0)2/2 (33) dependent on two free parameters: k (that controls the typical energy of one Higgs quantum), and h0 (that controls the intensity of the wave).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 6 shows the evolution using quanta with k = htop i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' energies around htop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In the left panel we see how a spherical in-ward wave forms a bubble with h > htop that however does not 17 :trigger vacuum decay and dissolves into an out-ward wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Increasing the intensity of the wave crosses the critical value above which the bubble ignites vacuum decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This case is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 6b, and corresponds to roughly the minimal number N ∼ 1000 of Higgs quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 7 shows the similar result using quanta with lower energy k peaked around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='01htop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This only allows to form bubbles with large radius r0 ∼ 1/k ∼ 100htop, that ignite vacuum decay as soon as the Higgs field inside is slightly above the top of the SM potential barrier at htop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Compared to the previous case of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 6, the total energy of the configuration is lower, E ∼ 500htop, but a larger number of Higgs quanta N ∼ 2600 is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' While in principle colliders with k ∼ TeV can trigger vacuum decay, in practice this would need a huge number N ∼ (htop/k)3 of Higgs bosons that would decay too fast, with low boost factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' To avoid hitting a singularity at h = ∞, our numerical computations introduced an extra non-renormalizable term in the Higgs potential that creates a deep minimum at large field value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' If this is absent and/or deep, gravity becomes relevant as the Higgs falls towards the true vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' As shown in [52, 53], gravity cannot stop the explosion: a black hole can form inside the bubble, that anyhow expands at the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='3 Can a vacuum bubble be controlled, extracting energy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The technique proposed to ignite a vacuum bubble is loosely similar to the colliding beam technique that aims at nuclear fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' A key difference is that vacuum energy, unlike energy stored in atoms or nuclei, is not limited by the amount of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Does this imply that vacuum decay only produces an unstoppable explosion, so that vacuum energy cannot be extracted and used as an energy source?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In line of principle, the expansion of a vacuum bubble could be slowered or blocked by hitting all its surface with beams of SM particles intense enough to balance the vacuum pressure ∆V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This is possible because most SM particles get ultra-heavy inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' For theoretical simplicity we here focus on the possibility of surrounding the bubble with a thermal bath at temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The system would behave as an ultra-hot star, stabilised only by artificially tuning the temperature T to be comparable to the energy difference ∆V 1/4 between our vacuum and the unknown vacuum inside the bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The value of ∆V is currently unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In line of principle, ∆V could be determined (without creating the bubble) by measuring safe few-particle collisions at ultra-high energy, and interpreting the observations in Quantum Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Three extreme possibilities are: 1) ∆V ∼ M 4 Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In this case the bubble would be uncontrollable but scientifically interesting: the true vacuum reaches the Planck scale, probing the multiverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 2) ∆V ∼ λh4 top/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 3) ∆V ≪ h4 top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In this case the bubble would be more controllable, but this is a tuned possibility, perhaps motivated by the vague idea of Multi-Criticality [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 18 Let’s assume (for simplicity, and perhaps because it’s needed) that gravity remains weak, namely that the bubble remains smaller than its Schwarzschild or AdS radius, Rs <∼ MPl/∆V 1/2, corresponding to Buchdahl mass |M| ∼ ∆V R3 s ∼ M 3 Pl/∆V 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' As a numerical example, in case 2) the vacuum energy density is ∼ 37 orders of magnitude larger than nuclear energy, Rs <∼ 10−16 m, releasing |M| ∼ 1028 J as energy with power W ∼ R2 sT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The stabilised bubble would have negative mass, but it would would soon become an im- practical source of energy: after a time of order Rs preventing the bubble explosion while keeping the bubble in the weak-gravity regime costs more energy than what it released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The bubble could be closed by ‘burning’ it with more intense beams, but this would cost at least all energy it released, because of energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' To extract vacuum energy, one needs to go in the strong-gravity regime: one could ‘dispose’ the bubble behind a pre-existing black hole horizon before the vacuum bubble explodes, or perhaps add extra energy such that the bubble gets screened behind its own black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 6 Conclusions We reconsidered vacuum decay stimulated by particle collisions, focusing on the possible in- stability of the SM Higgs potential extrapolated up to ultra-high field values htop ∼ 1010 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This instability is suggested by current values of the top mass and strong coupling, but more accurate measurements of these quantities are needed to establish if this instability would really exist, and to infer a precise value of htop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In section 2 we found that Higgs vacuum decay can be triggered by an initial Higgs field configuration (for simplicity spherical and static) with energy E >∼ 4πhtop/ � |λ| ∼ 500 Joule that contains a large number N >∼ Nmin ∼ 4π/|λ| ∼ 1000 of Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 1 shows the precise threshold on the size and intensity of the initial Higgs configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The critical configuration with minimal energy, the sphaleron, contains a number of Higgs quanta mildly higher than the critical configuration with minimal number of quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Since N ≫ 1 these are semi-classical configurations, and the quantum amplitude for forming a critical bubble out of collisions of few particles is exponentially suppressed by exp(−O(N)), as typical for quantum transitions between classically different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This exponential suppression was more precisely computed in thin-wall approximation by Voloshin [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In section 3 we verified this result: our result in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 2 confirms that particle collisions lift the exponential suppression only partially, because the reduced exponential sup- pression of tunnelling gets compensated by the exponential suppression for forming the needed semi-classical configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' However, the thin-wall approximation is not applicable to SM Higgs vacuum decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Going beyond this limit, in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='2 we argued that the lifting of its exponential suppression is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We provided arguments based on the approximate scale invariance of the Higgs potential, and assuming special field configurations that allow a simple computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Furthermore, in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='3 we addressed a related issue: the cross section σN for producing N Higgs bosons grows factorially with N, and some authors argue that σN might become 19 large at N >∼ 4π/λ, a possibility dubbed ‘Higgsplosion’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' This number of quanta (extended to relativistic Higgs bosons) would also form the semi-classical Higgs configuration that stimulates SM Higgs vacuum decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We thereby critically considered the issue, reviewing recent works that find that σN remains exponentially suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Having clarified that stimulated vacuum decay remains exponentially suppressed, in sec- tion 4 we compared its rate with the rate of ultra-high energy collisions of cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' By considering collisions happened in relatively compact and possibly short-lived astrophysical production sites of ultra-high energy cosmic rays, we estimated that the number of collisions with √s ∼ htop can be tens (up to 60) orders of magnitude larger than the minimal number of collisions usually estimated in outer space, as illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' The CR collision rate can be so high that ultra-high energy collisions among N > 2 cosmic rays (but not N ≫ 2) occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Furthermore, the √s of cosmic-ray collisions around their production sites could extend beyond the GZK cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Nevertheless, we find that these enhancements cannot compensate for the exponential suppression of Higgs vacuum decay stimulated by particle collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Finally, in section 5 we discussed how the exponential suppression can be bypassed classi- cally by using futuristic ultra-high energy colliders to artificially engineer an in-going wave of N >∼ 1000 highly boosted Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We proposed a collision scheme, illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 5, that exploits ℓ−ℓ+ → h on-shell production to form an inward-going Higgs wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We simu- lated the classical evolution of the Higgs wave, finding the critical threshold above which it triggers vacuum decay, rather than dissolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' Results are shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' 6 and 7 for sub- and super-critical waves, and for two different energies of the Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content='3 we wildly speculate about a basic issue: is it possible (at least in theory) to control a vacuum bubble slowering its expansion and using it as an energy source?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' We suggest a technique based on pressing beams and on disposing the dangerous negative-mass remnant behind a black-hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} +page_content=' References [1] D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfDAaO/content/2301.03620v1.pdf'} diff --git a/VNFLT4oBgHgl3EQfRy9S/content/tmp_files/2301.12038v1.pdf.txt b/VNFLT4oBgHgl3EQfRy9S/content/tmp_files/2301.12038v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c8d9069bc88d8ee4a8cae4165d17be7578b627e --- /dev/null +++ b/VNFLT4oBgHgl3EQfRy9S/content/tmp_files/2301.12038v1.pdf.txt @@ -0,0 +1,3114 @@ +arXiv:2301.12038v1 [cs.LG] 28 Jan 2023 +STEERING: Stein Information Directed Exploration for +Model-Based Reinforcement Learning +Souradip Chakraborty 1 Amrit Singh Bedi 1 Alec Koppel 2 Mengdi Wang 3 Furong Huang 1 Dinesh Manocha 1 +Abstract +Directed Exploration is a crucial challenge in re- +inforcement learning (RL), especially when re- +wards are sparse. Information-directed sampling +(IDS), which optimizes the information ratio, +seeks to do so by augmenting regret with in- +formation gain. +However, estimating informa- +tion gain is computationally intractable or re- +lies on restrictive assumptions which prohibit +its use in many practical instances. +In this +work, we posit an alternative exploration incen- +tive in terms of the integral probability metric +(IPM) between a current estimate of the tran- +sition model and the unknown optimal, which +under suitable conditions, can be computed +in closed form with the kernelized Stein dis- +crepancy (KSD). Based on KSD, we develop +a novel algorithm STEERING: STEin infor- +mation dirEcted exploration for model-based +Reinforcement LearnING. To enable its deriva- +tion, we develop fundamentally new variants of +KSD for discrete conditional distributions. We +further establish that STEERING archives sublin- +ear Bayesian regret, improving upon prior learn- +ing rates of information-augmented MBRL, IDS +included. Experimentally, we show that the pro- +posed algorithm is computationally affordable +and outperforms several prior approaches. +1. Introduction +Exploring effectively is a major challenge in reinforce- +ment learning (RL), particularly when the rewards are +sparse (Rengarajan et al., 2022; Achiam & Sastry, 2017). +Recent research using model-based reinforcement learn- +1Department of Computer Science, University of Mary- +land, College Park, USA. +2JP Morgan Chase AI Research, +USA. +3Department of Electrical Engineering, Princeton Univer- +sity/Deepmind, Princeton, NJ, USA. Correspondence to: Amrit +Singh Bedi . +This research was supported by Army Cooperative Agreement +W911NF2120076. +Figure 1. (Directed +exploration) +This +figure +illustrates +that +information-directed sampling (IDS) focuses on the distance be- +tween the current sample at episode k and the mean of the poste- +rior. In contrast, we focus on the distance to the true model. +ing (MBRL) with intrinsic curiosity (Pathak et al., 2017a; +Burda et al., 2018b; Pathak et al., 2019) has offered a po- +tential solution, but its theoretical understanding is rel- +atively immature. +On the other hand, posterior sam- +pling reinforcement learning (PSRL) offers an efficient +framework for balancing exploration and exploitation that +is conceptually well-substantiated (Osband et al., 2013; +Osband & Van Roy, 2014). However, PSRL may struggle +in scenarios where many trajectories are uninformative, due +to, e.g., reward sparsity (Russo & Van Roy, 2014a). +To incentivize exploration in MBRL, Lu & Van Roy +(2019b) proposed a design principle of information di- +rected sampling (IDS), which optimizes the tradeoff +between regret and information. +Tight information- +theoretic Bayesian regret bounds for IDS are derived in +(Lu & Van Roy, 2019b; Lu et al., 2021), under the specific +choice of Dirichlet priors for the transition model, inspired +by earlier work on bandits (Russo & Van Roy, 2014b). +Recently, Hao & Lattimore (2022) alleviated any require- +ments on the prior based upon the development of a surro- +gate environment estimation procedure via rate-distortion +theory. +Unfortunately, the existing IDS approaches face two main +challenges (1) they are computationally intractable due to +the need to estimate the information gain, (2) they do not +induce exploration directed towards the optimal true tran- +sition dynamics (see Fig. 1). By directed exploration, we +mean the algorithm moves in a direction toward the opti- + +Posterior over +Model at episode k +Proposed +IDS +M +M* True Model +Sample +at episode kStein Information Directed Exploration for Model-Based Reinforcement Learning +2 +mal transition dynamics rather than collecting all the infor- +mation about the underlying environment, which is the fo- +cus of information gain-based exploration in IDS. The first +challenge can be partially addressed by instead optimizing +the evidence lower-bound (Achiam & Sastry, 2017), but +ends up restricting focus to the posterior variance, which +may be insufficiently informative about the underlying tar- +get distribution. This motivates us to pose the following +question: +Can we develop a computationally tractable posterior +sampling-based RL algorithm that exhibits efficient di- +rected exploration with provable guarantees? +We provide an affirmative answer to this question by con- +sidering an alternative measure of information. That is, we +propose Stein information gain, which is the integral prob- +ability metric (IPM) difference between the estimated and +true (unknown) transition dynamics (Sriperumbudur et al., +2012), hence inducing directed exploration. +Under the +assumption that the transition model lies in the Stein +class, we employ Stein’s identity (Efron & Morris, 1973; +James & Stein, 1992) to evaluate this IPM between the true +(unknown) and estimated transitions in closed-form using +kernelized Stein discrepancy (KSD) (Gorham & Mackey, +2015; Liu et al., 2016; Hawkins et al.). This is the key nov- +elty that alleviates a major drawback of prior approaches +that require evaluating mutual information. Thereby we in- +troduce the Stein-information ratio, which may be seen as a +modification of the information ratio in (Russo & Van Roy, +2018; Lu et al., 2021), and incentivizes exploration. We +emphasize that our notion of KSD-based Stein informa- +tion gain empowers us to evaluate the distance to the true +transition dynamics. +Doing so permits us to derive the +best-known prior-free information-theoretic Bayesian re- +gret bounds. Towards this end, we also develop the first +KSD for conditional discrete distributions and employ it in +tabular RL settings. Appendix A provides a detailed con- +text of related works. +Contributions: Our main contributions are as follows. +⊲ We formalize the setting of model-based episodic RL +with Bayesian regret incorporating a notion of distance +to the ground-truth MDP with KSD, and hence achieves +directed exploration towards the optima. +⊲ We introduce discrete conditional KSD (DSD) in tabu- +lar RL for the first time and use it to analyze distribu- +tional distance, which empowers us to evaluate the ex- +ploration incentive towards the true transition dynam- +ics. Through this definition, we introduce a specific +exploration-incentivized modification of posterior sam- +pling, called STEERING (Algo. 1). +⊲ We establish prior-free Bayesian regret bounds for +STEERING that are sublinear both in the number of +Table 1. This table compares the Bayesian regrets (cf. +(3)) +of different existing approaches in the literature with or with- +out information augmentation (IA). We note that the mini- +max bounds +˜O( +√ +H3SAK) achieved in prior work apply to +the case without augmentation. +The baseline approaches are +PSRL (Osband et al., 2013), PSRL2 (Osband & Van Roy, 2017a), +TSDE (Ouyang et al., 2017), IDS-RL1 (Lu & Van Roy, 2019a), +and IDS-RL2 (Hao & Lattimore, 2022) respectively. +Approaches +Regularization +Regret +IA +PSRL +No +˜O( +√ +H3S2AK) +No +PSRL2 +No +˜O( +√ +H3SAK) +No +TSDE +No +˜O( +√ +H3S2AK) +No +IDS-RL1 +Mutual Inf. +˜O( +√ +H3SAK) +Yes +IDS-RL2 +Mutual Inf. +˜O( +√ +H4S3A2K) +Yes +STEERING +Stein Inf. +˜O( +√ +H4S2AK) +Yes +episodes and action space cardinality (see Table 1 for +detailed comparison). +⊲ We provide extensive experimental evidence for the pro- +posed STEERING algorithm in sparse reward settings +and show improved regret as well as efficient directed +exploration compared to all existing approaches. +2. Problem Formulation +We consider the problem of learning transition dynam- +ics in an episodic finite-horizon time-homogeneous tabular +Markov Decision Process (MDP) setting. We define the +unknown MDP as M := {S, A, R, P, H, Rmax, ρ}, where +S is finite state-space with S = |S|, A is the finite action +space with A = |A|, and H is the episode length. Here, +P : S × A → ∆S represents the transition dynamics for +the state-action transitions and R is the rewards distribution +where ∆S denotes the set of probability distributions over +a finite set S. After every episode of length H, state will +reset according to the initial state distribution ρ. At time +step i ∈ [H] within an episode, the agent observes state +si ∈ S, selects action ai ∈ A according to a stochastic +policy π : S → ∆A, receives reward ri ∼ R(si, ai) and +transitions to a new state si+1 ∼ P(·|si, ai). In this work, +we consider M as a random process, as is often the case in +Bayesian RL (Russo & Van Roy, 2014a). +Value Function and Bayesian Regret. For a given MDP +M, the value for time step i is the reward accumulation +during the episode, given by +V M +π,i(s) = E +� H +� +j=i +[¯rM(sj, aj)|si = s, aj ∼ π(·|sj)] +� +, +(1) +where j denotes the time-step within the episode and +¯rM(s, a) = Er∼RM(s,a)[r]. Without loss of generality, we +assume |¯rM(s, a)| ≤ Rmax for all s ∈ S, a ∈ A, which + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +3 +(a) Dense rewards. +(b) Sparse rewards. +(c) Variable sparsity (PSRL). +Figure 2. This figure compares the performance of various RL algorithms for Dense reward (Fig. 2(a)) and Sparse reward (Fig. 2(b)) +DeepSea environment (DSE) (Osband & Van Roy, 2017a). As we move from dense to sparse reward settings, we note that the regret +becomes almost linear for all algorithms except PSRL (Osband et al., 2013). We further show the performance degradation even for the +PSRL algorithm with different sparsity levels in Fig. 2(c) achieved by varying N (denotes the number of states in DSE). +implies that |V (s)| ≤ HRmax, for all s ∈ S. Next, for a +given MDP M, an optimal policy π⋆ is +π⋆ = argmax +π +V M +π,i(s), +(2) +for all s and i ∈ [H]. We emphasize that since π⋆ is a func- +tion of M, it is also a random variable. An RL agent would +interact with the environment over K number of episodes +with policy {πk}K +k=1. +The performance of the learning +agent with respect to environment M can be quantified by +Bayesian regret: +BRK := E +� K +� +k=1 +(V M +1,π∗(sk +1) − V M +1,πk(sk +1)) +� +, +(3) +where the expectation is with respect to the randomness in +the policy πk and the prior distribution of M. Typically, +one focuses on ensuring the sublinear growth of Bayesian +regret in (3) as a way to quantify the learning perfor- +mance of a given model-based estimate Mk and the resul- +tant policy πk := argmaxπ V Mk +π +(s). Posterior sampling- +based reinforcement learning (PSRL) operates this way +(Osband & Van Roy, 2017a), as does upper-confidence +bound (Ayoub et al., 2020), and Gittin’s index (Edwards, +2019). However, regret [cf. (3)] alone may under-perform +in contexts where the expected value function is a sparse +function of the initial state, which can occur when the re- +ward function is sparse, which is true for different applica- +tions (Weerakoon et al., 2022; Chakraborty et al., 2022c). +To emphasize this point, we consider the DeepSea environ- +ment of Osband & Van Roy (2017a) and compare the per- +formance of different existing methods in dense and sparse +reward settings in Fig. 2. The performance degradation +as we go from dense to sparse settings is evident (from +Fig. 2(a) to Fig. 2(b)). We note that all the algorithms ex- +hibit almost linear regret except PSRL, which establishes +the efficient exploration aspect of PSRL. But as we investi- +gate PSRL further for different sparsity levels in Fig. 2(c), +we conclude that even PSRL suffers badly in very sparse +reward settings. To deal with such challenges, information- +directed sampling has been developed in the literature to +jointly quantify value function sub-optimality with state +space coverage. +Information Directed Sampling (IDS) augments PSRL +by introducing the information ratio at episode k defined +as +Γk(π, M) := +� +Ek[V M +1,π∗(sk +1) − V M +1,π(sk +1)] +�2 +Iπ +k(M; Hk,H) +, +(4) +whose numerator is the Bayesian regret and the denomina- +tor is the information gain (Russo & Van Roy, 2014a). The +information gain Iπ +k(M; Hk,H) = H(M) − H(M|Hk,H) +quantifies the reduction in the entropy of the learning target +environment M after conditioning on the state-space cover- +age Hk,h := {s1 +1, a1 +1, r1 +1, . . . , sk +h, ak +h, rk +h} in episode k. The +policy selection becomes πk = arg maxπ Γk(π, M). Ob- +serve that policies π which have a small information ratio +(4) not only minimize regret but do so while maximizing +per unit information gain, which can yield efficient explo- +ration. Following Hao & Lattimore (2022, Lemma A.1), to +gain further insight, it is useful to rewrite the information +gain as +Iπ +k (M; Hk,H) +(5) += +H +� +h=1 +EkEM k +π +� +DKL +� +P M(·|sk +h, ak +h)||P M k(·|sk +h, ak +h) +�� +, +where EM k +π +is taken with respect to sk +h, ak +h, and Ek is with +respect to π and environment M. The expression in (5) +makes it clear that incorporating Iπ +k (M; Hk,H) into the ob- +jective motivates the agent to visit the state action region + +00 +5000Sparsity N +N=20~ +200 +N=8 +Regret to Oracle +150 +N=7 +100 +N=4 +50 +0 +0 +1000 +2000 +3000 +40 +TimeSteps350 +UBE-1 +300 +Q-Learning +Regret to Oracle +250 +UBE-2 +MM +200 +BQL +150 +PSRL +100 +50 +0 +0 +1000 +2000 +3000 +4000 +5000 +TimeStepsStein Information Directed Exploration for Model-Based Reinforcement Learning +4 +where DKL(P M||P M k) is higher which acts as an intrinsic +reward to explore state action pairs with high uncertainty. +2.1. Limitations of IDS +The ratio objective in (4) exhibits practical limitations re- +lated to the fact that the KL divergence in (5) cannot be +evaluated. We next detail why this is so and how a modifi- +cation can alleviate this issue. +(L1) Computational Intractability: A major challenge +in prior approaches including (Hao & Lattimore, 2022; +Lu & Van Roy, 2019b) lies in an accurate estimation of +mutual information. One of the first prior-free IDS anal- +yses by Hao & Lattimore (2022) relies on constructing a +covering set for KL divergence with cover radius ǫ = +1/KH. This implies that the information gain term grows +unbounded as the number of episodes K increases. More- +over, for practical implementation, to make the KL di- +vergence in (5) tractable, Hao & Lattimore (2022) sub- +stitutes this quantity by its lower bound via Pinsker’s +inequality: +Ek[DKL(P M(·|sk +h, ak +h)||P M k(·|sk +h, ak +h))] +≥ +� +s′ Var(P M(s′|sk +h, ak +h)). +However, Ozair et al. (2019) +shows that any high-confidence lower bound requires expo- +nential samples in the mutual information, which is a crit- +ical concern. Hence even the variance lower bound with +Pinsker’s inequality runs into the computational limits of +estimating mutual information. +(L2) Not Truly A Directed Exploration: +In the +majority of the prior research on information-directed +RL (Hao & Lattimore, 2022; Lu & Van Roy, 2019b), the +mutual information or KL divergence serves as the +information-theoreticregularization or intrinsic curiosity to +induce directed exploration to deal with the hard explo- +ration challenges as detailed in (Russo et al., 2017). How- +ever, as we note from (5), since the mutual information +must be substituted by the variance of the current estimate +of the posterior distribution over M for tractability pur- +poses, it only encourages the agent to explore trajectories +with high variance. Ideally, we would want our exploration +to be directed towards the true ground truth model M ⋆ (see +Fig. 1) from which the data (s, a, s′) samples are collected +in practice. A directed exploration towards M ⋆ is crucial +to avoid random wandering in the environment. For in- +stance, consider a setting where we start with the strong +belief prior, then, since the variance is already low, IDS +based approach will not add any benefit on top of PSRL- +based approaches. +To address the above limitations, we propose a novel notion +of Stein information gain to achieve directed exploration in +the next section. +3. Proposed Approach and Algorithm +In this work, we develop a novel Bayesian regret analy- +sis that incorporates a notion of distance to the true opti- +mal MDP and provides a computationally tractable alterna- +tive to the notion of information ratio in Hao & Lattimore +(2022). Before presenting the proposed approach, let us +discuss the technical development as follows. +3.1. Kernelized Stein Discrepancy +Integral probability metrics (IPM) have gained trac- +tion in Bayesian inference and generative modeling +(Arjovsky et al., 2017) for their ability to quantify the merit +of a given posterior distribution with respect to an unknown +target without specifically having knowledge of that target. +In particular, when one suitably assumes the class of poste- +riors over which the search is conducted to the Stein class +(Liu et al., 2016), IPMs admit a closed-form evaluation in +terms of Stein discrepancies. Please refer to Appendix B +for detailed discussion and derivation. To this end, Liu et al. +(2016) define kernel Stein discrepancy (KSD) between two +distributions p and q as +KSD(p, q) = Ex,x′∼p +� +uq(x, x′) +� +, +(6) +where uq(x, x′) is the Stein kernel defined as +uq(x, x′) := sq(x)⊤κ(x, x′)sq(x′) + sq(x)⊤∇x′κ(x, x′) ++ ∇xκ(x, x′)⊤sq(x′) + trace(∇x,x′κ(x, x′)), +(7) +where κ(x, x′) is the base kernel (any positive definite ker- +nel, for instance, Hamming Kernel for discrete rvs) The +Stein kernel in (7) measures the similarity between two +samples x and x′, which comes from p, using the score +function of q. For the setting in this work, we have p = P ∗ +(transition dynamics corresponding to true model M ∗) and +q = P Mk (transition dynamics corresponding to posterior +Mk ∼ φ(·|Hk)). Interestingly, KSD empowers us to eval- +uate the distance KSD(P Mk, P ∗). In the next subsection, +we present the main idea of this work. +3.2. Stein Information Directed Sampling +Now, after the introduction of KSD, we note that the distri- +butional distance to the unknown target can be computed in +closed form. We use this fact to address the limitations of +IDS discussed in Sec. 2.1, and propose to replace informa- +tion gain Iπ +k(M; Hk,H) in the denominator of (4) with what +we call Stein information gain Kπ +k (M; Hk,H) given by +Kπ +k (M; Hk,H) +(8) +:= +H +� +h=1 +EkEM∗ +π +� +KSD +� +P M(·|sk +h, ak +h), P ∗(·|sk +h, ak +h) +�� +. + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +5 +where EM∗ +π +is taken with respect to (sk +h, ak +h), and Ek is with +respect to π and environment M. There are two main differ- +ences here as compared to information gain defined in (5). +First, we use KSD to characterize if two transition models +are close or not. Second, we use the distributional distance +to true model M ∗ in (8) in contrast to divergence to pos- +terior mean M k in (5). Hence, the ratio objective in (4) +would modify to Stein information ratio as +ΓKSD +k +(π) := (Ek[V M +1,π∗(sk +1) − V M +1,π(sk +1)])2 +Kπ +k (M; Hk,H) +, +(9) +and we select πk = argminπ ΓKSD +k +(π). We remark that (8)- +(9) are the point of departure from the existing IDS-based +methods (Hao & Lattimore, 2022). Stein information gain +term of (8) is different from the reduction in entropy (as +in information gain) but instead characterizes the distribu- +tional distance to the true MDP transition dynamics P M∗. +So it forces the algorithm to move the model estimate to- +wards the optimal than focusing on the coverage of space +induced by information gain term in (4) (addressing L2). +Another interesting aspect of the modified ratio objective +in (9) is that it is computationally tractable due to the use +of KSD and we no longer need to utilize Pinsker’s inequal- +ity to approximate uncertainty via posterior variance which +is unavoidable in IDS based approaches for practical imple- +mentation (Hao & Lattimore, 2022) (addressing L1) . +Unfortunately, although KSD is well-established, the pre- +existing machinery (3.1) does not directly apply to our set- +ting in (8). The impediments are twofold: firstly, (6) holds +for the continuous smooth densities, but our setting is tabu- +lar MDP; secondly, (6) applies to estimating unconditional +target distributions, but in an MDP context, we require con- +ditional distributions. To adapt Stein discrepancy to our +setting, we need to first address both of these issues next. +3.3. Discrete Conditional KSD +In this subsection, we introduce the kernelized conditional +discrete stein operator and Discrete conditional kernelized +Stein Discrepancy (DSD) for analyzing the distributional +distance between P M for a given M and P ∗. This is a +unique contribution of this work which may be of indepen- +dent interest in mathematical statistics. In tabular RL set- +ting, if we are in state (s, a), then we know s′ ∼ P ∗(·|s, a), +and up to kth episode, we collect samples in dictionary +Dk +:= +{((s1, a1), s′ +1), ((s2, a2), s′ +2), · · · ((sk, ak), s′ +k)}. +Now, the objective is to derive DSD between P M(·|s, a) +and ground truth P ∗(·|s, a) leveraging the recent literature +on conditional independence testing with Kernel Stein’s +method (Jitkrittum et al., 2020) (defined only for continu- +ous smooth densities). For simplicity and analysis in this +subsection, let us denote the state-action pair (s, a) → x ∈ +X := S × A and the corresponding next state s′ → y ∈ S. +To write DSD, we start by defining the Stein operator (cf. +Appendix B for unconditional case) as +κM((x, y), ·) = G(x, ·) +� +sP M (y)l(y, ·) − △∗l(y, ·) +� +, (10) +where M signifies the dependence on P M, function l : +S × S → R is a positive definite kernels, and G(x, ·) is +a real-valued kernel associated with the RKHS Fk and ex- +plicitly defined in Proposition 3.1. Further in (10), we de- +fine score function sP M (y) for P M and △∗ as the differ- +ence operator w.r.t inverse permutation denoted by ∧ for +the set S. We denote △ as the cyclic permutation ∨ for +the set S. For example, with S = {+1, −1}, ∨s = −s, +∀s ∈ S. On the other hand, inverse permutation satisfies +∨(∧(s)) = ∧(∨(s)) = s (see (Yang et al., 2018) more de- +tails). Therefore, we expand the terms in (10) as +sP M (y)i =△yiP M(y|x) +P M(y|x) += 1 − P M(∨iy) +P M(y|x) +(11) +△∗ +yil(y, ·) =l(y, ·) − l(∧iy, ·), +(12) +for i = 1, 2, · · · , S. Next, we provide the definition of +DSD between two conditional pmfs in Proposition 3.1. +Proposition 3.1. Let G(x, ·): X ×X → R be a real-valued +kernel associated with the RKHS Fk and l: S × S → R be +positive definite kernels. Assume Gx(x, x′) := k(x, x′)I +for a positive definite kernel k: X × X → R. Then, DSD +between P M(s′|s, a) and P ∗(s′|s, a) is given by +DSD(P M) = E[(x,y),(x′,y′)][κM((x, y), (x′, y′))], +(13) +where (x, y) and (x′, y′) are samples from the joint distri- +bution P ∗. +The proof of Proposition 3.1 is provided in Appendix C +and we show that DSD(P M(·|s, a)) = 0 iff P M(·|s, a) = +P ∗(·|s, a). +In the tabular RL setting, since both x ∈ +X and y ∈ S are discrete random variables, we con- +sider both l(y, y′) = exp{−H(y, y′)} and k(x, x′) = +exp{−H(x, x′)} as the exponential Hamming kernel +which is a positive definite kernel. However, the specific +selection of kernels and kernel design in tabular RL is the +scope of future work. We remark that since now DSD is +defined, we will use DSD in place of KSD in all the places +moving forward. +3.4. STEERING: Proposed Algorithm +Now, we are ready to present the proposed STEin in- +formation dirEcted sampling for model-based Reinforce- +ment LearnING (STEERING) algorithm for tabular set- +tings, summarized in Algorithm 1. We begin STEERING +by assuming a prior over the transition, and the rewards +model denoted by φD1 = {PD1, RD1}. +At episode k, +STEERING first samples a transition model P Mk and re- +wards model RMk from the posterior distribution φDk = + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +6 +Algorithm 1 STEERING: STEin information dirEcted +sampling for model-based Reinforcement LearnING +1: Input : Episode length H, Total timesteps T , Dictio- +nary D, prior distribution φ = {P, R} , policy π(a|s), +hyperparameter γ. +2: Initialization : Initialize dictionary D1 with random +π0(a|s), posterior φD1 = {PD1, RD1}, policy π1(a|s). +3: for Episodes k = 1 to K do +4: +Sample a transition P Mk ∼ PDk and reward model +RMk ∼ RDk and initialize empty C = [ ] +5: +Estimate +the +optimal +policy +by +minimizing +the +Stein +information +ratio +: +πk(a|s) +← +argminπ ΓDSD +k +(π, M) as in (9) +6: +Interact with the environment using the optimal +policy πk(a|s) to gather and initialize empty C = +{sk,1, ak,1, rk,1, · · · , sk,H, ak,H, rk,H} +7: +Select the subset of samples C′ ∈ C with least sim- +ilarity to sample in samples in Dk in terms of Stein +kernel +8: +Update dictionary Dk+1 ← Dk ∪ C′ and update the +posteriors PDk, RDk +9: end for +{PDk, RDk}. It then optimizes the policy under these sam- +pled models by minimizing the proposed Stein information +ratio argminπ ΓDSD +k +(π) as in (9). This inner optimization +procedure is a key aspect of STEERING, as it uses the Stein +information to guide exploration. Finally, the agent inter- +acts with the real environment using the resulting policy +πk(a|s) to collect new samples at episode k and store them +in C. Instead of just appending C to dictionary Dk, we pro- +pose to use an intelligent sample selection procedure (simi- +lar to SPMCMC (Chen et al., 2019)) on the collected sam- +ples in each episode k before updating the dictionary Dk. +This procedure selects new samples by minimizing their +similarity to existing samples in the dictionary Dk through +a local optimization procedure, as outlined in Appendix E +starting from equation (38). This selection procedure helps +to derive tighter convergence rates in the next section. +4. Regret Analysis +In this section, we derive the merits of incorporating Stein’s +method into the Bayesian regret of an MBRL method for +the first time. We start by deriving our key result in The- +orem 4.1, which connects Bayesian regret (cf. (3)) with +Stein information gain (cf. (8)). Then, we analyze the evo- +lution of the model-based estimates in terms of DSD [cf. +Sec. 3.3], which decreases with the number of samples pro- +cessed (Lemma 4.2). Next, we connect this bound to the +Stein information ratio objective and the total Stein infor- +mation gain in Lemma 4.3. Finally, we utilize Theorem +4.1, Lemma 4.2, and Lemma 4.3 to derive Bayesian regret +in terms of state and action space cardinality in Theorem +4.4 . We also extend our analysis to regularized settings in +Theorem 4.5. Next, we present our first Bayesian regret as +follows. +Theorem 4.1. (Stein Information Theoretic Regret) When +Algorithm 1 is run for K episodes of horizon length H, it +achieves the following Bayesian regret: +BRK ≤ +� +� +� +�E[Γ∗]K +K +� +k=1 +E[Kπ +k (M; Hk,H)] , +(14) +where Kπ +k (M; Hk,H) is the Stein information gain (cf. (8)) +and Γ∗ is the worst case Stein information ratio such that +ΓDSD +k +(π) ≤ Γ∗ for any k ∈ K and π. +The proof of Theorem 4.1 is provided in Appendix D. We +call the regret in Theorem 4.1 as the Stein information theo- +retic regret because it upper bounds the Bayesian regret in +terms of Stein information gain (cf. (8)), which is a DSD +between the estimated model and the true model. This is +the main point of departure as compared to information- +theoretic regret derived in Osband & Van Roy (2017b); +Hao & Lattimore (2022), which eventuates in substitution +of the information gain by the posterior variance as its un- +certainty quantifier due to the computational effort required +to estimate information gain. By contrast, we consider this +distributional distance instead in terms of integral probabil- +ity metrics, which under some hypotheses, are computable +as DSD (cf. Sec. 3.3). To the best of our knowledge, this is +the first time this notion of distance to ground truth, which +is typical of frequentist analysis of Bayesian methods, has +been incorporated into the Bayesian regret. +Next, we proceed toward deriving an absolute upper bound +on the regret in terms of S, A, and H. To achieve that, we +present two intermediate results in Lemma 4.2-4.3. +Lemma 4.2. (DSD Convergence Rate) With Algorithm 1, +we collect dictionary Dk for which it holds that +Ek +� +DSD(P M; Dk)2� += O +�S +k +� +, +(15) +for all k. +In (15), Ek is the expectation over M, and +DSD(P M; Dk)2 denotes the empirical approximation us- +ing dictionary Dk of discretized conditional kernelized +Stein discrepancy (cf. (13)) between P M and true P ∗. +The proof of Lemma 4.2 is provided in Appendix E. +Lemma 4.2 establishes the convergence of the transition +model estimation P M to the true model P ∗, which is an +important result to prove next Lemma 4.3. +Lemma 4.3. For Algorithm 1, after K episodes of horizon +length H, it holds that + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +7 +(a) Low sparsity. +(b) Medium sparsity. +(c) High sparsity. +Figure 3. This figure compares the performance of STEERING on DeepSea enviornment (Osband & Van Roy, 2017a) against the +existing RL baselines vanilla Q-learning with ǫ−greedy action selection (Watkins & Dayan, 1992), Bayesian Q-learning (BQL) +(Dearden et al., 1998), Uncertainty Bellman Equation (UBE) (O’Donoghue et al., 2017), Moment matching (MM) across Bellman equa- +tion (Markou & Rasmussen, 2019), Posterior Sampling RL (PSRL) (Osband et al., 2013), and IDS (Hao & Lattimore, 2022). We present +results for three sparsity levels and observe STEERING outperforms existing baselines. Interestingly, the performance of STEERING +is comparable (still much better shown in Fig. 3(a)-(b) to PSRL or IDS with low or medium sparsity, but for high sparsity in Fig. 3(c), +STEERING significantly outperforms the other methods. +(1) Stein information ratio (cf. (9)) is upper bounded as +E[ΓDSD +k +(π)] ≤ SAH3 for all k. +(2) Total Stein information gain (cf. (8)) is bounded as +�K +k=1E[Kπ +k (M; Hk,H)] ≤ HS(log K). +The proof is provided in Appendix F. The upper bounds +established in Lemma 4.3 are crucial to specialize the gen- +eral regret bounds developed in Theorem 4.1 in terms of +state-action cardinalities, as follows in Theorem 4.4. +Theorem 4.4. When Algorithm 1 is run for K episodes of +horizon length H, it achieves the following performance in +terms of Bayesian regret: +BRK = ˜O +�√ +H4S2AK +� +, +(16) +where ˜O absorbs the log factors, S and A are the state and +action space cardinalities, respectively. +The proof of Theorem 4.4 is provided in Appendix G. The- +orem 4.4 states that STEERING achieves Bayesian regret, +which is sublinear in terms of episode index K and the num- +ber of actions A, and linear in terms of the number of states +S. A detailed comparison to other existing results in the +literature is provided in Table 1. +Remark 1 (Insights for Improved Regret): Here we em- +phasize on certain insights regarding the improvements +of STEERING over prior research. +One key factor is +the use of distributional directed sampling achieved via +the integration of DSD and a point selection strategy in- +spired by SPMCMC (Chen et al., 2019). By utilizing an +intelligent point selection method (cf. +proof of Lemma +4.2 in Appendix E), we achieve better exploration and +tighter bounds, as previously demonstrated in different con- +texts by (Koppel et al., 2021; Chen et al., 2019). Specifi- +cally, our point selection approach (Appendix E Eq. 38) +inf(x,y) +� +(xi,yi)∈Dk−1 κM((xi, yi), (x, y)) involves choos- +ing a new sample (x, y) as the point that minimizes the sim- +ilarity (in RKHS) to the current samples in the dictionary +Dk−1 resulting in directed exploration. By implementing +an efficient point selection scheme, we derive Lemma 4.2, +which sharpens in Theorem 4.4 in the sense that the regret +dependence on the state-action cardinality is replaced by a +dependence on the samples. +Regularized STEERING. We also consider a regularized +Stein information gain based sampling objective and opti- +mize the policy as +πk +r = argmax +π +Ek[V M +1,µ(sk +1)]) + λKπ +k (M; Hk,H) , +(17) +where λ is the unknown regularization parameter. Next, we +prove that the new regularized objective incurs the same +Bayesian regret as the Stein information ratio in (9). +Theorem 4.5. +(Regularized Regret) When Algorithm +1 +(after +replacing +step +5 +with +(17)) +is +run +for +K +episodes of horizon length H, +it achieves the +following +performance in +terms +of Bayesian +regret +BRK ≤ +� +3 +2E[Γ∗]K �K +k=1 E[Kπ +k (M; Hk,H)], for λ = +� +KE[Γ∗]/�K +k=1E[Kπ +k (M; Hk,H). +The proof of Theorem 4.5 is provided in Appendix H. +5. Experiments +In this section, we evaluate the performance of the pro- +posed STEERING algorithm and compare it with other ex- + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +8 +isting state-of-the-art algorithms. Since we are interested +in developing algorithms for efficient directed exploration +for an RL agent, we consider a challenging tabular sparse +reward environment of DeepSea Exploration (DSE) intro- +duced in Osband & Van Roy (2017a) (Fig. 4). +Figure 4. DeepSea Exploration Environment. +The DSE environment tests the agent’s capability of di- +rected and sustained exploration. The agent starts from the +left-most state and can swim left or right from each of the +N states in the environment with near zero rewards every- +where except a reward of r = 1 only on a successful swim- +right to s = N (see Appendix I.1 for more details). Hence, +increasing the number of states N induces more sparsity +in the environment, making it extremely hard for the agent +to explore without directed exploration, as verified in Fig. +2(c). This motivates us to present improvements in the DSE +environment. Next, we test STEERING on four different +aspects of performance: (1) Regret to the Oracle, (2) Di- +rected exploration, (3) Robustness to prior belief, and (4) +Convergence to optimal value function. +(1) Regret to the Oracle: First, in Fig. 3, we compare +STEERING with other Bayesian/ non-Bayesian RL algo- +rithms in terms of the cumulative regret accumulated with +respect to an oracle agent following the optimal policy (re- +fer to Appendix I for details). We present results in Fig. 3 +for three different levels of sparsity: low (N = 8), medium +(N = 14), and high (N = 15). The results validate our +hypothesis that under highly sparse environments, general +existing RL methods fail to explore efficiently, resulting in +higher regret but STEERING outperforms. +(2) Directed Exploration by STEERING: To emphasize +the nature of effective directed exploration provided by +STEERING, we analyze its state-action space coverage in +Fig. +5. We compare the heatmaps of state-action occu- +pancy measure of the initial (top row) and final episodes +(bottom row) for PSRL, IDS, and STEERING. +(3) Robustness to Prior Belief: +The performance of +Bayesian algorithms depends upon the prior, which might +be a confident but mis-specified belief in practical settings. +To test against such scenarios, we perform an ablation study +in Fig. 6 to validate the robustness of STEERING to differ- +ent levels of confidence of the prior belief over the MDP. +Figure 5. (Directed Exploration) We plot heatmaps showing the +distribution of state-action visits in the initial and final episodes +of training for PSRL, IDS, and STEERING. The results reveal +that STEERING can effectively identify and visit the most im- +portant states indicated by the dark colors in the bottom row for +STEERING. By important states, we mean the states closer to +the goal state. In contrast, PSRL and IDS exhibit less directed +exploration (see lighter colors in the bottom row for PSRL and +IDS), with their heatmaps showing less concentration on impor- +tant states. These findings demonstrate the superior performance +of STEERING in guiding the exploration process towards opti- +mality. +(4) Convergence to Optimal Value function: We per- +form additional experiments in Appendix J.1 to analyze and +compare the convergence of the predicted ˆQ values for by +STEERING. +In Fig. 7, we also compare STEERING with baselines on +WideNarrow MDP to validate its performance under fac- +tored posterior approximations and PriorMDP to validate +its performance in general and practical environments with- +out special structures (Markou & Rasmussen, 2019). +6. Conclusions +Information-directed sampling (IDS) provides a way to +induce exploration incentives into model-based reinforce- +ment learning (MBRL). But IDS approaches suffer from +computational tractability issues. To make such ratio-based +approaches computationally tractable and efficient, we pro- +pose a novel measure to quantify directed exploration +through a distributional distance to the optimal model +via kernelized Stein discrepancy. To this end, we intro- +duced a novel notion of Stein information gain and Stein +information-directed sampling in MBRL. We theoretically +established prior-free Bayesian regret bounds that are sub- +linear both in the number of episodes and action space di- +mensions. Experimentally, we demonstrate favorable per- +formance in practice. + +Start +Goal +Important +States +ERING (proposed)Initial Episodes +Final Episodes +PSRL +Var-IDS +STE1-1/N +1-1/N +SN_-1 +SN +1/N +1/N +1 +11/N +1 +1一 +1/N +=1/N +S1 +S2 +1/N +1/N +1 +1 +1-1/NStein Information Directed Exploration for Model-Based Reinforcement Learning +9 +(a) Strong belief prior. +(b) Weak belief prior. +Figure 6. 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Detailed Context of Related Works +Model based and Model Free RL. Most RL algorithms fall into two categories: +model-free (Schulman et al., +2017; +Mnih et al., +2013; +Haarnoja et al., +2018; +Lillicrap et al., +2015) +and +model-based +(Fan & Ming, +2021; +Deisenroth & Rasmussen, 2011; Chua et al., 2018; Janner et al., 2019). In model-free approaches, the agent learns di- +rect policy mapping from states to action with approximate dynamic programming methods. In contrast, in model-based +approaches, an agent learns the approximate model of the environment itself and trains a policy under the learned dynamics. +Recently, probabilistic model-based RL algorithms have shown superior performance in practice relative to their model- +free counterparts, despite the strong conceptual guarantees for model-free approaches (Janner et al., 2019). We focus on +model-based RL in this work. +Performance Measure. To understand why this may be so, it’s important to assess the convergence criteria of RL meth- +ods: Probably approximate correct (PAC) bounds (Valiant, 1984), Frequentist regret (Jin et al., 2018; 2020), Bayesian +regret (Osband et al., 2013), Mistake (MB) Bound (Littlestone, 1987), KWIK (Knows What It Knows) (Li et al., 2008) +& convergence in various distributional metrics (Amortila et al., 2020; Borkar & Meyn, 2002; Chakraborty et al., 2022a; +Chowdhury & Gopalan, 2019) all exist. A crucial challenge lies in deciding the optimal selection criteria to evaluate the +algorithm’s computational and statistical efficiency. Rather than comment on the specific merit of a particular convergence +criterion as a motivation for our restriction of focus to Bayesian regret, we note a few of its salient attributes: it imposes min- +imal requirements on access to a generative model underlying state transitions (Osband & Van Roy, 2017a; Osband et al., + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +14 +2013; Osband & Van Roy, 2014), and respects the inherent uncertainty associated with the optimal policy, rather than +supposing that it can be effectively captured by confidence sets based on a few moment-based estimates of the transition +dynamics, which can lead to undesirable behavior in the presence of sparse rewards (Jin et al., 2020; Yang & Wang, 2020; +Zanette et al., 2020). +Information-theoretic Approaches. To encapsulate the inherent uncertainty present in the optimal policy, one may aug- +ment notions of regret to quantify distance to the optimal occupancy measure or other information-theoretic quantities +(Russo & Van Roy, 2018; Hao & Lattimore, 2022). That this is advantageous may be seen by honing in on the sparse +reward setting: consider the traditional definition of regret E[V ∗(s) − V k(s)] for any kth episode in an environment with +near-zero rewards. Suppose one policy incorporates exploration based on a distributional estimate of the environment, +whereas the other only consider moments of the distribution of returns, such as UCB (Lai et al., 1985). In this case, the +traditional notion of regret may not encourage exploration in a way that yields increased state-space coverage, as the value +distribution for this case would be near-null. This issue is well-documented in the bandit setting (Russo et al., 2017). Hence, +there is an intrinsic motivation to consider augmentations of regret that are well-calibrated to the inherent uncertainty of the +optimal policy. Motivated by (Russo & Van Roy, 2018; Hao & Lattimore, 2022), we consider convergence criteria from +information theory and Bayesian inference to define an appropriate notion of regret. To understand the exact manner in +which these modifications are incorporated into model-based RL (MBRL), we contrast them with the model-free setting. In +such settings, incorporating exploration bonuses is well-established (Jin et al., 2020; Eysenbach & Levine, 2021; Liu et al., +2019; Ahmed et al., 2018), either in the form of augmenting the reward, the value function (Jin et al., 2018; Osband et al., +2019), or the policy gradient (Gelfand & Mitter, 1991; Raginsky et al., 2017). However, such methods sample uniformly +with respect to a value or policy rather than in pursuit of reducing the estimation error to the optimal transition dynamics, +which can yield spurious behavior (Weerakoon et al., 2022; Shyam et al., 2018). By contrast, information-theoretic regular- +isation in model-based RL is an active area of research. Empirical advancements based on intrinsic curiosity (Burda et al., +2018a; Pathak et al., 2017b), i.e., modifying the sampling probabilities driven by uncertainty estimates in the transition +dynamics or forward model prediction error, can improve performance in practice but lack conceptual guarantees. +Information Directed Sampling. +To substantiate these approaches conceptually, the resultant algorithms have re- +cently been rewritten in a way that their performance can be quantified by information-theoretic or Bayesian regret +(Lu & Van Roy, 2019b; Lu et al., 2021), under a specific choice of Dirichlet priors for the transition model, inspired +by earlier work on bandits (Russo & Van Roy, 2014b). +Extensions that alleviate any requirements on the prior for +MBRL also exist based upon the development of a surrogate environment estimation procedure via rate-distortion the- +ory (Hao & Lattimore, 2022). Unfortunately, the resultant algorithm requires estimating the information gain, which +is generally intractable. This issue can be partially addressed by instead optimizing the evidence lower-bound (ELBO) +(Achiam & Sastry, 2017), but exhibits exponential dependence on the mutual information with respect to the optimal occu- +pancy measure (Ozair et al., 2019). Related approaches replace mutual information by Bregman divergence; however, this +necessitates inverting a Fisher information matrix per step which can be computationally costly (Lattimore & Szepesvari, +2019; Zimmert & Lattimore, 2019). Hence, previous efforts to incorporate information-theoretic bonuses in MBRL either +impose restrictive assumptions on the prior or yield computationally heavy objectives whose algorithmic solutions exhibit +scalability problems. +B. Preliminaries: Kernelized Stein Discrepancy +Consider the notion of Integral probability metric to measure the deviation between the estimated distribution q and the +unknown target distribution p defined as +dF(q, p) = sup +f∈F +|Eq[f(X)] − Ep[f(X)]|, +(18) +where the supremum is over a class of real-valued test functions f ∈ F. By adjusting the function class F, we can recover +the well-known metrics such as Total variation distance, Wasserstein distance (Sriperumbudur et al., 2010), etc. However, +the major challenge in evaluating the IPM in (18) is that it requires an integration under the true distribution p which is +intractable. A seminal idea to alleviate this issue is called Stein’s method, which restricts the class of distributions F to +functions such that Ep[f(X)] = 0. Building upon this idea, (Liu et al., 2016) develops a tractable way to evaluate the IPM +by restricting distributions to the Stein class, associated with a reproducing kernel Hilbert space (RKHS) over Stein kernels +(Berlinet & Thomas-Agnan, 2011). In this case, the IPM can be evaluated in terms of the Stein kernel as the kernelized +stein discrepancy (Gorham & Mackey, 2015). Stein’s method provides a generalised framework for studying distributional + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +15 +distances and relies on the fact that two smooth densities p(x) and q(x) are identical iff they satisfy the Stein’s identity +given by +max +f∈F +� +Ep[sq(x)f(x) + ∇xf(x)] +�2 = 0, +(19) +where sq(x) denotes the score function of q(x) given by sq(x) = ∇x log q(x). As an example, Stein’s identity in (19) +holds for smooth functions f lying in the Stein class of p. A function f is in the Stein class of p if it’s smooth and satisfies +� +x ∇x(f(x)p(x))dx = 0. Hence, for any function f in the Stein class of p, we can say Ep[Apf(x)] = 0 where Ap is the +Stein operator of p which is a linear operator. +From here the Stein discrepancy between p and q is defined as (Liu et al., 2016) +KSD2(p, q) = max +f∈F +� +Ep[sq(x)f(x) + ∇xf(x)] +�2, +(20) +where F is a class of smooth functions satisfying Stein’s identity (19). However, the above definition is computationally +intractable as it requires solving a complex variational optimization. To this end, (Liu et al., 2016) define a computationally +tractable modification as +KSD(p, q) = Ex,x′∼p +� +uq(x, x′) +� +, +(21) +where uq(x, x′) is the Stein kernel defined as +uq(x, x′) := sq(x)⊤κ(x, x′)sq(x′) + sq(x)⊤∇x′κ(x, x′) ++ ∇xκ(x, x′)⊤sq(x′) + T r(∇x,x′κ(x, x′)), +where κ(x, x′) is the base kernel. +For the setting in this work, we have p = P ∗ (transition dynamics corresponding to true model M ∗) and q = P Mk +(transition dynamics corresponding to posterior Mk ∼ φ(·|Hk)). Interestingly, KSD empowers us to evaluate the distance +KSD(P Mk, P ∗). +C. Proof of Proposition 3.1 +Proof. Let us first define the compact notation such that Gx := G(x, ·) and ξP M +y|x(y, ·) := sP M (y)l(y, ·) − △∗l(y, ·). Here, +we derive our proposed DSD as defined in (13). Further, for simplicity of analysis, we denote P ∗(s′|s, a) → P ∗ +(y|x)(y|x), +P ∗(s, a) → P ∗ +x(x) and P ∗(s, a, s′) → P ∗ +(x,y)(x, y) due to the notation state-action pair (s, a) → x ∈ X := S × A and the +corresponding next state s′ → y ∈ S. To start the proof, let us consider DSD(P M, P ∗) and write +DSD(P M, P ∗) = +��E(x,y)∼P ∗ +x,yGxξP M +y|x(y, ·) +��2 +(22) += +� +E(x,y)∼P ∗ +(x,y)GxξP M +y|x(y, ·), E(x′,y′)∼P ∗ +(x,y)Gx′ξP M +y′|x′ (y′, ·) +� +(23) += E(x,y)∼P ∗ +(x,y)E(x′,y′)∼P ∗ +(x,y) +� +GxξP M +y|x(y, ·), Gx′ξP M +y′|x′ (y′, ·) +� +(24) += E(x,y)∼P ∗ +(x,y)E(x′,y′)∼P ∗ +(x,y) +� +Gx′GxξP M +y|x(y, ·), ξP M +y′ |x′ (y′, ·) +� +(25) += E(x,y)∼P ∗ +(x,y)E(x′,y′)∼P ∗ +(x,y)[k(x, x′)κP ((x, y), (x′, y′))] +(26) += E(x,y)∼P ∗ +(x,y)E(x′,y′)∼P ∗ +(x,y)[κM((x, y), (x′, y′))]. +(27) +Here we have applied the reproducing property of kernels with the linearity of expectations to derive the equations. where, +Gx′Gx = G(x, x′) = k(x, x′)I. This proves the derivation for an equivalent Stein operator for our scenario. Now, we +need to show a version of Stein’s identity to complete the derivation of our proposed Kernelized Conditional Discrete Stein +Discrepancy. We note that +Ex,y∼P ∗ +(x,y)κM((x, y), ·) = Ex,y∼P ∗ +(x,y)GxξP M +y|x(y, ·) +(28) += Ex∼P ∗ +x GxEy∼P ∗ +(y|x)ξP M +y|x(y, ·). +(29) + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +16 +Now, we show that if P M(y|x) = P ∗(y|x), Ey∼P ∗ +y|xξP M +y|x(y, ·) = 0 which proves an equivalent notion of Stein’s identity +for our Discrete conditional case. Replacing P M = P ∗. +Ey∼P ∗ +(y|x)ξP M +y|x(y, ·) = Ey∼P ∗ +(y|x)[sP ∗ +(y|x)(y)l(y, ·) − △∗l(y, ·)] +(30) += +� +y +[sP ∗ +(y|x)(y)l(y, ·)P ∗(y|x) − △∗l(y, ·)P ∗(y|x)] +(31) += +� +y +[△P ∗(y|x)l(y, ·) − △∗l(y, ·)P ∗(y|x)]. +(32) +Here the first equality holds by denoting ξP M +y|x = +� +sP M (y)l(y, ·) − △∗l(y, ·) +� +from equation (10). Then expanding upon +the expectation and replacing the expression of the score function from equation (11), we get the final expression. For each +i we can write the first and second part of (32) from the definition of difference operators in equation (11) as +� +y +[△yiP ∗(y|x)l(y, ·)] = +� +y +[l(y, ·)P ∗(y|x) − l(y, ·)P ∗(∨iy|x)], +(33) +� +y +[△∗l(y, ·)P ∗(y|x)] = +� +y +[l(y, ·)P ∗(y|x) − l(∧iy, ·)P ∗(y|x)]. +(34) +The two equations are equal since ∨ and ∧ are inverse cyclic permutations on S with ∧i(∨iy) = ∨i(∧iy) = y and hence +substituting equation (33) into equation (32) we get Ey∼P ∗(s′|s,a)ξy|x(y, ·) = 0. So, this proves an equivalent notion of +Stein’s identity which completes the proof. +D. Proof of Theorem 4.1 +Proof. Let us start with the definition of Bayesian regret defined in (3) as follows +BRK = +K +� +k=1 +E +� +Ek +� +V M +1,π∗(sk +1) − V M +1,πk(sk +1) +�� +, +(35) +where the inner expectation is over the posterior distribution, and the outer expectation is over the stochastic policy and +environment M ∗. For brevity of notation, let us define Rk := Ek +� +V M +1,π∗(sk +1) − V M +1,πk(sk +1) +� +. Next, we introduce the Stein +information ratio via multiplying and dividing by Kπ +k (M; Hk,H) as follows +BRK = +K +� +k=1 +E + + +� +(Rk)2 +Kπ +k (M; Hk,H)Kπ +k (M; Hk,H) + + . +After applying Cauchy–Schwartz inequality, using the linearity of expectations, and considering the definition of ΓDSD +k +(π) +in (9), we can get +BRK ≤ +� +� +� +�E +� K +� +k=1 +ΓDSD +k +(πk) +�� +� +� +�E +� K +� +k=1 +Kπ +k (M; Hk,H) +� += +� +� +� +�E +� K +� +k=1 +Kπ +k (M; Hk,H) +� K +� +k=1 +E[ΓDSD +k +(πk)] . +(36) +From definition of ΓDSD +k +(πk), we have ΓDSD +k +(πk) ≤ Γ∗ for any k ∈ [K]. Hence, from (36), we can write (14). +E. Proof of Lemma 4.2 +Proof. We run a local optimization procedure by dividing the total number of samples H in an episode into batches of size +Z with Z′ := H +Z batches and select Stein optimal points per batch using an SPMCMC style update. We begin the analysis +by representing the dictionary till the kth episode as Dk and we expand upon the definition of DSD (13) as + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +17 +|Dk|2DSD(P M; Dk)2 = +� +(xi,yi)∈Dk +� +(xj,yj)∈Dk +κM((xi, yi), (xj, yj)) +(37) +=|Dk−1|2DSD(P k−1; Dk−1)2 ++ +Z′ +� +z=1 +� +κM((xz +k, yz +k), (xz +k, yz +k)) + 2 +� +(xi,yi)∈Dk−1 +κM((xi, yi), (xz +k, yz +k)) +� +. +(38) +In the above expression, equality in (37) comes from the empirical definition of DSD, where κM is the Stein kernel (cf. (10)) +which depend upon the score function of P M. Next, for each z, we select the sample (xz +k, yz +k) from Yz := {(xl +k, yl +k)}Z +l=1 +using an SPMCMC style local optimization procedure as in (Chen et al., 2019, Appendix A.1)). Now, from the SPMCMC +based selection, we can write +κM((xz +k, yz +k), (xz +k, yz +k)) + 2 +� +(xi,yi)∈Dk−1 +κM((xi, yi), (xz +k, yz +k)) += +inf +(xz +k,yz +k)∈Ym κM((xz +k, yz +k), (xz +k, yz +k)) + 2 +� +(xi,yi)∈Dk−1 +κM((xi, yi), (xz +k, yz +k)) +≤ B2 + 2 +inf +(xz +k,yz +k)∈Yz +� +(xi,yi)∈Dk−1 +κM((xi, yi), (xz +k, yz +k)). +(39) +The inequality in (39) holds because we restrict our attention to regions for which it holds that κM((x, y), (x, y)) ≤ B2 +for all (x, y) ∈ Yz +k for all k and z. Utilizing the upper bound of (39) into the right hand side of (38), we get +|Dk|2DSD(P M; Dk)2 ≤|Dk−1|2DSD(P M; Dk−1)2 + Z′B2 ++ 2 +Z′ +� +z=1 +inf +hz +k∈Yz +� +(xi,yi)∈Dk−1 +κM((xi, yi), (xz +k, yz +k)). +(40) +From the application of Theorem 5 (Hawkins et al.) for our formulation with H new samples in the dictionary. +2 +inf +(xm +k ,ym +k )∈Ym +� +(xi,yi)∈Dk−1 +κM((xi, yi), (xm +k , ym +k )) ≤ rk∥fk∥2 +K0 + DSD(P M; Dk−1)2 +rk +, +(41) +for any arbitrary constant rk > 0 and any fk which lies in RKHS corresponding to state space S. Since the cardinality of S +is bounded and is given by S, we can assume an upper bound on ∥fk∥2 +K0 ≤ C0S, where C0 is a positive constant. Hence, +we can use the upper bound in (41) to the right hand side of (40), to obtain +|Dk|2DSD(P M; Dk)2 ≤ |Dk−1|2 +� +1 + Z′ +rk +� +DSD(P M; Dk−1)2 + Z′(B2 + rkC0S). +(42) +Next, we divide the both sides by |Dk|2 = (|Dk−1| + Z′)2 to obtain +DSD(P M; Dk)2 ≤ +|Dk−1|2 +(|Dk−1| + Z′)2 +� +1 + Z′ +rk +� +DSD(P M; Dk−1)2 + Z′ � +B2 + rkC0S +� +(|Dk−1| + Z′)2 +(43) +It is interesting that a novel aspect in analysis lies in establishing a recursive relationship for the DSD amongst iterations +which eventually paves the way to establish the DSD convergence results. After unrolling the recursion in (43), we can +write +DSD(P M; Dk)2 ≤ +k +� +i=1 +� +Z′� +B2 + riC0S +� +(|Di−1| + Z′)2 ++ ǫi +�  + +k−1 +� +j=i +|Dj| +|Dj| + Z′ + + +2  + +k−1 +� +j=i +� +1 + Z′ +rj+1 +� + . +(44) + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +18 +Applying the log-sum exponential bound �k−1 +j=i +� +1 + +Z′ +rj+1 +� +≤ exp +� +Z′ �n +j=1 +1 +rj +� +, we can write (44) as +DSD(P M; Dk)2 ≤ exp + +Z′ +k +� +j=1 +1 +rj + + +k +� +i=1 +� +Z′(B2 + riC0S) +(|Di−1| + Z′)2 + +�  + +k−1 +� +j=i +|Dj| +|Dj| + Z′ + + +2 +. +(45) +Next, we consider the inequality in (45). By replacing, rj = +k +Z′ , we get rid of the constant exponential term and obtain +DSD(P M; Dk)2 ≤ +k +� +i=1 +� +Z′(B2 + riC0S) +(|Di−1| + Z′)2 +�  + +k−1 +� +j=i +|Dj| +|Dj| + Z′ + + +2 += +k +� +i=1 +� +Z′(B2 + riC0S) +(|Dk−1| + Z′)2 +�  + +k−1 +� +j=i +|Dj| +|Dj−1| + Z′ + + +2 +, +(46) +where Equation (46) corresponds to the sampling error and represents the bias incurred at each step of the SPMCMC point +selection scheme. The equality in the second line holds by rearranging the denominators in the multiplication and pulling +(|Dk−1| + Z′)2 inside the first term. Next, from the fact that |Dj| = |Dj−1| + Z′ which implies that the product will be 1, +we can upper bound the right hand side of (46) as follows +DSD(P M; Dk)2 ≤ +k +� +i=1 +� +Z′(B2 + riC0S) +(|Dk−1| + Z′)2 +� +. +(47) +From the dictionary update, we note that |Dk−1| + Z′ = |Dk| = O(k), which implies that 1/ (|Dk−1| + Z′)2 = 1/k2, +which we utilize in the right hand side of (47) to write +DSD(P M; Dk)2 ≤ +k +� +i=1 +�Z′(B2 + riC0S) +k2 +� +. +(48) +We note that the above bound holds for any given M. And hence we can conclude that after taking an expectation over +posterior M ∼ φ(·|Dk), it holds that +Ek +� +DSD(P M; Dk)2� +≤ +k +� +i=1 +�Z′(B2 + riC0S) +k2 +� += O +�S +k +� +, +(49) +where we absorb the constants into O(·) notation. Hence proved. +F. Proof of Lemma 4.3 +Proof. Proof of statement (1): We start by analyzing the regret decomposition of the value function as follows +Ek +� +V M +1,π∗(sk +1) − V M +1,πk(sk +1) +� += Ek +� +V M +1,π∗(sk +1) − V M∗ +1,πk(sk +1) +� +� +�� +� +I1 ++ Ek +� +V M∗ +1,πk(sk +1) − V M +1,πk(sk +1) +� +� +�� +� +I2 +. +(50) +Note that here we introduce the notion of M ∗, the ground truth MDP for the first time in the Bayesian regret analysis and +is a clear departure from the traditional Bayes regret analysis +Upper Bound on I1 +Now, first we derive the upper bound on I1 +∆M +h (s, a) := Es′∼P M(·|s,a)[V M +h+1,π∗(s′)] − Es′∼P M∗(·|s,a)[V M +h+1,π∗(s′)] . +(51) +It is important to observe that we have replaced the second term of Equation (51) Es′∼P M∗(·|s,a)[V M +h+1,πk(s′)] = +Es′∼P M∗(·|s,a)[V M +h+1,π∗(s′)] since conditioned on the history Dk, the law of π∗, πk are the same as in (Hao & Lattimore, + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +19 +2022) +Now, using the definition in (51), we can write I1 as +I1 = Ek +� H +� +h=1 +EM∗ +π∗ +� +∆M +h (sk +h, ak +h) +� +� +(52) += +H +� +h=1 +Ek + +� +(s,a) +dM∗ +h,π∗(s, a)∆M +h (s, a) + + , +(53) += +H +� +h=1 +Ek + +� +(s,a) +dM∗ +h,π∗(s, a) +(Ek[dM∗ +h,π∗(s, a)])1/2 (Ek[dM∗ +h,π∗(s, a)])1/2∆M +h (s, a) + + . +(54) +The equality in (53) holds by introducing the state action occupancy measure, and in (54), we divide and multiply with +dM∗ +h,π∗(s, a) > 0. From the Cauchy–Schwartz inequality and using the fact that dM∗ +h,π∗(s, a) ≤ 1, we can write +I1 ≤ + + +H +� +h=1 +Ek + +� +(s,a) +(dM∗ +h,π∗(s, a))2 +Ek[dM∗ +h,π∗(s, a)] + + + + +1/2  + +H +� +h=1 +Ek + +� +(s,a) +Ek[dM∗ +h,π∗(s, a)](∆M +h (s, a))2 + + + + +1/2 +≤ + + +H +� +h=1 +Ek + +� +(s,a) +dM∗ +h,π∗(s, a) +Ek[dM∗ +h,π∗(s, a)] + + + + +1/2  + +H +� +h=1 +Ek + +� +(s,a) +Ek[dM∗ +h,π∗(s, a)](∆M +h (s, a))2 + + + + +1/2 +. +(55) +First, from the first part in the right-hand side of (55), we note that +H +� +h=1 +Ek + +� +(s,a) +dM∗ +h,π∗(s, a) +Ek[dM∗ +h,π∗(s, a)] + + = +H +� +h=1 + +� +(s,a) +Ek[dM∗ +h,π∗(s, a)] +Ek[dM∗ +h,π∗(s, a)] + + ≤ HSA . +(56) +Then, from the second part on the right-hand side of (55), we note that given Dk, we have dM∗ +h,π∗(s, a) and ∆M +h (s, a) +independent. This implies +Ek + +� +(s,a) +dM∗ +h,π∗(s, a) + + Ek +� +(∆M +h (s, a))2� += Ek + +� +(s,a) +dM∗ +h,π∗(s, a)(∆M +h (s, a))2 + + += Ek +� +EM∗ +π∗ +� +(∆M +h (sk +h, ak +h))2�� +, +(57) +From the definition in (51), we can write +Ek +� +EM∗ +π∗ +� +(∆M +h (sk +h, ak +h))2�� += H2Ek +� +EM∗ +π∗ +�� +Es′∼P M(·|sk +h,ak +h) +� +V M +h+1,πk(s′)/H +� +− Es′∼P M∗(·|sk +h,ak +h) +� +V M +h+1,π∗(s′)/H +��2�� += H2Ek +� +EM∗ +π∗ ∥P M∗(·|sk +h, ak +h) − P M(·|sk +h, ak +h)∥2� +. +(58) +Next, combining (56), (57), and (58), can upper bound I2 in (55) as +I1 ≤ +� +� +� +�H3SA +H +� +h=1 +Ek +� +EM∗ +π∗ ∥P M∗(·|sk +h, ak +h) − P M(·|sk +h, ak +h)∥2� +). +(59) +Next, we take square on both sides and introduce the Kernelized Stein discrepancy between probability measures P M and +P M∗ by upper-bounding the total variation norm in equation (59) 1 (Gorham & Mackey, 2015) which is a clear departure +1In ((Liu et al., 2016)), the Kernelized Stein discrepancy has been defined as KSD2 and in (Gorham & Mackey, 2015) as KSD, +hence it can be used interchangeably. For ease of the analysis, we proceed by considering KSD2 as the Stein discrepancy. + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +20 +from the traditional Bayesian regret analysis as in (Osband et al., 2013; Osband & Van Roy, 2017a; 2014; Osband et al., +2019). to obtain +I2 +1 ≤ H3SA +H +� +h=1 +Ek +� +EM∗ +π∗ DSD2(P M(·|sk +h, ak +h), P M∗(·|sk +h, ak +h)) +� +≤ H3SA +H +� +h=1 +Ek +� +EM∗ +π∗ DSD2(P M(·|sk +h, ak +h)) +� +. +(60) +The second line in the equation is due to the definition of KSD which requires only the unnormalized density/pmf of one +and samples from the other. Next, from the definition in (8), we can finally write +I2 +1 ≤ H3SA · Kπ +k (M; Hk,H) . +(61) +From (50), we can write +� +Ek +� +V M +1,π∗(sk +1) − V M∗ +1,πk(sk +1) +��2 +≤ H3SA · Kπ +k (M ∗; Hk,H) . +(62) +The upper bound for I2 is similar to that of I1 and applying the similar steps as I1, we get +� +Ek +� +V M +1,π∗(sk +1) − V M∗ +1,πk(sk +1) +��2 +≤ H3SA · Kπ +k (M ∗; Hk,H) . +(63) +Hence, adding equation (62) & equation (63) and using the definition of Stein information ratio in (9) and inequality in +(62), it is clear that we can upper bound E[ΓDSD +k +] as E[ΓDSD +k +] ≤ 2SAH3. Now from the definition of Γ∗ ≤ ΓDSD +k +(π) we +have Γ∗ ≤ 2SAH3 that which completes the proof. +Proof of statement (2): In this section, we derive an upper-bound on the total Stein information for the K episodes given +by �K +k=1E[Kπ +k (M; Hk,H)]. We start by deriving an upper bound for E[Kπ +k (M; Hk,H)] with an SPMCMC style local +optimization method proposed originally in (Chen et al., 2019) and later used in sequential decision-making scenarios by +(Chakraborty et al., 2022b; Hawkins et al., 2022). +We start with the definition in (8) to write +Ek[Kπ +k (M; Hk,H)] = +H +� +h=1 +E +� +DSD2(P M(·|sk +h, ak +h) +� +. +(64) +From the upper bound in Lemma 4.3, we can write +E[Kπ +k (M; Hk,H)] ≤ +H +� +h=1 +O +�S +k +� += O +�HS +k +� +. +(65) +Finally, we take summation over k and obtain upper bound as +�K +k=1E[Kπ +k (M; Hk,H)] ≤ HS +K +� +k=1 +1 +k ≤ HS +� K +1 +1 +xdx = HS(log K). +(66) +Hence Proved. +G. Proof of Theorem 4.4 +Proof. Consider the expression in (14) and we note that the Bayesian regret depends on the Stein information ratio E[Γ∗] +and the total Stein information gain �K +k=1 E[Kπ +k (M ∗; Hk,H)]. From Lemma 4.3, we can upper bound the right hand side +of (14) as +BRK ≤ +� +SAH3 · K · HS(log K) +(67) +=H2� +S2AK log K = ˜O(H2√ +S2AK), +(68) +where ˜O absorbs the log factors and we obtain the final result. + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +21 +H. Proof of Theorem 4.5 +Proof. Following the inequality 2ab ≤ a2 + b2, for any policy π, we can write +Rk +� +λKπ +k (M; Hk,H) +� +λKπ +k (M; Hk,H) ≤ +R2 +k +2λKπ +k (M; Hk,H) + λ +2 Kπ +k (M; Hk,H) , +(69) +where Rk := E +� +V M +1,π∗(sk +1) − V M +1,πk(sk +1) +� +, where M ∼ φ(·|Dk). Now, recollecting the definition of Bayesian regret from +(3) and after adding and subtracting the regularization term, we can write +BRK = E +� K +� +k=1 +Rk − λ +K +� +k=1 +Kπ +k (M; Hk,H)) + λ +K +� +k=1 +Kπ +k (M; Hk,H)) +� +. +(70) +Utilizing the upper bound in (69), we can write +BRK ≤ 1 +2λ +K +� +k=1 +E +�� +Ek +� +V M +1,π∗(sk +1) − V M +1,π(sk +1) +��2 +Kπ +k (M; Hk,H)) +� ++ λ +K +� +k=1 +E[Kπ +k (M; Hk,H) +=KE[Γ∗] +2λ ++ λ +K +� +k=1 +E[Kπ +k (M; Hk,H)]. +(71) +Now, select λ in (71) as λ = +� +KE[Γ∗]/�K +k=1E[Kπ +k (M; Hk,H) to obtain the final expression. +I. Detailed Information of Experimental Setup +I.1. Description of the Environments +DeepSea Environment: The DeepSea exploration environment (a slightly modified version of Osband & Van Roy (2017a) +as used in Markou & Rasmussen (2019)) is an extremely challenging environment (see Fig. 8) to test the agent’s capability +of directed and sustained exploration. As shown in Fig. 8, there are total N states, the agent starts from the left-most state +and can swim left or right from each of the N states in the environment. The agent gets a reward of r = 0 (red transitions) +for the left action. On the other hand, the right action from s = 1, · · · , (N−1) succeeds with probability (1−1/N), moving +the agent to the right and otherwise fails and moving the agent to the left (blue arrows), giving r ∼ N(−δ, δ2) regardless +of whether it succeeds. A successful swim-right from s = N moves the agent back to s = 1 and gives r = 1. Hence, as +we increase the number of states N, it will increase the amount of sparsity in the environment, making it extremely hard +for the agent to explore (we provided this evidence in Fig. 2). +Figure 8. DeepSea Exploration Environment (Osband & Van Roy, 2017a; Markou & Rasmussen, 2019). This environment tests the +agent’s capability of sustained and directed exploration. This figure is same as Fig. 4 and we repeat it here for quick reference. +WideNarrow MDP Environment: +This is another challenging environment (see Fig. +9) presented in +Markou & Rasmussen (2019), which has 2N + 1 states with deterministic transitions. In WideNarrow MDP environ- +ment, odd-numbered states except s = (2N + 1) have W actions, out of which one gives r ∼ N(µl, σ2 +l ) whereas all other +actions result in r ∼ N(µh, σ2 +h), with µl < µh. Even-numbered states have a single action which results in r ∼ N(µh, σ2 +h). +In the experiments, we use muh = 0.5, µl = 0, σl = σh = 1. The WideNarrow MDP helps understand and compare the +performance of STEERING under factored posterior approximations. + +1-1/N +1-1/N +SN_-1 +SN +1/N +1/N +1 +11/N +1 +1一 +1/N +=1/N +S1 +S2 +1/N +1/N +1 +1 +1-1/NStein Information Directed Exploration for Model-Based Reinforcement Learning +22 +Figure 9. WideNarrow MDP environment (Markou & Rasmussen, 2019). The purpose of this environment is to test the agent’s perfor- +mance under factored posterior approximations. +PriorMDP Environment: The previous environments, DeepSea Exploration and WideNarrow MDP, have a special +structure within them that tests different aspects of the agent/algorithm. +In contrast, the PriorMDP environment of +Markou & Rasmussen (2019) provides a more general environmental setup to test the algorithms where the dynamics +are sampled from a Dirichlet prior with concentration k = 1 and reward from Normal prior with mean and precision drew +from a Normal-Gamma prior with parameters ⟨0, 1, 1, 4⟩. +I.2. Baselines and Evaluations +We compare our proposed algorithm STEERING with various Bayesian/ non-Bayesian and distributional RL baselines. +The baselines include +• Q-learning: Vanilla Q-learning with ǫ−greedy action selection (Watkins & Dayan, 1992), +• BQL: Bayesian Q-learning (Dearden et al., 1998), +• UBE: Uncertainty Bellman equation (O’Donoghue et al., 2017), +• MM: Moment Matching across Bellman equation (Markou & Rasmussen, 2019), +• PSRL: Posterior Sampling Reinforcement Learning (Osband et al., 2013). +1. Bayesian Q-Learning : BQL introduced in Dearden et al. (1998) models the distribution over state-action returns Z∗, +which is assumed to follow Gaussian (ergodic MDP) and updates the posterior belief of P(θZ∗|D) using Bayesian update +rule. BQL considers Normal-Gamma prior on the parameters (mean and precision) of the Gaussian. However, there is a +factored posterior assumption that restricts its generalisability and hinders the performance, as shown in Fig. 13 +2. Uncertainty Bellman Equation: UBE proposed in O’Donoghue et al. (2017) is a model-based reinforcement learning +approach designed primarily for modeling the epistemic uncertainty in µz = Ez[Z|θz] but with a strong assumption of +MDP being a directed acyclic graph with bounded mean rewards. In other words, each state-action can be visited at most +once per episode, which is restrictive and requires sparse design (repeating state-action multiple times) to make it work +even for toy problems. Under these assumptions and a suitable Bellman operator, it holds that UBE has a unique solution +which upper-bounds the epistemic uncertainty VarθT ,θR[µz] where θT , θR are the parameters of the transition and rewards +model, respectively. However, the strong assumptions restrict the generalisability of the UBE-based methods to various +practical scenarios where the inherent structure of the MDP is not acyclic. +3. Moment Matching across Bellman Equation: An interesting approach proposed recently by Markou & Rasmussen +(2019) also uses the Bellman equation to estimate the epistemic uncertainty by comparing the moments (first and second +moments) across the Bellman equation. While the first-order moments give the standard V (s) & Q(s, a), the second-order +moments can be decomposed into aleatoric and epistemic terms without the need to compute upper bounds as in UBE-based +methods. Similar to prior methods, the policy is optimized w.r.t P(θT |D), P(θR|D) and for the epistemic uncertainty µz. +However, as with existing methods, MM also approximates with a factored posterior leading to performance loss. +4. Posterior Sampling Reinforcement Learning: Posterior Sampling reinforcement learning (PSRL) introduced by +Osband et al. (2013); Osband & Van Roy (2014) is primarily built upon Thompson sampling or probability matching prin- +ciple. PSRL provides an efficient and tractable solution to model-based RL with provable guarantees. The algorithm works + +S1 +S3 +S2N-1 +S2N +S2N+1Stein Information Directed Exploration for Model-Based Reinforcement Learning +23 +by sampling a transition and rewards model from the posterior distribution at any kth episode θk +T ∼ P(θT |Dk), θk +R ∼ +P(θR|Dk) and the optimal policy πk is obtained by solving the Bellman equation under the sampled transition, rewards +model. The agent then follows the policy πk to interact with the environment and gather data and update the dictionary Dk. +The primary advantage of PSRL lies in its computational tractability, as the policy needs to be optimized under a single +sampled transition and rewards model. For PSRL, state-of-the-art Bayesian regret bounds exist under minor assumptions +(Osband & Van Roy, 2014; 2017b). However, even the performance of PSRL degrades for complex environments such as +under sparse reward scenarios, as empirically proved in Fig. 2(c). +We evaluate all the algorithms by comparing their performance in terms of cumulative regret. Further, for the Bayesian +methods, we also evaluate the algorithms in terms of the posterior representation and concentration on the true Q∗ values. +I.3. Implementation Details of STEERING +Here we present the implementation details of STEERING, as outlined in Algorithm 1. Our approach begins by sampling +transition and reward models from the posterior distribution using Categorical-Dirichlet and Normal-Gamma distributions +for the transition and rewards model, respectively, as in Markou & Rasmussen (2019). To ensure a fair comparison, we +considered the same setup for all categorical and continuous distributions for the other baselines. In the second phase, +our algorithm represents a distinct deviation from prior PSRL and IDS-based methods by quantifying the distributional +distance between the true MDP and the current estimated MDP through the use of Stein discrepancy. Specifically, we +utilize the regularized Stein information sampling as outlined in Theorem 4.5 for empirical analysis. Next, we compute the +Stein-discrepency for each (s, a) pair using the formulae for DSD as defined in (13). +I.4. Hyperparameters +For all Dirichlet priors in the algorithms we use hyperparameters η(s,a) = 1 and for Normal-Gamma priors we use +(µ, Λ, α, β)(s,a) = (0, 4, 3, 3) as in Markou & Rasmussen (2019). For both STEERING and Var-IDS we use the same +regularization constant (IA) λ = 0.5. For all the environments, we run the algorithms for T = 5000 timesteps with a buffer +length (max) of size N, where N denotes the number of states and run the policy iteration for 2N iterations. We have also +run PSRL, Var-IDS and STEERING for T = 10000 in Figure 1 to observe the effect of prior with convergence. For the +baseline implementation of the algorithms including Q-Learning, UBE, BQL, MM, PSRL we leverage 2. We utilize 3 and +4 for the DSD computation. We thank the authors for the open-source repositories. +J. Additional Experimental Results & Discussions (Intuitive Insights) +J.1. Evolution of Posterior Representations for DeepSea Environment +To gain more insights about the improvements and directed exploration behavior achieved by STEERING, we perform a +detailed ablation study to analyze the evolution of posterior representation learned over iterations for all the algorithms. As +a metric to plot, we plot the mean and variance of Q values calculated from the learned posterior and compare them against +the true value denoted by Q∗ (see Fig. 10 for STEERING). We remark that as the posterior concentrates with the progress +in training, estimated Q values would also concentrates to the true Q∗ (shown by dotted red line in Fig. 10). Since N +denotes the number of states for DeepSea environment, and we obtain an estimated Q value for each state action pair, we +choose to plot the evolution of the last 4 states -action pairs (left action in top row and right action in bottom row in Fig. +10). +Fig. 10 is of critical importance, as it gives a clear understanding of whether the agent is over-exploring or under-exploring +actions based on its sub-optimality. To make the advantage clear as compared to existing approaches, we plot similar +figures for all the existing algorithms in Fig. 11 to Fig. 19. Interestingly, in all the comparison plots from Fig. 11 +to Fig. 19, STEERING exhibits a superior posterior representation which results in directed exploration, even with +increased sparsity. The most interesting aspect of STEERING over baselines is that it does not over-explore actions once +it is confident that those are sub-optimal which is an important feature helps in achieving directed exploration. +2https://github.com/stratisMarkou/sample-efficient-bayesian-rl +3https://github.com/jiaseny/kdsd +4https://github.com/colehawkins/KSD-Thinning + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +24 +(a) +(b) +Figure 10. This plot analyzes the concentration of the predicted ˆQ to the true Q∗ (ground-truth) with iterations for STEERING. (a) This +figure shows the performance for DeepSea environment with N = 4. (b) This figure shows the performance for DeepSea environment +with N = 8. +(a) +(b) +Figure 11. Comparison of STEERING and Q-learning on DeepSea Exploration with sparsity N = 4. This plot shows the concentration +of the predicted ˆQ to the true Q∗ (ground-truth) versus iterations. It is evident that STEERING converges to true Q∗ much faster than +Q-learning. Remark: As right actions are optimal for DeepSea environment, STEERING stops exploring left actions beyond a point, +leading to a comparatively higher variance in ˆQ for left actions, thus providing directed exploration. +(a) +(b) +Figure 12. Comparison of STEERING and Q-learning on DeepSea Exploration with sparsity N = 8. This plot shows the concentration +of the predicted ˆQ to the true Q∗ (ground-truth) with iterations. Remark: As the sparsity is increased, the performance of Q-learning +degrades drastically (potentially due to lack of efficient exploration) whereas STEERING converges to true Q∗ efficiently. Further, we +note that since right actions are optimal for DeepSea environment, STEERING stops exploring left actions beyond a point leading to a +comparatively higher variance in ˆQ for left actions, thus providing directed exploration. + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +25 +(a) +(b) +Figure 13. Comparison of STEERING and Bayesian-Q learning (BQL) on DeepSea Exploration with sparsity N = 4. This plot shows +the concentration of the predicted ˆQ to the true Q∗ (ground-truth) versus iterations. Remark: STEERING converges to true Q∗ much +faster than Bayesian-Q learning which fails to concentrate on the true Q∗. This is because BQL does not have an efficient forgetting +mechanism in its update rule leading to high dependence on inaccurate past updates. This leads to the observation that the posterior is +overconfident about incorrect predictions in BQL. However, STEERING stops exploring left actions beyond a point as right actions are +optimal for DeepSea environment, leading to a comparatively higher variance in ˆQ for left actions, but low variance for right actions, +thus providing directed exploration. +(a) +(b) +Figure 14. Comparison of STEERING and Bayesian-Q learning (BQL) on DeepSea Exploration with sparsity N = 8. This plot shows +the concentration of the predicted ˆQ to the true Q∗ (ground-truth) versus iteration with more sparsity. Remark: STEERING converges +to true Q∗ much faster than Bayesian-Q learning which fails to concentrate on the true Q∗. A major reason can be that BQL doesn’t +have an efficient forgetting mechanism in its update rule leading to high dependence on inaccurate past updates. Hence, we observe that +the posterior for BQL is overconfident about incorrect predictions. In contrast, STEERING stops exploring left actions beyond a point +as right actions are optimal for DeepSea Exploration environment, leading to a comparatively higher variance in ˆQ for left actions, thus +providing directed exploration. + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +26 +(a) +(b) +Figure 15. Comparison of STEERING and uncertainty Bellman equation (UBE) on DeepSea Exploration with sparsity N = 4. This +plot shows the concentration of the predicted ˆQ to the true Q∗ (ground-truth) versus iterations. Remark: STEERING converges much +more efficiently to true Q∗ (or µ∗ +z) compared to UBE which fails to concentrate properly. For UBE, even if the predicted mean is closer +to optima, the variance is too high which leads to sub-optimal exploration. In contrast, STEERING stops exploring left actions beyond +a point as right actions are optimal for the DeepSea Exploration environment, leading to a comparatively higher variance in ˆQ for left +actions, thus providing directed exploration. +(a) +(b) +Figure 16. Comparison of STEERING and uncertainty Bellman equation (UBE) on DeepSea Exploration with sparsity N = 8. This +plot analyzes the concentration of the predicted ˆQ to the true Q∗ (ground-truth) versus iterations with more sparsity. Remark: As the +sparsity is increased (N = 8), the variance in estimated µ∗ +z for UBE increases significantly and the performance degrades drastically +leading to random action selection. In contrast, STEERING converges much more efficiently to true Q∗ (or µ∗ +z). Also, STEERING +stops exploring left actions beyond a point as right actions are optimal for DeepSea Exploration environment, leading to a comparatively +higher variance in ˆQ for left actions, thus providing directed exploration. + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +27 +(a) +(b) +Figure 17. Comparison of STEERING and moment matching (MM) on DeepSea Exploration with sparsity N = 4. This plot analyzes +the concentration of the predicted ˆQ to the true Q∗ (or µ∗ +z) versus iterations. Remark: Although MM performs well in this setting, +STEERING converges more efficiently and faster to true Q∗. Also, STEERING unlike MM stops exploring left actions beyond a point +as right actions are optimal for DeepSea Exploration environment, leading to a comparatively higher variance in ˆQ for left actions and +low variance for right actions, thus providing directed exploration. +(a) +(b) +Figure 18. Comparison of STEERING and moment matching on DeepSea Exploration with sparsity N = 8. This plot analyzes the +concentration of the predicted ˆQ to the true Q∗ (or µ∗ +z) (ground-truth) versus iterations. Remark: As the sparsity is increased (N = 8), +the performance of MM degrades with increased variance of predicted µz and also converges to sub-optimal muz. While STEERING +converges to true Q∗ efficiently. Also, since right actions are optimal for DeepSea Exploration environment, STEERING stops exploring +left actions beyond a point leading to a comparatively higher variance in ˆQ for left actions, thus providing directed exploration. + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +28 +(a) +(b) +Figure 19. Comparison of STEERING and posterior sampling reinforcement learning (PSRL) on DeepSea Exploration with sparsity +N = 4. This plot analyzes the concentration of the predicted ˆQ to the true Q∗ versus iterations. Remark: The performance of PSRL +is comparable to STEERING, but still the convergence is faster with lesser variance for STEERING. We also note that the sparsity level +for this setting is low (N = 4). +(a) +(b) +Figure 20. Comparison of STEERING and posterior sampling reinforcement learning (PSRL) on DeepSea Exploration with sparsity +N = 8. This plot analyzes the concentration of the predicted ˆQ to the true Q∗ (ground-truth) versus iterations. Remark: As the +sparsity is increased (N = 8), the performance of PSRL degrades with slower convergence to Q∗ with higher variance in ˆQ prediction. +In contrast, STEERING converges much more efficiently to true Q∗. Also, STEERING stops exploring left actions beyond a point as +right actions are optimal for DeepSea Exploration environment, leading to a comparatively higher variance in ˆQ for left actions, thus +providing directed exploration. + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +29 +J.2. Convergence Plots for DSD +Finally, we show the correctness of our approach by observing the convergence of DSD in Fig. 21. We note that for the +optimal actions (orange), the DSD converges to much lower values than sub-optimal actions (blue) which in-turn implies +the effectiveness of our Stein-information based proposed approach. +(a) +(b) +(c) +(d) +Figure 21. This figure provides evidence of DSD convergence (mean plot over 5 seeds) for different state-action pairs and sparsity levels +with iterations for the DeepSea environment. Additionally, the plot also provides an indication of directed exploration through the DSD +convergence to lower values for right actions which moves agent towards goal than left in states s = 3 and s = 4. + +Stein Information Directed Exploration for Model-Based Reinforcement Learning +30 +J.3. Additional Comparisons for WideNarrow MDP and PriorMDP +Here in Fig. +22, we also compares the performance of STEERING on WideNarrow MDP and PriorMDP envi- +ronments (Markou & Rasmussen, 2019; Osband & Van Roy, 2017a) against the existing RL baselines : +vanilla Q- +learning with ǫ−greedy action selection (Watkins & Dayan, 1992), Bayesian Q-learning (BQL) (Dearden et al., 1998), +Uncertainty Bellman Equation (UBE) (O’Donoghue et al., 2017), Moment matching (MM) across Bellman equation +(Markou & Rasmussen, 2019), Posterior Sampling RL (PSRL) (Osband et al., 2013), and IDS (Hao & Lattimore, 2022). +We approximated information gain with variance to implement IDS, hence denoted by Var-IDS in Fig. 22. +(a) WideNarrow MDP +(b) PriorMDP +Figure 22. This figure compares the performance of STEERING on (a) WideNarrow MDP and (b) PriorMDP environments +(Markou & Rasmussen, 2019; Osband & Van Roy, 2017a). Remark: WideNarrow MDP tests the algorithm’s ability under factored +posterior approximations, whereas PriorMDP tests the algorithm’s ability to more general environments without specific structures. We +note that STEERING outperforms existing baselines in both the environments. + diff --git a/VNFLT4oBgHgl3EQfRy9S/content/tmp_files/load_file.txt b/VNFLT4oBgHgl3EQfRy9S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eaf276124c8416a1903d5542512c42c817ff4cae --- /dev/null +++ b/VNFLT4oBgHgl3EQfRy9S/content/tmp_files/load_file.txt @@ -0,0 +1,1856 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf,len=1855 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content='12038v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content='LG] 28 Jan 2023 STEERING: Stein Information Directed Exploration for Model-Based Reinforcement Learning Souradip Chakraborty 1 Amrit Singh Bedi 1 Alec Koppel 2 Mengdi Wang 3 Furong Huang 1 Dinesh Manocha 1 Abstract Directed Exploration is a crucial challenge in re- inforcement learning (RL), especially when re- wards are sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' Information-directed sampling (IDS), which optimizes the information ratio, seeks to do so by augmenting regret with in- formation gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' However, estimating informa- tion gain is computationally intractable or re- lies on restrictive assumptions which prohibit its use in many practical instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' In this work, we posit an alternative exploration incen- tive in terms of the integral probability metric (IPM) between a current estimate of the tran- sition model and the unknown optimal, which under suitable conditions, can be computed in closed form with the kernelized Stein dis- crepancy (KSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' Based on KSD, we develop a novel algorithm STEERING: STEin infor- mation dirEcted exploration for model-based Reinforcement LearnING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' To enable its deriva- tion, we develop fundamentally new variants of KSD for discrete conditional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' We further establish that STEERING archives sublin- ear Bayesian regret, improving upon prior learn- ing rates of information-augmented MBRL, IDS included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' Experimentally, we show that the pro- posed algorithm is computationally affordable and outperforms several prior approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' Introduction Exploring effectively is a major challenge in reinforce- ment learning (RL), particularly when the rewards are sparse (Rengarajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' Achiam & Sastry, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' Recent research using model-based reinforcement learn- 1Department of Computer Science, University of Mary- land, College Park, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' 2JP Morgan Chase AI Research, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' 3Department of Electrical Engineering, Princeton Univer- sity/Deepmind, Princeton, NJ, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFLT4oBgHgl3EQfRy9S/content/2301.12038v1.pdf'} +page_content=' Correspondence to: Amrit Singh Bedi 95%), +indicating that those are indeed novel structures. Additionally, a search in the ChEMBL +database does not reveal any structurally related molecules (Tanimoto similarity > 70%), +indicating that no topologically similar molecules have been evaluated against BTK. The +topological similarity with the seed molecule extracted from GDC-0853 (Figure 4c) is also +low. In contrast, in terms of pharmacophore and shape, most input pharmacophore features +are covered inside these generated molecules, including hydrogen bonding with Lys430 and +Met477 and hydrophobic interactions at the two ends of each molecule. Synthesis paths +are also proposed by DeepLigBuilder+ for each generated molecule, making retrosynthetic +analysis of generated molecules easier. For example, Figure 6a shows the proposed synthetic +16 + +path for Structure 4. The path for Structure 2 and 3 are shown in Figure S12. In summary, +by combining MCTS with the conditional generative model, DeepLigBuilder+ can enrich +molecules with high binding affinity based on the pharmacophore constraint. +3.3 +Case study: designing inhibitors targeting the NAD+ pocket +of PHGDH +Targeting cancer metabolism represents an important strategy for cancer drug develop- +ment.64 Human phosphoglycerate dehydrogenase (PHGDH), a key enzyme in the serine +biosynthesis pathway, has been demonstrated to have crucial roles in tumorigenesis,65 mak- +ing it a promising cancer-related target. One strategy for targeting PHGDH is to design +inhibitors that bind to its NAD+ pocket. Multiple such inhibitors have been reported in pre- +vious works, with most of them containing an indole-based scaffold. In this case study, we +use DeepLigBuilder+ to design potential binders for the NAD+ pocket with novel structures. +Figure 5a shows the structure of PHGDH (PDB ID: 6plg) together with compound 15, +a potent inhibitor of the target developed by Mullarky et al.66 Figure 5b demonstrates the +interaction between the ligand and the target. The nitrogen atom in the amide group in +compound 15 acts as a hydrogen bond donor and interacts with Asp175. The carboxyl +group in compound 15 can form hydrogen bonds with backbone nitrogen atoms. It also +forms charge-charge interaction with Arg155. The indole structure of compound 15 resides +inside a hydrophobic region, as shown by the orange arrow in Figure 5b. A pharmacophore +model is constructed based on those interactions, as shown in Figure 5c. We use the amide +structure as the seed for molecule growth, as shown in Figure 5c and Figure S13b. The +shape of compound 15 is also used as an input feature. +Next, we use DeepLigBuilder+ to generate molecules based on the pharmacophore and +shape information and the target structure. Similar to the previous section, we first evaluate +whether the conditional model offers more enriched results based on the provided informa- +tion. Indeed, Figure 5d-e shows that molecules sampled from the conditional transformer +17 + +Figure 5: a. The structure of PHGDH with compound 15. b. The interaction between +compound 15 with the NAD+ binding pocket of PHGDH. c. +The pharmacophore and +shape condition extracted based on the interaction pattern. As well as the seed atom used for +molecule growth. d-f. The distribution of: d. the similarity with the given pharmacophore, +e. the similarity with the given shape, f. Smina docking scores inside the NAD+ binding +pocket of PHGDH, among generated molecules. g. The best rewards among all generated +molecules at each step of MCTS. h-j. +Several generated molecules with high predicted +binding affinity. +The conformation generated by the model is shown in grey, and that +produced by redocking the molecule is shown in white. +18 + +b. +a. +C. +Hydrophobicity +ILE 156 +ASP 175 +O HBD +○ HBA ○ Hydrophobic +ARG 155 +d. +f. +e. +g. +10 - +MCTS +Conditional +8 - +rollout +Reward +Conditioned on +6 +shape and + pharmacophore +Unonditional +Unconditional +rollout +0.0 +0.1 +0.2 +0.3 +0.2 +0.4 +0.6 +-8 +-6 +10 +100 +Pharmacophore +Number of MCTS steps +Shape similarity +Smina score (kcal/mol) +similarity (Aign-it) +(log scale) +h. +i. +j. +Structure 5 +Structure 6 +Structure 7 +Morgan: +0.184 +Morgan: +0.165 +Morgan: +0.180 +Pharmacophore: +Pharmacophore: +Pharmacophore: +0.227 +0.203 +0.294 +0 +:0 +Shape: +0.770 +Shape: +0.658 +Shape: +0.704 +HN +Smina score: -10.5 +HN +Smina score: -9.56 +Smina score: -9.74 +HN +0 +ILE 156 +ILE 156 +ILE 156 +ASP 175 +ASP 175 +ASP 175 +ARG 155 +ARG 155 +ARG 155match better to the input pharmacophore(Figure 5d) and shape(Figure 5e) compared with +the unconditional model. Figure 5g shows that introduction conditions help to accelerate +the MCTS search, as demonstrated by the blue curve. When evaluating the benefit of the +MCTS module, we found that MCTS search helps the model to generate molecules with bet- +ter pharmacophore and shape matches (Figure 5d-e), and also helps to improve the docking +score of the result, with an average improvement of Smina score of 0.53 kcal/mol. 11% of +molecules generated using MCTS have a Smina score < -9.5 kcal/mol, compared to the value +of 2% for those directly sampled from the conditional model. +Figure 5h-j shows several molecules generated by DeepLigBuilder+ with high predicted +binding affinity. The reaction paths generated by the model are shown in Figure 6b and +Figure S14. Those molecules have low topological similarities with compound 15, but share +pharmacophore features such as hydrogen bond donors that interact with Asp175 and accep- +tors that interact with Ile156 or Arg155. A search in PubChem does not reveal results with +high topological similarity with those molecules (Tanimoto similarity >95%), indicating that +those are indeed novel structures. Also, the ChEMBL dataset does not contain topologically +related compound records (Tanimoto similarity >70%), which means that similar molecules +are not yet evaluated against PHGDH. Interestingly, those molecules contain cyclobutane +structures that are similar to the oxetane structure in compound 15. This structural motif +helps to form a turn in the molecule shape for better accommodation with the pocket, and +also creates a hydrophobic interaction with Ile177. +3.4 +Ablation studies and the effects of different hyperparameters +It is important to understand how different architectural and hyperparameter choices affect +the performance of the proposed model. In this section, we demonstrate the impact of several +important network features and hyperparameters. Details about the configurations explored +are shown in Table S1. The performances of the model under different configurations are +shown in Table S4. +19 + +Figure 6: a. The synthetic path of Structure 4 proposed by DeepLigBuilder+. It involves a +two-step process, which first connects the amide bond using the Schotten-Baumann reaction, +and then forms the double bond using the Wittig reaction. Enamine IDs of reactants are +also given. +Note that the Schotten-Baumann reaction requires an additional activation +step that transforms the carboxyl group into the acyl chloride. Also, the Wittig reaction +requires the formation of the ylide. +b. +The synthetic path of Structure 5 proposed by +DeepLigBuilder+. The first two reactants are connected using the Grignard reaction. The +third reactant is connected by forming a urea structure using the amine group and the +isocyanate group. Note that before the first step, reactant 2 needs to be transformed into +the Grignard reagent. Additionally, the amine group in reactant 2 (marked grey) needs to +be protected before carrying out other reactions. +20 + +a. +Schotten-Baumann +reaction +EN300-27296 +Structure 4 +Wittig reaction +F3C +EN300-88479 +EN300-729603 +b. +Grignard reaction +EN300-82407 +Structure 5 +Urea formation +EN300-313611 +EN300-3789413.4.1 +The effect of changing the set of accessible building blocks +A major feature of DeepLigBuilder+ is its capability to suggest synthetic paths with accessi- +ble building blocks along with its generated molecules. However, the accessibility of building +blocks is a constantly changing factor. On one hand, due to technical advances, the number +of synthesizable building blocks is rapidly growing over the years. On the other hand, in- +stock supply of such building blocks may vary between times and locations, and most require +on-demand synthesis, increasing the cost. Ideally, DeepLigBuilder+ should allow the user +to choose a building block set that fits their need, without the need to perform re-training. +Here, we simulate such scenarios and report how the choice of building blocks impacts the +quality of generated molecules. +To simulate the lack of in-stock availability for certain building blocks, we restrict the +building blocks to the EU stock, which is a smaller set with 81,235 compounds. In terms +of the quality of topological structures, although Figure 3e shows that there is an increase +in the MMD value after the restriction, Table S2 confirms that the generated molecules still +maintain a high drug-likeness, with an average QED value of 0.60, close to the value before +changing the building blocks. In terms of the quality of 3D conformation, Table S4 and +Figure 3f indicates an increase of RMSD from 0.69Å to 0.72Å, but still much lower than +1Å. From the results, we believe that although using a smaller building block may indeed +impact the performance of the network, such an effect should be minor and still allows for +regular application of DeepLigBuilder+ in drug design tasks. +Then, to demonstrate how increasing the building block set may affect the generation +result, we expand to include the comprehensive catalog, which contains more than 1 mil- +lion (1,162,033) compounds, some may be synthesized on-demand. Figure 3e-f shows that +this change induces little impact on the performance of the output. Only 28.1% of molecules +generated have used the newly added building blocks. In conclusion, we believe that DeepLig- +Builder+ still offers promising performance when the building block set is changed, but a +re-training may be required if we want to fully utilize newly added building blocks. +21 + +3.4.2 +The effect of different ways to encode 3D structural information +Besides using the relative 3D positional encoding module to incorporate 3D information, +DeepLigBuilder+ also uses invariant point attention (IPA), which offers a geometrically- +aware way to pass information between atoms. To understand the benefit of including IPA, +we disable the two modules consecutively and investigate the impact of those changes. The +results are shown in Figure 3e-f and Table S4. We found that removing IPA has little impact +on the quality of 3D conformation, as measured by RMSD, showing that IPA is not essential +for maintaining 3D structure quality. However, the quality of 2D structure, as measured by +2D MMD, is reduced. If we remove the 3D positional embedding and keep IPA, we observe a +significant improvement in 2D structural quality, but the RMSD value increased to 0.718Å. +The results above demonstrated a trade-off between 2D and 3D structural quality, and +that the two ways of including 3D information, IPA and relative positional encoding, have +different emphases on the two aspects. One way to understand the result is to view IPA as +a more regularized way of encoding geometric information. Our previous work has demon- +strated that the model can not reliably generate correct 2D structures if it overly relies on +accurate 3D information since it reduces the model’s ability to recover from errors during +generation.33 The highly structured way to communicate 3D information in IPA can act as +a form of regularization on how the model uses the 3D information. On the other hand, +there is no limitation on how 3D relative positional embedding will be processed by the net- +work. Therefore, although the model can still generate accurate 3D structures without IPA, +a lack of regularization will reduce the quality of 2D structures. Using both modules acts +as a compromise, with an improved 3D conformation quality and a balanced 2D structure +quality. +22 + +4 +Conclusion +We have developed a new de novo drug design tool, DeepLigBuilder+, that generates +synthesis-driven 3D molecules for a given target. DeepLigBuilder+ uses geometric trans- +former combined with an MCTS-based reinforcement learning module to navigate the space +of synthesizable 3D molecules to identify potential bioactive compounds. This method aims +to address two major challenges faced by deep molecule generative models: (1) the design of +3D molecules based on 3D constraints, and (2) the design of molecules with high synthetic +accessibility. DeepLigBuilder+ has shown promising performances in overcoming these chal- +lenges. DeepLigBuilder+ is capable of generating 3D molecules with high drug-likeness and +geometric quality under the synthetic accessibility constraint. +In the case study related +to BTK and PHGDH, DeepLigBuilder+ significantly enriches molecules with high docking +scores and favorable interaction patterns with the target pocket, using the 3D information +provided. For each generated molecule, DeepLigBuilder+ proposes a synthetic route with +explicit reactions and building blocks that can be directly queried from the supplier, making +retrosynthetic analysis much easier. +DeepLigBuilder+ takes advantage of recent developments in 3D generative networks33 +and the idea of synthetically aware de novo design.37 To restrict the model to the chemical +space of synthesizable molecules, we develop a method that calculates a stepwise constraint of +the generation trajectory to ensure that it leads to purchasable building blocks. Compared to +other approaches that require a fragment-based generation scheme, our method can in theory +be applied to various atom-based molecule generative models. In addition, it offers better +scalability to large building block datasets by organizing them into a tree-based structure +and avoids full database scans at each step. +To achieve structure-based generation, we +constructed a new dataset of pharmacophore-ligand pairs using large-scale 3D alignment of +molecules, and then use it to develop a novel SE(3)-equivariant transformer conditioned on +3D information. This network is then combined with MCTS as the rollout policy, and it is +demonstrated that the combination results in a significant improvement in the search speed +23 + +of MCTS. +DeepLigBuilder+ could be improved in the following aspects in the future. First, the +present molecule assembling process relies on simple SMARTS rules, which may be limited +in precision. We are planning to include a more dedicated model for yield and selectivity pre- +diction so that we can further improve the synthesizability of the assembled molecules by the +model. Second, we are planning to update the reaction set to include broader, more modern +reactions. In addition, the current version of DeepLigBuilder+ requires user-provided seed +structures for molecule growth, and we are planning to develop methods to automate the +seed selection process. Finally, we are planning to build a transformer model that is directly +conditioned on the 3D pocket structure, without the need for pharmacophore extraction. In +summary, due to its unique capability of generating highly synthesizable molecules with 3D +structures, DeepLigBuilder+ provides a powerful tool to generate bioactive molecules and +to accelerate the process of structure-based drug design. +5 +Acknowledgements +This work has been supported in part by the National Natural Science Foundation of China +(22033001). 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Bioorganic & Medicinal +Chemistry Letters 2019, 29, 2503–2510. +31 + +Supporting Information: +Synthesis-driven design of 3D molecules for +structure-based drug discovery using +geometric transformers +Yibo Li,† Jianfeng Pei,∗,‡ and Luhua Lai∗,†,‡,¶ +† Center for Life Sciences, Academy for Advanced Interdisciplinary Studies, Peking +University, Beijing 100871, China +‡ Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking +University, Beijing 100871, China +¶ BNLMS, College of Chemistry and Molecular Engineering, Peking University, Beijing +100871, China +E-mail: jfpei@pku.edu.cn; lhlai@pku.edu.cn +1 +Supplementary Methods +1.1 +Constructing the synthon dataset +We use the building block sets provided by Enamine as the set of purchasable reactants. +The global stock, which contains 238,980 compounds at the time of access (July 2022), is +used to assemble the training set molecules. To investigate the impact of changing available +reactants on the model’s performance, we also downloaded the EU stock and comprehensive +catalog, which contains 81,235 and 1,162,033 molecules respectively (by Oct 2022). +S-1 +arXiv:2301.00167v1 [q-bio.QM] 31 Dec 2022 + +For reactions, we use the 58 SMARTS rules collected by Hartenfeller et al.,S1 which +represents a set of robust chemical reactions relevant to drug design. Based on the reaction +set, we constructed a set of SMARTS rules to convert reactants to synthons. The conversion +is performed using the following procedure: +(1) For a given building block and reaction rule, we predict the product structure using +RDKit. If the reaction involves two reactants, the other reactant is set to be a minimum +structure with the required function group. +(2) The substructure in the product molecule that corresponds to the building block is ex- +tracted. The open valences resulting from the bond break in the extracted substructure +are labeled with the reaction type and its role in the reaction. +(3) If the substructure extraction results in multiple valences, this indicates that new rings +are formed after the reaction. In this case, we either include the new ring structure +inside this synthon or leave the ring to the synthon corresponding to the other reactant. +A visual demonstration of this process is given in Figure S1 and Figure S2. To simplify +the generative model, we require that each reaction will result in at most one open valence +in synthon structures, and such valence must correspond to a single bond in the product +molecule. Most reactions satisfy this requirement. As a result, we kept 52 reactions and +constructed 108 SMARTS rules for converting reactants to synthons. +The Enamine building blocks are then converted to synthons using those rules. +For +each building block, we enumerate all possible synthon structures by iterating through the +reaction rules. At most two reactions are allowed to happen in one reactant. To ensure +that the generated synthons are fragment-like and relevent to drug discovery, we apply the +following rules to filter the results: +(1) Element types of atoms inside each molecule are restricted to the set {C, O, N, P, S, F, +Cl, Br, I} and bond types are restricted to single, double, triple, and aromatic bonds. +S-2 + +(2) Synthons are required to be “fragment-like” based on the rule of two (Ro2).S2 +(3) Each fragment can have at most 4 rings, and the size of each ring should not be larger +than 7. +(4) Since we are focusing on designing non-covalent binders, we filter structures with the +potential of forming covalent bonds with the protein, using a set of SMARTS rules.S3 +The filtering results in a dataset containing 241,310 synthons from the global stock, +103,385 synthons from the EU stock, and 783,195 synthons from the comprehensive catalog. +The synthons from the global stock are later used to assemble the training set molecules +(See Section S1.5). +1.2 +Performing molecule generation +DeepLigBuilder+ generates 3D molecules using a synthon-based method. Each molecule is +composed of 3 synthon structures, which is equivalent to combining three building blocks with +two reaction steps. When generating each synthon, DeepLigBuilder+ uses a graph-based +approach similar to our previous work.S4 Specifically, the model generates 3D synthon struc- +tures by producing molecular graphs. We write the output graph as G = (V, E, A, B, X), +where V and E are the set of nodes (atoms) and edges (bonds), A = {av}v∈V and B = +{buv}{u,v}∈E are labels representing the type of each atom and bond, and X = {xv}v∈V are +the 3D coordinates of each atom. +When generating each synthon, the model starts with an empty graph G0 = (, ..., ), +iteratively updates its structure Gt = at(Gt−1), and outputs the graph as a new synthon +fragment when it is ready. During generating, additional information is attached to each +node in the graph to record the generation history, which includes: +(1) The currently focused node, denoted as v∗ +t , where t represents the step ID. The defini- +tion of a “focused” node is similar to that in our previous works.S4 Briefly, all edits to +S-3 + +the molecular graph, whether to add new atoms or new bonds, happen on the focused +node. +(2) The parent of each node Pt = {pv}v∈Vt. A node pv is called the “parent” of another node +v if pv is the focused node when v is generated. Note that the parent-child relationship +induces a spanning tree of the molecular graph Gt, which can be used to calculate +tree-based distances between atoms (or nodes) in the graph as input features to the +transformer network (as detailed in Section S1.4). +We denote the state of the molecular graph at step t with the additional information as +G′ +t = (Vt, Et, At, Bt, Xt, v∗ +t , Pt). The graph structure is iteratively modified based on actions +at sampled from the neural network. +The following types of actions are allowed during +generation: +(1) Initialization, which adds the first atom to the molecular graph; +(2) Append, which attaches a new atom to the focused atom using a new bond. For this +action, the model needs to decide the type of the new atom and bond, as well as the 3D +position of the new atom. When generating the position, the model uses a spherical +coordinate frame attached to the focused atom, following our previous work.S4 Besides +the element type of the new atom, the model also needs to decide whether there will +be branches on this atom and whether the atom will be a target of future ring closure, +similar to Ahn et al.S5 +(3) Backtracking, which requires the model to move the focused atom to its closest ancestor +that allows branching. The generation terminates if the focused node has no parents. +(4) Search loop target. This action indicates that a new ring will be formed, and the model +should examine the closest ancestor of the focused node to see whether it is a suitable +target during the ring closure. The network can generate this action multiple times +until a suitable target is found for ring closure. +S-4 + +(5) Start loop. This action happens after a “search loop target” action when the appropriate +target of ring closure is found. In this action, the model determines the size of the +ring. After this action, a series of “append” actions should be issued by the model to +complete the ring formation. +(6) Close loop. This action happens after all ring atoms have been generated, and the +model is ready to connect the ring to the target atom determined in the “search loop +target” step. The model decides the type of bond used to close the ring during this +action. +The process of molecule generation can be represented using a finite state machine, as +shown in Figure S3. A full path for generating an example molecule is shown in Figure S4. +A major difference between the generation scheme proposed in this work compared with +the previous version of DeepLigBuilderS4 is its emphasis on ring generation. Before gener- +ating explicit ring structures, DeepLigBuilder+ will first determine the size of the ring, as +well as the location the ring will be closed. This information will help to guide the process +of ring generation. It will also help to avoid problems when the user changes the synthon +dataset (to be discussed in the next section, also see Figure S5). +1.3 +Constraining the model to generate structures inside the syn- +thon database +To ensure synthetic accessibility, the synthons generated by DeepLigBuilder+ must be re- +stricted to the synthon dataset derived from purchasable building blocks. To achieve this +goal, most previous methods use reactants as basic units for molecule generation and ap- +ply neural networks to parametrize scoring functions that filter the reactant dataset for +appropriate candidates at each generation step. In this work, we propose a radically dif- +ferent approach. Instead of adopting a generation scheme based on reactants, we still use +atoms as basic generation units, building on the foundation of previous works.S4,S6 To en- +S-5 + +force constraints on the chemical space, we apply masks on the action space at each step of +generation, so that we can ensure that the output topological structure can be found in the +synthon dataset. +Next, we show how such action masks can be calculated at each step. First, for each +synthon in the synthon database s ∈ S, we represent it as a list of actions that can be +used to generate its molecular graph, which is indicated as (a1, ..., aT) → s. The definition +of the action space and the generation process follows Section S1.2. Since we only need +to constrain the topological structures, 3D action information, such as the position of new +atoms, is removed from the action space. For convenience, we refer to such action sequences +as trajectories and write them as τ. In this way, we convert the synthon dataset into a +collection of trajectories T (S) = {τ|∃s ∈ S s.t. τ → s}. +To ensure synthons generated by the model lie in S, we need to ensure that the generation +trajectories lie in T (S). At each step t, given the generation history τt = τ[1..t − 1] = +(a1, ..., at−1), in order to ensure that the final trajectory will lie inside T (S), we need to +make sure that the next action at is inside the following set: +at ∈ A(τt, S) = {a|∃τ ′ ∈ T (S) s.t. τ ′[1..t] = τt · a} +It is easy to prove that as long as this requirement holds at each step, we can guarantee +that the resulting synthon will be inside S. In order to efficiently calculate A(τt, S), the +trajectories in T (S) are organized into a prefix tree. Inside the tree, each node represents +a prefix of some trajectory in T (S), and each edge represents an action. During retrieval, +we descend the tree to find the node that equals τt, and then collect all its outgoing edges. +It can be seen that those edges form the set A(τt, S). In this way, we can construct action +masks at each step to ensure that the resulting synthon can be found in the synthon dataset. +As a method to constrain the chemical space of the generative model, our method has +some advantages compared with previous methods: +S-6 + +(1) The complexity of generating each synthon does not scale with the size of the synthon +dataset. Searching inside the prefix tree has an average cost of O(L), where L is the +average number of steps required to generate a synthon. Other approaches generally +require a full scan of the reactant dataset, which has more limited scalability for larger +synthon databases. Constructing the prefix tree will cost O(LN), where N is the size +of the synthon, but we only need to construct the tree once, and it can then be reused +for subsequent generation tasks. +(2) Our method only constrains the 2D (topological) structure of molecules, while the 3D +coordinates are generated by the neural network. This eliminates the need of build- +ing a 3D fragment database, as done in several previous works.S7,S8 Such approaches +may cause some technical issues. First, enumerating 3D conformers will significantly +increase the size of the fragment dataset, especially when the dataset contains large +flexible structures. Second, the conformation of a fragment depends on its environ- +ment, and enumerating its conformation in isolation may result in inaccurate results +when the fragment is attached to another molecule. +Some practical issues need to be considered when applying this method: +(1) There are generally multiple ways to generate a molecular structure. To reduce com- +plexity and computational cost, we follow the approach in previous works,S4,S6 which +uses a depth-first, canonically ordered way to generate molecules. In this way, a molec- +ular graph will correspond to exactly one trajectory, when the starting atom for gen- +eration is given. To make the model more flexible, we allow multiple starting points +for the generation as in our previous work.S4 +(2) When using the proposed method to constrain the chemical space, there may be issues +related to ring conformation when the synthon dataset is changed without model re- +training. An illustrative example is given in Figure S5. To alleviate this problem, we +S-7 + +adopt a more refined ring-generation scheme, which allocates the size and location of +the ring before its structures are generated, as discussed in the previous section. +1.4 +Network architecture +In this section, we give a detailed description of the neural network architecture. We first +describe the inputs required by the network. Then, we show how the inputs are embedded +before feeding to the neural network. Next, we detail the architecture of the neural network. +Finally, we demonstrate how the output features from the transformer are used to generate +the action at each step with a MADE-based policy network. +1.4.1 +Input features +The model receives previously generated synthons as inputs, as well as shape and pharma- +cophore information for structure-based generation tasks. +Graph inputs +All graph-related inputs, including previously generated synthons and the +intermediate synthon structures, are represented as their generation trajectories τ. This acts +as a sequence-based representation of molecular graphs, similar to the concept of a “sentence” +in NLP tasks. Each action in the sequence corresponds to a “token” or “word” in the sentence. +The following information is included in each token: +(1) The current step id (t); +(2) Action performed at this step, including the action type (actt) and the type of new +bonds added (nbtt); +(3) Information of the focused node after the action is applied, such as its index (idt, or- +dered based on the step each atom is generated), element type (elt), formal charge(fct), +and the number of explicit hydrogens (neht) attached. Information about whether the +S-8 + +node allows for branching (bt) or whether the node can act as a target for ring closure +is also included(rt). +(4) If a ring is being generated, we also input the expected size of the ring (rst) and the +expected target for ring closure (rtt). +In addition to the information above, each token is attached with a 3D coordinate frame +(ot, Rt) using the method developed in our previous work.S4 Those frames have several +functionalities. +(1) The coordinates of newly generated atoms are defined under those frames. The spheri- +cal coordinate values of the local frames correspond to bond lengths, bond angles, and +torsion angles. +(2) Those frames are used in the IPA modules to communicate 3D information between +the tokens. +(3) Those frames are used to define the relative 3D positional embeddings between tokens +(to be discussed below). +Besides input features for each token, we also include features for each action pair as +relative embedding to increase the performance of the model. Those pair features include: +(1) The topological distance between focused atoms at each step (toptt′). +(2) The distance between the focused atoms in the spanning tree induced by the generation +trajectory (treett′, recall descriptions in Section S1.2). +(3) The relative 3D positions between coordinate frames attached to each action (∆xtt′). +Specifically, for the action pair (at, at′), we first calculate the displacement between the +origins of each frame and then transform the coordinate values to the local coordinate +frame attached to ai. +S-9 + +Pharmacophore inputs +The input pharmacophore model can be represented as a se- +quence of individual pharmacophore features, with definitions adopted from Align-it.S9 Each +pharmacophore p contains information about its type (ptp) and radius (prp). We use the +Euclidean distances (pdpp′) between pharmacophore pairs (p, p′) as relative positional em- +beddings. +In the decoder, we need to communicate between the pharmacophore model and the +synthon structure. Therefore, we feed the model with the following information about the +3D relationship between each pharmacophore-action pair: +(1) The position of each pharmacophore in the local coordinate system attached to each +action (∆xpt); +(2) The direction for each pharmacophore (only HBDs and HBAs) in the local coordinate +systems attached to each action (ˆnpt). +Shape inputs +The shape input can originate from known active ligands or directly from +the target pocket using programs such as PANTHER.S10 In both cases, we represent the +input shape as a set of 3D spheres. Most previous models use 3D-CNN to encode shape +information.S11 This method is not equivariant and induces additional computational costs. +In comparison, DeepLigBuilder+ uses a more compact, SE(3) equivariant representation for +3D shapes based on 3D Zernike coefficients. For an input shape composed of 3D spheres, we +can represent it using a 3D scalar function following Grant et al.:S12 +f(x) = +N +� +i=1 +pi exp(−αi|x − xi|2) +Where xi is the location of each sphere, and pi and αi are parameters related to the radius +of each sphere. We define the function in the coordinate frame placed in the center of the +S-10 + +spheres xc = 1 +N +�N +i=1 xi. The function is then decomposed as: +f(x) = +� +nlm +cm +nlZm +nl(x) +Where Zm +nl(x) are 3D Zernike polynomials.S13 In this work, we use a truncated series with +n ≤ 9. Those functions are generalizations of Zernike polynomials defined in 2D space and +act as the orthogonal basis for 3D functions (defined in a ball with radius 1). Additionally, +those coefficients c = {cm +nl}nlm changes in an equivariant manner when a rotation R ∈ SO(3) +is applied to the function f: +f ′(x) = f(R−1x) = +� +nlm +� +m′ +Rl +mm′cm′ +nl Zm +nl(x) +Where Rl +mm′ are the matrices that can be used to “rotate” the coefficients of the function f. +Those matrices can be efficiently calculated using the methods proposed by Ivanic et al.S14 +The representation of the shape can then be written as (xc, c). When feeding into the +model, we rotate it into the local coordinate frames of each action (xc +t, ct), and concatenate it +with other action features. In other words, the shape information is used by the transformer +similar to a positional embedding for each token. +1.4.2 +Embedding layers +Several types of embedding layers are used for the input information discussed above, in- +cluding: +(1) Lookup tables with trainable parameters. This type of layer is used to embed atom, +bond, and pharmacophore types, as well as topological distances. The full list of inputs +includes actt, nbtt, elt, fct, neht, bt, rt, rst, toptt′, treett′, and ptp. +(2) Positional embedding using sine and cosine functions.S15 This type of embedding is +widely used in transformer models to embed position-related data. In this work, we +S-11 + +use it to embed time, position, and distance-related information. The full list includes +t, idxt, rtt, ∆xtt′, pdpp′, ∆xtp and xc +t. +(3) Some inputs with continuous representations are input to the model as-is. This in- +cludes: prp, ˆnpt, and ct. +After the inputs are embedded, for each token (action or pharmacophore) and token pair, +we concatenate all input information into a vector and use a linear layer to project the inputs +to a predefined dimension, which is then used as transformer inputs. We write the size of +the dimension as F for each token and F ′ for each token pair. In this work, we have F ′ = F +2 +and two values {512, 256} are experimented for F. +1.4.3 +The transformer architecture +The transformer is responsible for processing the input features to generate a state embedding +at each step. Later, this state embedding will be used by the policy network for action +sampling. The transformer consists of multiple encoders responsible for processing different +inputs, and a decoder used to generate the state embedder. The decoder and encoders are +composed of transformer layers, each containing one or more attention layers and a tokenwise +dense layer. The attention and dense layers are wrapped inside residue blocks. +Attention layers +Two types of attention layers are used in this work. The first is the +widely used scaled dot-product attention(SDPA),S15 with additional relative positional bias. +Given the features of the source sequence {hs +i}ls +i=1 , the target sequence {ht +i}lt +i=1, and the +source-target token pairs {hp +ij}j=1,...,ls +i=1,...,lt , the attention layer performs the following operations +to calculate the output feature {h′i}lt +i=1 for each target token: +[kh +j , vh +j ]H +h=1 = Linearkv(hs +j), [qh +i ]H +h=1 = Linearq(ht +i), [bh +ij, zh +ij]H +h=1 = Linearp(hp +ij) +S-12 + +ah +ij = softmaxj( 1 +√ +d +qh +i · kh +j + bh +ij) +h′ +i = Linearout([ +ls +� +j=1 +ah +ijvh +j , +ls +� +j=1 +ah +ijzh +ij]H +h=1) +where i = 1, ..., lt; j = 1, ..., ls; h = 1, ..., H +H denotes the number of attention heads, which is set to be 16 in this work. d is the +dimension of each query, key, or value vector, which is set to be d = +F +H , the [·] operator +represents concatenation or unpacking respectively when it appears in the right or left side +of the equations. +The second type of attention is invariant point attention (IPA), initially proposed in Al- +phaFold2.S16 Similar to SDPA, we first calculate vector-based queries for the target sequence, +as well as the keys and values for the source sequence: +[kh +j , vh +j ]H +h=1 = Linearkv(hs +j), [qh +i ]H +h=1 = Linearq(ht +i), [bh +ij, zh +ij]H +h=1 = Linearp(hp +ij) +where i = 1, ..., lt; j = 1, ..., ls +Different from SDPA, IPA also calculates keys, queries, and values based on 3D points: +[⃗kph +i ,⃗vph +i ]h=1,...,H +p=1,...,P = Linear3D +kv (hs +i), [⃗qph +j ]h=1,...,H +p=1,...,P = Linear3D +q (ht +j) +where i = 1, ..., lt; j = 1, ..., ls +Where P is the number of points for each attention head and is set to be 8 in this work. +Those points are defined on the local coordinate system attached to each token and can be +transformed into the global coordinate system as: +⃗k′ph +j = Rs +j⃗kph +j + os +j, ⃗v′ph +j = Rs +j⃗vph +j + os +j, ⃗q′ph +i += Rt +i⃗qph +i ++ ot +i +S-13 + +where i = 1, ..., lt; j = 1, ..., ls; h = 1, ..., H; p = 1, ..., P +Where {(ot +i, Rt +i)}lt +i=1 and {(os +i, Rs +i)}ls +i=1 are coordinate frames attached to each source and +target tokens. Attention maps are then calculated as: +ah +ij = softmaxj(wL( 1 +√ +d +qh +i · kh +j + bh +ij − γhwC +2 +P +� +p=1 +|⃗q′ph +i − ⃗k′ph +j |2)) +where i = 1, ..., lt; j = 1, ..., ls; h = 1, ..., H +In which wL = +� +1 +3 and wC = +� +2 +9P . The values are then used to calculate the output +features: +f h +i = +ls +� +j=1 +ah +ijvh +j +˜f h +i = +ls +� +j=1 +ah +ijzh +ij +⃗f hp +i += (Rt +i)−1( +ls +� +j=1 +ah +ij⃗vph +j − ot +i) +h′ +i = Linearout([f h +i ,˜f h +i , [⃗f ph +i ]P +p=1]H +h=1) +where i = 1, ..., lt; h = 1, ..., H; p = 1, ..., P +Note that compared with the original implementation, we do not include the norm of ⃗f hp +i +in the input of the linear projection. SDPA and IPA are used in different situations in the +transformer network. +The encoder for previous synthons uses IPA. In the decoder, self- +attention layers and outer-attention layers with previous synthons use IPA. The encoder for +pharmacophores and the decoder outer-attention layers with pharmacophores use SDPA. +The token-wise dense layers (MLP layers) +MLP layers consist of two dense layers, +each with a normalization-activation-linear architecture. In this work, we use layer nor- +malizationS17 and ELUS18 as activation units. The number of hidden features is set to be +S-14 + +2F. +Transformer layers +The attention layers and MLP layers are composed of transformer +layers that are later stacked into the encoder and decoder networks. Each transformer layer +in the encoder consists of one attention layer and one MLP layer. Each transformer layer +in the decoder consists of two attention layers, one for self-attention and the other for outer +attention, and an MLP layer. The attention and MLP layers are all wrapped inside residual +blocks. +Encoders, decoders, and the transformer network +Multiple transformer layers are +stacked to form the encoder and decoders. For unconditional generation tasks, we use 6 +blocks for the encoder of previous synthons and 6 blocks for the decoder. For structure-based +generation tasks, the model also receives shape and pharmacophore-based information, which +uses 3 more encoder and decoder layers for processing. A shallow configuration is also exper- +imented with in unconditional generation tasks, as a way to demonstrate the performance +using different network scales. In this configuration, the encoder and the decoder each uses +3 blocks of transformer layers. +Two versions of geometric transformers are developed in this work. An unconditional +transformer is used to access the ability of this method to generate drug-like, geometrically +valid molecules with high synthesizability. A conditional one, which receives user-provided +pharmacophores and shapes as extract inputs, is used as the rollout policy to accelerate +MCTS in SBDD problems. +1.4.4 +The policy network +Using the state embedding generated by the transformer network, a policy network is then +applied to generate the action for the next step. Before specifying the architecture of this +network, we need to first define the action space. Two types of decisions need to be made at +each step of the generation. The first one relates to the topological structure of the molecule, +S-15 + +including the type of action to be carried out, the type of the new atom and bond, the size of +the new ring, etc. The second one relates to the 3D molecular structure, that is the position +of the new atom. +For topological decisions, we iterate through the synthon dataset to collect the actions +that are needed to produce all the synthon structures. This result in the topological action +space Atopo. For 3D actions, we write the location of the new atom added to each step in the +local spherical coordinate system attached to the focused node (r, θ, φ). We then discretize +r, θ and φ, each using two integers. Take the φ coordinate for example. We first split its +domain (−π, π] into N1 equal-sized intervals (−π+ 2π +N1i, −π+ 2π +N1(i+1)]; i = 1, ..., N1, and find +the one containing the coordinate value φ. The interval found is named φcrude. To achieve +further precision, φcrude is then divided into N2 smaller chunks. We find the one containing +φ, and name it φrefined. Similar procedures are applied for r and θ. +Following the definition above, we can now write the action as: +a = (atopo, rcrude, rrefined, θcrude, θrefined, φcrude, φrefined) +where atopo ∈ Atopo; +rcrude, θcrude, φcrude ∈ {1, ..., N1} +rrefined, θrefined, φrefined ∈ {1, ..., N2} +In this work, N1 is set to be 30 and N2 to be 32. The task of the policy network is to +parametrize the distribution of a using the neural network: pη(a|h), where η is the parameter +of the network, and h is the state embedding. To efficiently model the joint distribution of +discrete variables in a, we factorize pη autoregressively and use MADE (masked autoencoder +for density estimation) as the model architecture. The network contains 3 layers and 630 +hidden units for each layer. +The general idea of the policy network is similar to the previous version of DeepLig- +S-16 + +Builder.S4 The major difference is that we now use a discretized action space for the 3D +positioning of new atoms. Previously, we found that modeling the continuous distribution +of atom positions faces numerical issues, and proposed SoftMADE to address those issues. +However, SoftMADE works by adding noise to the 3D coordinates, which reduces the ac- +curacy of the model. Here in DeepLigBuilder+, we use a two-step discretization process, +which ensures the precision of the distribution and can also avoid numerical instabilities in +continuous distributions. +1.5 +Dataset and network training +1.5.1 +Training the unconditional model +We use a dataset containing drug-like molecules randomly assembled using the building +blocks in the Enamine global stock as the training set. The assembling process follows a +step-wise procedure, which is initialized with a random building block sampled from the +dataset. At each step, the possible reactions that can happen to the molecule are enumer- +ated. The reaction type is then randomly selected from the results, and the next reactant +is sampled from the building block set based on the selected reaction type. Molecules are +assembled with three reactants combined using two reaction steps. Several filters are applied +to obtain drug-like molecules, including (1) Lipinski’s rule-of-5 (Ro5),S19 (2) Veber’s rule,S20 +(3) PAINS patterns,S21 and (4) a QEDS22 threshold of 0.5. After this process, we obtained +approximately 1 million (974,917) molecules, with 4/5 of which used as the training set, and +the rest used for validation and testing. The 3D conformers of those molecules are generated +using RDKit, by first using ETKDG to embed the molecules into 3D space, and then opti- +mizing them using MMFF94s. To create more stable conformers, at most 10 conformers are +generated for each molecule, and the one with the lowest energy is used for model training. +Finally, those molecular structures are converted to synthons and subsequently transformed +into generation trajectories to train the unconditional transformer. +The network is implemented using PyTorch, and Adam is used for model optimization,S23 +S-17 + +with a linear learning rate warm-up of one epoch to 0.001, followed by an exponential learning +rate decay. The decay rate is 0.01, and several decay frequencies are experimented with (see +Table S1). The batch size is set to 1024, and the model is trained for 100 epochs using 4 +A100 GPUs, which may take 1-2 days to complete. +To train the shape and pharmacophore-conditioned model, a dataset of input-output +pairs is constructed. First, we extracted a set of ligand-based pharmacophore models and +shapes from the 3D ligands in the PDBBind 2020 dataset.S24 We use PDBBind as the +data source due to its ligand diversity. +It not only contains drug-like ligands, but also +metabolites, peptides, fragments, and other types of ligands that lack drug-likeness but are +still frequently used in pharmacophore extraction and interaction analysis. In this way, we +can increase the diversity of the pharmacophore and shape inputs. Note that we do not +use the protein structure and bioactivity values inside the PDBBind dataset, therefore the +extracted information is fully unlabeled. Future research may also consider that information +to improve the quality of the extracted pharmacophores. +An overall 12,456 pharmacophores and shapes are extracted. Next, we align the assem- +bled molecules to the extracted pharmacophores and filter those with a good match to form +the training set. This process requires 974, 917 × 12, 456 3D alignment operations, and due +to its time cost, we utilize an approach based on sequential filtering: +(1) First, we perform similarity calculation based on USRCAT fingerprints, and filter the +top 10,000 most similar molecules to each pharmacophore and shape query. +Since +USRCAT similarity does not involve 3D alignment, it can be carried out with high +efficiency inside GPU; +(2) Next, we perform 3D alignments between the 10, 000 × 12, 456 pairs of molecules and +queries based on 3D shape. The shapes are represented as a combination of 3D Gaus- +sian functions, as described previously,S12 each “colored” with a pharmacophore type +assigned using a set of SMARTS patterns defined in RDKit. PCA is then performed on +the point sets to obtain the principle axes, and those axes are aligned to form the initial +S-18 + +pose. 4 candidate poses are created by rotating 180◦ around each axis. A gradient- +based optimization process is then used to tune the rotation and translation to achieve +the best overlap between two shapes. The alignment process is accomplished using an +in-house PyTorch program. Shape similarity is computed using the aligned pose, and +the top 100 most similar molecules are retained for each pharmacophore-shape query. +(3) After shape-based alignment and filtering, a more refined pharmacophore-based filter- +ing is used to further enrich molecule-query pairs with a good match. At this step, +we retain the top 10 most similar molecules for each pharmacophore and shape-based +query. +The filtering process described above creates a dataset containing 10×12, 456 = 124, 560 +input-output pairs for model training. When training the conditional model, we use the pre- +trained unconditional model as the base model and add pharmacophore and shape-related +layers at the tail of the transformer. +To avoid overtraining, only the parameters of the +newly added layers are allowed to change. The training is performed for 160 epochs and the +learning rate decay is performed for every 30 steps. Other hyperparameters for training the +conditional model are similar to that used to train its unconditional counterpart. +1.6 +Monte Carlo tree search +Monte Carlo tree search is a widely used technique in reinforcement learning which finds +promising solutions for a given problem by strategically expanding the search tree. In this +work, we combine MCTS with the pharmacophore and shape-conditioned transformer for the +design of synthesizable 3D molecules inside a given pocket. In order to search for promising +molecular structures, MCTS maintains a look-ahead tree T and iteratively builds T using +four steps: selection, expansion, simulation, and backpropagation. DeepLigBuilder+ uses +a variant of MCTS that includes several modifications to better suit it to the 3D molecule +generation tasks. In this section, we first describe the data structure of the look-ahead tree +S-19 + +T , then discuss how the tree is updated at each step, and finally specify the details of the +hyperparameters used during MCTS runs. +The look-ahead tree in MCTS is used to store the history of previous visits, with each node +representing an intermediate state during molecule generation, and each edge representing +an action carried out at each step. +In DeepLigBuilder+, we introduced several custom +modifications in the data structure of nodes and edges: +(1) Edges in the tree contain topological and 3D actions applied to the molecular graph at +each step. The 3D action is discretized using the method introduced in Section 1.4.4. +Note that the edge only stores the value of the φ coordinate, or torsion angle. This +is because the bond lengths r and bond angles θ are largely determined by the bond +types and ring sizes, which are already stored in the topological actions. +(2) The torsion angles of new atoms are stored in a coarse-grained form, that is φcrude. +In this way, nodes in T now represent sets of molecule structures with similar 3D +conformations. This has a similar effect of clustering intermediate states using the +torsion fingerprint. +At each step of MCTS, the following operation are carried out consecutively: +(1) The selection operation, which chooses a promising state from the tree based on its +estimated value function. We follow MENTSS25 and use E2W (Empirical Exponential +Weight) to generate the selection policy. As mentioned that each node represents a +cluster of states with the same topological structure and similar 3D conformation. We +randomly select one of the states from the cluster; +(2) The expansion operation, which enumerates all possible actions that can be carried out +given a state. In practice, we found that although the allowed action space for the 3D +generative model is large, generally only a small subset of actions will be selected by +the model. Based on this observation, we first perform multiple independent sampling +S-20 + +of actions given the state selected using the transformer, and cluster the actions based +on their topological action and torsion angle, as described previously. This procedure is +similar to pruning branches in the search tree with a small probability of being chosen +by the transformer; +(3) The simulation operation, in which a full rollout is performed based on the selected +state, and the results are evaluated using the Smina scoring function. Note that all new +states created in the expansion operation are used to perform the rollout, which acts +as a form of leaf-level parallelism.S26 Additionally, all subsequent actions generated at +each rollout are added to the tree, which may help to reduce the instability during the +tree search. +Compared with the previous version of DeepLigBuilder, during the rollout, we uses a +shape and pharmacophore-conditioned generative model +When evaluating the generated outcome, DeepLigBuilder+ uses a soft version of the +Smina score as the reward function: +R(m) = softplus(−S(m)) + �3 +i=1 softplus(−S(si)) +2 +Where m is generated molecule, si, i ∈ {1, 2, 3} are the synthons fragments composed +of the molecule, S(·) is the Smina score (evaluated directly without minimization), and +softplus(x) = ln(1+exp(x)). Adding softplus to the equation helps to reduce large penalties +from the clashes with the pocket, making the reward function softer. +(4) The backtracking operation, in which the calculated rewards are used to update the Q +value estimates for each edge. Following MENTS, we use the soft-bellman backup as +the operator to update the Q values. +The number of rollout steps for each case study is set to 100. +To fully utilize GPU +resources, tree-level, root-level, and leaf-level parallelism are applied.S26 During selection, +S-21 + +an overall of 16 nodes is selected at once for simulation. Also, 20 trees are constructed +independently for each case study. During expansion, 32 actions are sampled from the full +action space for clustering, and those actions are all used to form updated states for rollout. +There are two major parameters controlling the balance between exploration and exploitation +in MENTS,S25 the temperature parameter τ and the exploration parameter ϵ in E2W. In +this work, we have τ = 0.25 and ϵ = 0.05. The MCTS program uses one NVIDIA 3070 +graphics card with 1 CPU core. +1.7 +Model evaluation +Several evaluations are performed to examine the performance of DeepLigBuilder+ in dif- +ferent aspects. +1.7.1 +The unconditional generation tasks +For the unconditional model, we investigate whether it can generate drug-like molecules +with high-quality 3D structures. +To this end, a dataset containing molecules randomly +assembled from the building blocks without drug-likeness filters is constructed as the target +of comparison, and the following evaluation metrics are employed: +(1) Metrics related to the molecular properties. +For each molecule, a series of 2D or +3D properties are calculated, including molecular weight, LogP, QED, the number of +hydrogen bond donors and acceptors, the number of rotatable bonds, total and po- +lar solvent accessible surface areas, and the radius of gyration. The distributions of +those properties are then compared between the generated molecules, the assembled +molecules, and the test set molecules. To make the comparison, the mean and stan- +dard deviation values are calculated for each property in each dataset. We also use a +metric calculated using the RMSD between mean and standard deviation values for +a quantitative measurement: W = +� +(µ1 − µ2)2 + (σ1 − σ2)2. Mathematically, this +S-22 + +metric is equivalent to the Wasserstein distance between Gaussian approximations of +two distributions. +(2) For a more quantitative measurement of whether the model correctly constructed the +drug-like chemical space of synthesizable molecules, we evaluate the MMDS27 between +generated molecules and test set molecules using 2D (morgan) and 3D (USRCAT) +fingerprints. +MMD, which stands for the maximum mean discrepancy, is a widely +used technique for evaluating the differences between distributions, and are applied in +several previous works for the evaluation of molecule generative models.S4,S28 MMD +can be calculated as follows: +MMD2 +u(X, Y ) = +1 +m(m − 1) +� +i̸=j +k(xi, xj) + +1 +n(n − 1) +� +i̸=j +k(yi, yj) − 2 +mn +� +ij +k(xi, yj) +Where X = {xi}m +i=1 and Y = {yi}n +i=1 are two datasets to be compared. k is the kernel +function based on either the Morgan or USRCAT fingerprint. +(3) To evaluate the quality of 3D structures, we first examine the quality of local geometries +of generated molecules. Similar to the previous work,S4 we compare the distribution +of torsion angles between generated molecules and the test set molecules. The en- +vironments are described using torsion SMARTS patterns by Schärfer et al.S29 The +difference between torsion angle distributions is quantized using MMD. We use the +cosine values of torsion angle difference as kernel function when calculating the MMD. +(4) To further access the quality of generated conformers by the model, we optimize each +generated molecule using the MMFF94s force field and calculate the RMSD value +between conformers before and after the optimization. To provide a context of the +model’s performance, we also perform this evaluation on conformers generated using +the ETKDGS30 method. This method is initially proposed as a faster alternative for +forcefield-based conformation optimization. +S-23 + +1.7.2 +The structure-based generation tasks +When evaluating the performance of DeepLigBuilder+ in structure-based generation tasks, +we mainly focus on answering the following two questions: +(1) Can MCTS help enrich molecules with high docking scores? +(2) Can shape-based and pharmacophore-based conditional rollout policy help MCTS to +discover better results faster? +A series of ablation studies are performed to answer those two questions. First, in terms +of the benefit of MCTS-based sampling, we compare the distribution of Smina docking scores +for molecules generated with or without MCTS. When calculating the Smina scores, we first +move the generated molecules out of the protein pocket, perform local relaxation for each +molecule using the MMFF94s forcefield, and then re-dock the relaxed conformers back into +the target pocket using Smina. +To access the benefit of the conditional rollout policy, the following studies are performed: +(1) We determine whether the conditional model can achieve enrichment in pharmacophore +and shape compared with its unconditional counterpart. This question can be answered +by examining the distribution of shape and pharmacophore similarity between gener- +ated molecules and input queries. +(2) We investigate whether the extra conditional inputs can help MCTS to achieve faster +search speeds. +To this end, ablation studies are performed by enabling and then +disabling the conditional inputs for the rollout policy. At each MCTS step, we record +the reward value for the best molecule found so far. We then compare whether the +conditional rollout policy can help MCTS reach solutions with higher rewards faster. +2 +Supplementary Figures +S-24 + +Figure S1: The process of converting a reactant in the building block dataset into a synthon. +S-25 + +Reactant: +Reaction +Schotten-Baumann +0H +EN300-729603 +Convert to the reaction product +个 +CH,CH,NH, +NHCH,CH3 +Extract substrcture +as synthon +Anchor label +Reaction: Schotten-Baumann +NHCH,CH, +Role: carboxylic acid +SynthonFigure S2: Special treatment is needed when converting reactants to synthons when the +reaction involves ring formation. +S-26 + +Huisgen reaction +OH +H3CH2C- +NN +EN300-107725 +H3CH2C +H3CH2C. +H3CH2C +N +orFigure S3: A representation of the molecule generation process using a finite state machine. +From the figure, we can see that the model moves back and forth between ring generation +(represented as the “searching ring target” state and the “generating ring” state) and chain +generation (represented as the “generating chains” state and the “backtracking” state). +S-27 + +START +END +Initialize +Terminate +Backtrack +Append +Generating chains +Backtracking +Backtrack +Append +CloseLoop +SearchLoopTarget +Searching ring target +StartLoop +Generating rings +SearchLoopTarget +AppendFigure S4: The process of generating a 3D molecule using DeepLigBuilder+ +S-28 + +APPEND +APPEND +APPEND +Search +StartLoop +APPEND +INIT +(C,-,C,X。) +(C,=,C,X) +(C,-,r,x2) +LoopTarget +size=5 +(C,ar,C,X,) +2/5 +3/5 +4/5 +5/5 +APPEND +APPEND +APPEND +APPEND +APPEND +CloseLoop +Backtrack +(C,ar,C,X4) +(C,ar,b,x,) +(C,ar,C,X。) +(C,ar,c,X,) +(C,-,b,Xg) +b +b +APPEND +APPEND +Backtrack +Backtrack +Backtrack +Backtrack +TER +(N,-,C,Xg) +(O,=,C,X1) +Carbon +Focused atom +Loop target +Single bond +Nitrogen +Branching atom +Loop count down +Double bond +0/5 +Aromatic bond +Oxygen +Potential loop targetFigure S5: Issues related to ring generation when the synthon dataset is changed, and how +the refined ring-based generation scheme can alleviate this problem. a. The original synthon +dataset, and the structures of generated molecules. b. The six-membered ring is removed +from the dataset. Without retraining, the model will not acknowledge this change. As a +result, the model will still attempt to generate six-membered rings, but the generation will be +terminated in advance due to the constraints on available synthons, resulting in problematic +structures shown below. c. However, if the ring size is determined before generating its +structures, the model will acknowledge the change in the synthon dataset, since the action of +generating a six-membered ring is masked due to the imposed synthon constraints, resulting +in molecule structures with higher quality. +S-29 + +a. +DeepLigBuilder+ +Synthon dataset +Generated structure +b. +Removed +DeepLigBuilder+ +Synthon dataset +Generated structure +Removed +Blocked +DeepLigBuilder+ +Synthon dataset +Generated structureFigure S6: Examples of several generated 3D molecules using the unconditional transformer, +along with the building blocks and proposed reactions to synthesize the molecules. +S-30 + +Structure S1 +Structure S2 +Structure S3 +Topological +structure +HC +Building +HaN +blocks +HO +N +NH2 +N +(1) +(2) +(3) +(1) +(2) +(3) +(1) +(2) +(3) +(1) + (2): +(1) + (2) & (1-2) + (3): +(1) + (2): +Grignard reactiont +Reactions +Oxadiazole formation +Schotten-Baumann reaction +(1-2) + (3): +using hydroxylamine +(1-2) + (3): Wittig reaction +Schotten-Baumann reaction +Generated +structuresFigure S7: The distribution of 2D and 3D molecular properties of the generated (blue), test +set (grey), and randomly assembled (red) molecules. 2D properties includes: a. Molecular +weight, b. LogP, c. QED, d. the number of hydrogen bond acceptors (HBA), e. the number +of hydrogen bond donors, f. The number of rotatable bonds (ROT). 3D properties includes: +g. The total amount of solvent accessible surface area (SASA), h. polar solvent accessible +surface areas (PolarSASA), i. the radius of gyration. +S-31 + +b. +C. +a. +Density +Generated +Assembled +Test set +300 +400 +500 +600 +0 +2 +4 +6 +0.2 +0.4 +0.6 +0.8 +1.0 +Molecular Weight +LogP +QED +f. +d. +e. +Density +Generated +Assembled +Test set +0 +5 +10 +0 +2 +4 +6 +5 +10 +0 +15 +HBA +HBD +ROTB +h. +i. +g. +Density +Generated +Assembled +Test set +500 +600 +700 +800 +900 +100 +200 +300 +4003 +4 +5 +6 +7 +SASA +Polar SASA +Radius of gyrationFigure S8: A t-SNE visualization of the distribution of a. Morgan fingerprint and b. US- +RCAT fingerprint. Blue dots represent generated molecules, grey dots represent test set +molecules, and red dots represents molecules randomly assembled from the building blocks. +Figure S9: +The distribution of RMSD values after relaxation with MMFF94s. +Blue: +molecules with conformations generated by the network. Grey: molecules with conforma- +tions generated by the ETKDG method provided by RDKit +S-32 + +b. +a. +Generated + Assembled +• Test set1.2 +1.0- +Probability Density +0.8 +0.6 +0.4 +0.2 +0.0 - +ETKDG +Generated +0 +2 +3 +4 +5 +RMSD (A)Figure S10: The detailed structure of the search tree for the first synthon in the BTK’s +ATP-binding pocket (only containing nodes with visit count larger than 25). Some gener- +ated synthon structures are shown below the tree, with important pharmacophore features +highlighted. The leftmost example shows a state with a low estimated Q-value, which largely +resulted from the unfavorable anchor position. As the result, this state is not as frequently +visited as other states with higher Q-values. +S-33 + +Seed +Initial state +Q-values: +1.0 +3.0 +5.0 +7.0 +Anchor atom +Hydrophobic +Unfavorable +anchor position +HBD +HBAFigure S11: a. The topological structure of GDC-0853 (Fenebrutinib). b. The part of GDC- +0853 used extract pharmacophore, shape, and seed for molecule generation. Note that some +part of GDC-0853 inside the solvent-exposed region is not considered for pharmacophore +and shape extraction. +Figure S12: The proposed synthetic path for Structure 2 (a) and Structure 3 (b) by the +model. Functional groups marked in grey needs to be protected before the reactions. +S-34 + +a. +GDC-0853 (Fenebrutinib) +Kd = 0.91 nM +Smina score: -14.1 +b. +The sub-structure used to generate +the shape and pharmacophore +conditions, as well as the seed +Solvent +structure for molecule growth +exposed +region +Smina score: -13.6 +O HBD +● HBA +○ Hydrophobic + Shape +o Seed atomNegishi +reaction +EN300-6496494 +Grignard reaction +EN300-72460 +EN300-50850 +Structure 2 +b +Huisgen +reaction +EN300-79402 +Buchwald-Hartwig +reaction +EN300-20284 +EN300-107725 +Structure 3Figure S13: a. The topological structure of Compound 15, a potent inhibitor targeting the +NAD pocket of PHGDH. b. The pharmacophore features extracted from the binding mode +of the molecule, as well as the seed location for molecule growth. +Figure S14: The proposed synthetic path for Structure 6 (a) and Structure 7 (b) by the +model. Functional groups marked in grey needs to be protected before the reactions. +S-35 + +a. +Cl +1 +Compound 15 (Mullarky et. al.) +OHBD + HBA● Hydrophobic +Kd = 15 nM +Negative charge :: Shape +Seed atom +Smina score: -9.9Sonogashira +reaction +EN300-82406 +Structure 6 +Urea formation +EN300-27699201 +EN300-139617 +b. +Grignard +HN +reaction +EN300-27745883 +Structure 7 +Urea formation +EN300-58858 +EN300-824063 +Supplementary tables +Table S1: A summary of different hyperparameter configurations experimented in this work (BBs: building blocks, LR: learning +rate) +Model +architecture +Training +parameter +Generation +parameter +Group +Variant name +IPA +3D pair +embedding +Width +(F) +DepthDecay frequency +(#steps) +Noise +probability +Noise +scale(Å) +Building +block +Default +configuration +✓ +✓ +512 +6 +150 +0.5 +0.1 +Global +BBs +EU stock +✓ +✓ +512 +6 +150 +0.5 +0.1 +EU +Comprehensive +catalog +✓ +✓ +512 +6 +150 +0.5 +0.1 +Comprehensive +3D en- +coding +Dropped IPA +× +✓ +512 +6 +150 +0.5 +0.1 +Global +Dropped 3D pair +embedding +✓ +× +512 +6 +150 +0.5 +0.1 +Global +Scale +Narrow network +✓ +✓ +256 +6 +150 +0.5 +0.1 +Global +Shallow network +✓ +✓ +512 +3 +150 +0.5 +0.1 +Global +LR +Fast learning +rate decay +✓ +✓ +512 +6 +70 +0.5 +0.1 +Global +S-36 + +Model +architecture +Training +parameter +Generation +parameter +Slow learning +rate decay +✓ +✓ +512 +6 +300 +0.5 +0.1 +Global +Noise +High noise +probability +✓ +✓ +512 +6 +150 +0.9 +0.1 +Global +Low noise +probability +✓ +✓ +512 +6 +150 +0.1 +0.1 +Global +High noise scale +✓ +✓ +512 +6 +150 +0.5 +0.2 +Global +Low noise scale +✓ +✓ +512 +6 +150 +0.5 +0.05 +Global +S-37 + +Table S2: The distribution of 2D molecular properties among molecules generated from models with different hyperparameters +(see Table S1 for a detailed description of each configuration). We report the mean and standard deviation for each property, +as well as the estimated Wasserstein distance to the test set. For the first row, we report the statistics of molecules randomly +assembled from building blocks without any drug-likeness filtering. +Model variant +MW +LogP +HBA +HBD +ROT +QED +mean +std +wd +mean std +wd +mean std +wd +mean std +wd +mean std +wd +mean std +wd +Randomly assembled +454.32 47.45 35.46 3.48 +1.37 +0.57 +6.81 +1.64 +0.7 +1.91 +1.12 +0.33 +6.96 +2.22 +0.93 +0.45 +0.15 +0.18 +Default configuration +412.31 42.39 7.64 +2.9 +1.2 +0.12 +6.13 +1.54 +0.08 +1.69 +0.98 +0.08 +6.05 +1.85 +0.12 +0.62 +0.12 +0.03 +EU stock +416.63 43.96 4.84 +3.02 +1.2 +0.11 +6.12 +1.51 +0.06 +1.76 +1.02 +0.14 +6.13 +1.93 +0.16 +0.60 +0.13 +0.05 +Comprehensive +catalog +415.46 42.65 4.85 +2.91 +1.21 +0.13 +6.15 +1.51 +0.05 +1.73 +0.99 +0.1 +6.12 +1.86 +0.09 +0.62 +0.12 +0.03 +Dropped IPA +413.22 42.51 6.82 +2.94 +1.19 +0.09 +6.07 +1.51 +0.08 +1.69 +0.97 +0.07 +6.07 +1.86 +0.12 +0.62 +0.12 +0.03 +Dropped 3D pair +embedding +414.91 41.63 4.93 +2.93 +1.16 +0.07 +6.11 +1.5 +0.05 +1.66 +0.97 +0.05 +6.07 +1.82 +0.08 +0.62 +0.11 +0.03 +Narrow network +411.01 42.46 8.91 +2.95 +1.2 +0.1 +6.04 +1.52 +0.11 +1.68 +0.98 +0.07 +6.03 +1.86 +0.14 +0.62 +0.12 +0.04 +Shallow network +412.79 42.25 7.14 +2.93 +1.2 +0.11 +6.1 +1.52 +0.07 +1.69 +0.98 +0.08 +6.06 +1.88 +0.13 +0.62 +0.12 +0.03 +Fast learning rate +decay +411.36 42.44 8.57 +2.94 +1.19 +0.1 +6.06 +1.54 +0.11 +1.7 +0.98 +0.08 +6.03 +1.89 +0.16 +0.62 +0.12 +0.04 +Slow learning rate +decay +413.53 42.63 6.58 +2.94 +1.19 +0.09 +6.12 +1.52 +0.06 +1.67 +0.99 +0.08 +6.11 +1.87 +1.87 +0.62 +0.12 +0.03 +High noise probability +414.14 41.36 5.61 +2.94 +1.17 +0.07 +6.09 +1.5 +0.06 +1.69 +0.99 +0.08 +6.07 +1.82 +0.09 +0.62 +0.12 +0.03 +S-38 + +Model variant +MW +LogP +HBA +HBD +ROT +QED +Low noise probability +416.82 45.94 6.43 +2.89 +1.26 +0.18 +6.3 +1.55 +0.19 +1.72 +1.03 +0.13 +5.91 +1.9 +0.26 +0.6 +0.14 +0.05 +Large noise scale +413.76 41.78 6.08 +2.94 +1.18 +0.08 +6.12 +1.5 +0.05 +1.71 +0.98 +0.09 +6.03 +1.81 +0.11 +0.62 +0.12 +0.03 +Small noise scale +413.24 42.9 +6.95 +2.93 +1.19 +0.1 +6.14 +1.53 +0.07 +1.7 +0.99 +0.09 +6.03 +1.86 +0.14 +0.62 +0.12 +0.04 +Validation set +419.52 40.27 - +2.98 +1.1 +- +6.13 +1.46 +- +1.65 +0.92 +- +6.14 +1.77 +- +0.63 +0.09 +- +Test set +419.72 40.13 - +2.97 +1.11 +- +6.14 +1.46 +- +1.46 +0.92 +- +6.14 +1.77 +- +0.62 +0.09 +- +0.09 +S-39 + +Table S3: The distribution of 3D molecular properties among molecules generated from models with different hyperparameters +(see Table S1 for a detailed description of each configuration). We report the mean and standard deviation for each property, +as well as the estimated Wasserstein distance to the test set. For the first row, we report the statistics of molecules randomly +assembled from building blocks without any drug-likeness filtering. +Model variant +SASA +Polar SASA +Radius of gyration +mean +std +wd +mean +std +wd +mean +std +wd +Randomly assembled +667.72 +60.84 +41.41 +185.77 +58.93 +26.24 +4.9 +0.71 +0.3 +Default configuration +617.86 +53.27 +10.07 +162.93 +49.92 +2.96 +4.56 +0.63 +0.05 +EU stock +624.14 +55.58 +6.01 +163.68 +51.66 +4.83 +4.61 +0.64 +0.03 +Comprehensive catalog +620.45 +54.08 +7.91 +163.53 +49.83 +3.05 +3.05 +0.63 +0.04 +Dropped IPA +620.16 +54.23 +8.24 +162.19 +49.44 +2.44 +4.6 +0.64 +0.03 +Dropped 3D pair embedding +621.13 +52.84 +6.8 +161.87 +49.19 +2.24 +4.59 +0.62 +0.02 +Narrow network +618.08 +53.9 +10.04 +161.95 +49.53 +2.56 +4.59 +0.64 +0.03 +Shallow network +619.66 +54.05 +8.62 +162.59 +50.19 +3.19 +4.6 +0.63 +0.02 +Fast learning rate decay +616.74 +53.97 +11.34 +161.7 +49.55 +2.63 +4.58 +0.64 +0.04 +Slow learning rate decay +619.5 +53.42 +8.54 +162.15 +50.4 +3.41 +4.57 +0.63 +0.05 +High noise probability +620.89 +52.67 +6.99 +162.3 +49.31 +2.3 +4.6 +0.63 +0.02 +Low noise probability +614.44 +56.41 +14.35 +166.52 +52.61 +6.96 +4.47 +0.6 +0.15 +Large noise scale +619.44 +52.9 +8.45 +162.75 +50.11 +3.12 +4.59 +0.63 +0.03 +Small noise scale +618.84 +54.05 +9.38 +162.16 +49.65 +2.65 +4.57 +0.63 +0.05 +Validation set +627.61 +50.87 +- +162.5 +47.12 +- +4.62 +0.62 +- +S-40 + +Model variant +SASA +Polar SASA +Radius of gyration +Test set +627.52 +50.49 +- +162.57 +46.89 +- +4.61 +0.61 +- +S-41 + +Table S4: Quantitative measurement of the quality of generated samples. The first two columns indicate the sample diversity +measured using Tanimoto and USRCAT fingerprints. The third and fourth column shows the 2D and 3D MMD values, which +measures the ability of the network to correctly model the distribution in the chemical space. +The final column contains +the average RMSD after conformers are relaxed using MMFF94s forcefield. Each row represents a different hyperparameter +configuration, as detailed in Table S1. The first row represents molecules randomly assembled from building blocks without any +drug-likeness filtering. +Model variant +Diversity(2D) +Diversity(3D) +MMD(2D) +MMD(3D) +Mean RMSD(Å) +Randomly assembled +0.120 +0.156 +0.003604 +0.003492 +1.142 (ETKDG) +Default configuration +0.121 +0.162 +0.000158 +0.000156 +0.692 +EU stock +0.123 +0.161 +0.000798 +0.000109 +0.725 +Comprehensive catalog +0.121 +0.162 +0.000134 +0.000122 +0.691 +Dropped IPA +0.121 +0.161 +0.000195 +0.000068 +0.696 +Dropped 3D pair embedding +0.122 +0.162 +0.000082 +0.000045 +0.718 +Narrow network +0.121 +0.161 +0.000256 +0.000066 +0.716 +Shallow network +0.121 +0.161 +0.000204 +0.000059 +0.707 +Fast learning rate decay +0.121 +0.161 +0.000224 +0.000126 +0.734 +Slow learning rate decay +0.121 +0.162 +0.000183 +0.000120 +0.672 +High noise probability +0.122 +0.162 +0.000171 +0.000060 +0.685 +Low noise probability +0.120 +0.165 +0.000475 +0.000954 +0.795 +Large noise scale +0.122 +0.162 +0.000163 +0.000080 +0.685 +Small noise scale +0.121 +0.162 +0.000212 +0.000143 +0.713 +S-42 + +References +(S1) Hartenfeller, M.; Eberle, M.; Meier, P.; Nieto-Oberhuber, C.; Altmann, K.-H.; Schnei- +der, G.; Jacoby, E.; Renner, S. 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The PDBbind Database: Collection of Binding +Affinities for Protein-Ligand Complexes with Known Three-Dimensional Structures. +Journal of Medicinal Chemistry 2004, 47, 2977–2980. +S-45 + +(S25) Xiao, C.; Huang, R.; Mei, J.; Schuurmans, D.; Muller, M. Maximum entropy monte- +carlo planning. Advances in Neural Information Processing Systems 2019, 32, 9520– +9528. +(S26) Chaslot, G. M. J. B.; Winands, M. H. M.; Herik, H. J. v. d. Parallel Monte-Carlo tree +search. Proceedings of 6th International Conference on Computers and Games (CG) +2008, 5131, 60–71. +(S27) Gretton, A.; Borgwardt, K. M.; Rasch, M. J.; Scholkopf, B.; Smola, A. A kernel +two-sample test. The Journal of Machine Learning Research 2012, 13, 723–773. +(S28) Li, Y.; Hu, J.; Wang, Y.; Zhou, J.; Zhang, L.; Liu, Z. DeepScaffold: A Comprehensive +Tool for Scaffold-Based De Novo Drug Discovery Using Deep Learning. 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Journal of Chemical Information and +Modeling 2015, 55, 2562–2574. +S-46 + diff --git a/YtAyT4oBgHgl3EQfWvcM/content/tmp_files/load_file.txt b/YtAyT4oBgHgl3EQfWvcM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..89f60e211d61fbf271a4a67bf24be871f0e37b2d --- /dev/null +++ b/YtAyT4oBgHgl3EQfWvcM/content/tmp_files/load_file.txt @@ -0,0 +1,2737 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf,len=2736 +page_content='Synthesis-driven design of 3D molecules for structure-based drug discovery using geometric transformers Yibo Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='† Jianfeng Pei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='‡ and Luhua Lai∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='¶ † Center for Life Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Academy for Advanced Interdisciplinary Studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' China ‡ Center for Quantitative Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Academy for Advanced Interdisciplinary Studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' China ¶ BNLMS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' College of Chemistry and Molecular Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' China E-mail: jfpei@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' lhlai@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='cn Abstract Finding drug-like compounds with high bioactivity is essential for drug discovery, but the task is complicated by the high cost of chemical synthesis and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' With their outstanding performance in de novo drug design, deep generative models represent promising tools for tackling this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In recently years, 3D molecule generative models have gained increasing attention due to their ability to directly utilize the 3D in- teraction information between the target and ligand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' However, it remains challenging to synthesize the molecules generated by these models, limiting the speed of bioactivity val- idation and further structure optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this work, we propose DeepLigBuilder+, a deep generative model for 3D molecules that combines structure-based de novo drug 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='00167v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='QM] 31 Dec 2022 design with a reaction-based generation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Besides producing 3D molecu- lar structures, the model also proposes synthetic pathways for generated molecules, which greatly assists the retro-synthetic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To achieve this, we developed a new way to enforce the synthesizability constraint using a tree-based organization of purchasable building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This method enjoys high scalability and is compatible with existing atom-based generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Additionally, for structure-based design tasks, we developed an SE(3)-equivariant transformer conditioned on the shape and pharmacophore-based inputs, and combine it with the Monte Carlo tree search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Us- ing the ATP-binding pocket of BTK and the NAD+ binding pocket of PHGDH for case studies, we demonstrate that DeepLigBuilder+ is capable of enriching drug-like molecules with high predicted binding affinity and desirable interaction modes while maintaining the synthesizability constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We believe that DeepLigBuilder+ is a powerful tool for accelerating the process of drug discovery, and represents an impor- tant step towards a fully automated design-synthesis-evaluation workflow for molecule design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1 Introduction The high financial cost and low success rate of drug discovery place tremendous challenges in finding treatments for important diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 To address those challenges, computational methods have been developed to find promising compounds from the vast space of chemical structures for subsequent biological validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Computational virtual screening (VS) have been widely used, which filters chemical libraries using scoring functions2 for favorable com- pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In spite of their success in finding bioactive molecules,3 VS is constrained by the screening library it uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Small libraries may have limited coverage of the chemical space, and large ones impose high computational costs for the screening process and require spe- cialized software and hardware platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4 De novo drug design, which uses computational algorithms to generate molecule structures from scratch,5 provides an option to explore new 2 chemical space beyond libraries of existing compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Over the decades, a variety of de novo drug design programs have been proposed, such as LEGEND,6 LUDI,7 CONCEPTS8 and LigBuilder,9–11 many of which have been used to design bioactive molecules with successful experimental validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='12–16 In recent years, deep molecule generative models have emerged as a new class of promis- ing methods for de novo drug design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='17 Using deep learning, models can automatically learn traits of desirable molecule structures from the training data, with little need for manual intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This contrasts significantly with traditional de novo design programs, which in general require extensive efforts to design the search rules and scoring functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The advantages of deep generative models have helped to spawn a series of research aiming to utilize them for drug discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Those works range from exploring different molecule repre- sentations, including SMILES18 and molecular graph,19–21 to testing with various training methods, such as VAE,22 GAN23,24 and RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='25,26 As a result, a wide range of models has been proposed to address various issues related to drug design based on molecular properties,22 pharmacophores,27 scaffolds28 and targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='29 Most deep generative models for molecules have been focused on designing the 2D (topo- logical) chemical structures, but the foundation of bioactivity lies in the interactions be- tween the 3D structures of targets and ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Directly generating 3D molecules inside the target binding pocket can help the model to better utilize the interaction information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Additionally, it can reduce the need for ligand-based information, potentially leading to molecules with higher novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Those benefits have led to growing attention in developing 3D generative models of molecules based on target information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Earlier approaches include models that convert 3D pocket information into SMILES strings, such as LiGANN30 and the pocket-conditioned RNN proposed by Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=',31 but the generated structure only con- tains topological information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In order to directly generate 3D structures, Masuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='32 proposed liGAN, which uses VAE based on 3D-CNN to generate atomic density grids, and later convert the grids to 3D molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To avoid the conversion between different represen- 3 tations, we previously proposed DeepLigBuilder33 to directly produce 3D molecules inside pockets using graph generative models and Monte Carlo tree search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Other graph genera- tive models, such as Pocket2Mol,34 use equivariant networks to encode pocket information as conditional inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' More recent works have experimented with diffusion models for 3D molecule generation, such as DiffLinker,35 which features a permutation invariant way of generation compared to autoregressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Despite progress in structure-based 3D deep generative models, there is still a critical issue to be addressed: the synthesizability of generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Chemical synthesis is a common rate-limiting step in medicinal chemistry research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Since de novo design programs are not constrained by any compound library, it is easier for these methods to propose molecules that are challenging to synthesize, especially in objective-directed situations,36 making experimental validation difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A solution to this problem is to use synthetically aware models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='37 Those models generate molecules by generating their synthetic path, us- ing explicit building blocks and chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Such approaches have been relatively common in traditional de novo design programs, such as SYNOPSIS38 and DOGS,39 but is largely absent from early deep learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' More recently, an increasing number of models have been proposed to integrate this approach with deep generative networks, includ- ing MoleculeChef,40 DoG-AE and DoG-Gen,41 PGFS42 and SynNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='43 Those methods have shown promising results, but they are largely focused on 2D molecule design, which, as dis- cussed before, inherits several limitations compared to recent 3D generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Based on the discussions above, we believe that it is highly beneficial to develop a deep generative model that can perform pocket-based 3D molecule design while ensuring the synthesizability of the generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this work, we combine geometric deep learning and synthesiz- ability constraints to develop a new de novo drug design program, DeepLigBuilder+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The program follows a reaction-based scheme for generating drug-like molecules, while at the same time produces their 3D conformations, making it easy to be applied to structure-based design tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Specifically, we use a transformer network to generate 3D molecular graphs 4 atom-by-atom, while at each step, we mask inappropriate atom and bond types from the action space so that the output structure is guaranteed to be inside the user-provided reac- tant dataset (represented as synthons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In order to incorporate 3D pocket information, we trained an SE(3)-equivariant transformer network conditioned on pharmacophore and shape information, and combine it with a reinforcement learning module based on Monte Carlo tree search (MCTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To demonstrate the capability of DeepLigBuilder+ in drug design applica- tions, we use it to design inhibitors targeting the ATP-binding pocket of Bruton’s tyrosine kinase (BTK), as well as the NAD+-binding pocket of human phosphoglycerate dehydroge- nase (PHGDH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In both cases, DeepLigBuilder+ generated molecules with high predicted binding affinity and favorable binding modes while maintaining the enforced synthesizability constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 2 Methods In this section, we give a brief account of the architecture of DeepLigBuilder+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The imple- mentation details for DeepLigBuilder+ are provided in the Supplementary Methods (Section S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To ensure high synthetic accessibility, DeepLigBuilder+ generates molecules one reac- tant at a time and then produces the resulting molecules using corresponding reaction rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Unlike previous methods, DeepLigBuilder+ also generates 3D conformation of the molecule for subsequent structure-based design tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Since many reactions involve large conforma- tional changes, instead of directly generating 3D structures of reactants, DeepLigBuilder+ first generates synthon structures and later covert them to the corresponding reactants, as shown in Figure 1a and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Synthons are hypothetical reactants that have one or more open valences with specific reactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A reactant can be converted to a synthon by extracting the substructure of the product that is derived from this reactant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this way, adding a new synthon to the molecule will not affect the conformation of previous synthons, making 5 Figure 1: An overview of DeepLigBuilder+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' It generates synthesizable 3D molecules follow- ing a reaction-based method using synthons (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' When producing each synthon, the model adopts a graph-based generation scheme (e) that iteratively edits the molecular graph by adding new nodes (atoms) or edges (bonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The decisions of how to perform those edits are made by a transformer network (a-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The encoders are used to process the input in- formation, such as previously generated synthons (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For structure-based generation tasks, the encoder also receives shape(b) and pharmacophore(c)-based inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The decoder (e) uses those input information to produce a state embedding, which is later used by the policy network (based on MADE blocks) to output a distribution in the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We apply action masks at each step to constrain the generation trajectory so that it only produces synthons that can be converted into purchasable building blocks (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Finally, the generated synthons can be converted to a synthetic route, with explicit reactants and reaction types (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 6 SE(3)-equivariant transformer d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Previous synthons Shape information Pharmacophore model Synthon dataset 3D Zernike coefficients Prefx tree Project to local frames 区区 Action mask p(action) Step i Stepi+ 1 Attention layer Attention layer Multi-layer perceptron Masked autoencoder Focused (IPA-based) (SDPA-based) (dense layers) for density estimation atom Synthon-based generation DeepLigBuilder+ 3D Molecule Proposed synthetic path g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' OHC (HO)2B NO2 Structure 1it more suitable for 3D generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this work, we use the global stock of Enamine building blocks as the reactant set and use the reactions collected by Hartenfeller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='44 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Each product molecule is assembled from three reactants using two reaction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Details related to the synthon dataset are given in Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' When generating each synthon structure, we adopt a graph-based approach similar to our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='33 Specifically, we treat the synthon structure as a 3D graph, and it is generated by iteratively refining the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' At each step, the model either adds a new atom or a new bond or performs other operations such as backtracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We also introduce various improvements compared to the previous method, including a more detailed treatment of ring generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Specifically, before generating each ring structure, DeepLigBuilder+ first specify the size of the ring, as well as the location the ring will be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This can better guide the generation process and can avoid potential issues when the user changes the synthon dataset (see Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The generation scheme is detailed in Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' During generation, we need to constrain the synthon structure to the space of purchasable building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To achieve this, we perform step-wise masking of the action space so that the generated structure will not leave the space of purchasable synthons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The mask is constructed by querying a prefix tree of synthon structures built from the building block dataset, as detailed in Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This new method of introducing chemical constraints offers better scalability compared to previous approaches,41,43 which usually requires a scan through the entire set of building blocks to generate the next action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' DeepLigBuilder+ uses an SE(3) equivariant transformer to decide which action to perform at each step of generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Specifically, we convert a 3D molecular graph as a sequence of actions that are used to generate its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This is equivalent to the concept of a “sentence” in NLP-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Correspondingly, each action represents a “word” in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The network adopts an encoder-decoder architecture, which is used to translate the input information, including previously generated synthons and pharmacophores (discussed 7 Figure 2: DeepLigBuilder+ uses Monte Carlo tree search (MCTS) to achieve structure-based molecule generation given the pocket structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The search starts from a user-provided seed atom (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this figure, we use the ATP-binding pocket of BTK as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' At each step, the model first selects a promising node from the look-ahead tree (b), then expands the tree by enumerating possible actions that can be performed on the 3D molecule (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Next, the model selects an expanded state and uses the conditional transformer to perform the rollout(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Finally, the generated molecule is evaluated using the Smina score, and the reward is backpropagated to parent nodes to update the Q-value estimates(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' below), into new synthon structures, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To incorporate 3D information in an equivariant manner, we attach a 3D coordinate system to the focused atom after each action and use invariant point attention (IPA)45 to communicate information between actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We also use relative 3D positional encoding to express the spatial relationship between action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For network training, we assemble a drug-like set of synthesizable molecules from the Enamine building blocks, and use the 3D structures of these molecules generated by RDKit to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' More information related to the network and its training are given in Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4 and Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To introduce 3D information of targets, Monte Carlo tree search (MCTS), a widely used algorithm in reinforcement learning, is applied to optimize the molecule structure inside the pocket (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We use a search method similar to MENTS,46 with custom modifications described in Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6, and a reward function based on the Smina score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='47 The generated 3D pose by DeepLigBuilder+ is directly used for scoring, eliminating the time-consuming docking step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 8 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Selection d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Simulation e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Backup c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Expansion Seed a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Initial state RewardsTo enhance the performance of MCTS, we developed a pharmacophore and shape-conditioned transformer as its rollout policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Pharmacophore models represent abstracted interaction patterns that can be used to explain the bioactivity of ligands and are widely used in computer-aided drug design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='48 Shape information can help the model by constraining the molecule to match the geometry of the pocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Those information are coded using SE(3) equivariant representations discussed in Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For model training, a dataset of pharmacophore-ligand pairs is created by aligning synthesizable 3D molecules to pharma- cophore features extracted from PDBBind ligands (Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The conditional rollout policy offers significant speed ups for MCTS search, as demonstrated in the following sec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 3 Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Performance of the unconditional generative model We first evaluate the performance of the transformer network in the unconditional setting, in a manner similar to our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='33 Specifically, we investigate whether the model is capable of generating drug-like and synthesizable molecules with valid 3D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Several generated molecules by the network are shown in Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A visual inspection of these molecules reveal that they all adopt reasonable 3D conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Local geometries are cor- rectly structured based on the hybridization state of their atomic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Neighbors of sp2-atoms are planarized, while that of sp3-atoms form tetrahedron structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Also, the overall conformations generated are relaxed and contain no significant clashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Those observations will be later confirmed using quantitative evaluation metrics (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' DeepLigBuilder+ is unique in that synthetic paths are also generated for each molecule along with its 3D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure 1e shows the proposed route for synthesizing structure 1, which contains explicit purchasable reactants with Enamine IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this way, the synthesis routes of generated molecules can be greatly simplified, potentially reducing the complexity 9 Figure 3: The performance of the unconditional generative network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' a-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The distribution of QED (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=') and SASA (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=') among generated (blue), randomly assembled (red), and test set (grey) molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The distribution of molecular shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A visualization of the distribution of 2D Morgan fingerprints among generated (blue), randomly assembled (red), and test set (grey) molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Boxes with dashed borders show locations in which randomly assembled (non-druglike) molecules are enriched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The 2D MMD values (negative log scale) of generated molecules using different model configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The red bar shows the values calculated using randomly assembled molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The RMSD values of generated molecules using different model configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The red bar shows the RMSD values for conformers generated using ETKDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 10 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Linear Spherical Probability density ANPR2 Generated Assembled NPR1 Planar Test set Generated molecules 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='0 500 600 700 800 900 Test set molecules QED SASA e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 10- d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Randomly assembled log MMD (2D) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 6- 4- ETKDG f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='142 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='73- nl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='71- RMSD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='67 ^ X2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='65- X1 Default No IPA Generated EU stock No relative (using different Comprehensive 3D poditional hyperparameter) Generated Assembled Test set embedding stock Frequency of the Generated g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' MMD environment in test set Test set 100 10-5 10-1 0 0 II- T Smallest MMD Rank (from best to worse) Largest MMDof wet-lab evaluations of those molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Distribution of 2D and 3D molecular properties To verify whether the model is capable of generating molecules with desirable drug-like properties, 10,000 structures are sampled from the model and several important 2D properties are calculated for each molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The properties include molecular weight, LogP, the number of rotatable bonds (ROT), hydrogen bond donors (HBD) and acceptors (HBA), as well as QED,49 which is a widely used metric for drug-likeness estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The distributions of those properties are visualized in Figure S7 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' It can be seen that the distribution of most properties matches well between the generated (blue) and test set molecules (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A majority of generated molecules (85%) have a QED value larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5, as shown in Figure 3a, indicating high drug-likeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We also compared the result with molecules randomly assembled from reactants with- out any drug-likeness filters (shown in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In general, the property distribution of these randomly assembled molecules differs significantly from that of the drug-like test set, with high molecular weight and low drug-likeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To offer a more quantitative evaluation, we calculate the sum of squared differences between the mean and the standard deviation statis- tics between generated and test-set molecules, as shown in Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Mathematically, this metric is equivalent to the Wasserstein distance between Gaussian approximations of two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The property distributions of the generated molecules are indeed more similar to the drug-like test set, compared to the randomly assembled molecules, confirming that the model can indeed significantly enrich drug-like molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Similarly, several 3D molecular descriptors are calculated to examine the method’s ability to model 3D properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure 3b and Figure S7g-i shows the distribution of solvent- accessible surface areas (SASA50), Polar SASA and the radius of gyration(Rg 51) for the generated and test set molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Like the 2D case, a close match between the property distributions is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In addition, we visualized the shape distribution of these molecules 11 using normalized PMI ratios (NPRs52), as shown in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' It can be seen that the generated molecules are enriched in the linear region, while tilted towards the planar region, following the distribution of test set molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Randomly assembled molecules show a very different distribution in most 3D properties, and quantitative measurements shown in Table S3 confirm that molecules generated by the model share higher similarity in 3D properties with the validation and test set compared with assembled ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 The ability for the model to correctly model the drug-like chemical space To access the network’s ability to model the drug-like space of synthesizable molecules, we visualize the distribution of 2D and 3D structures for the generated, test set, and assembled molecules, as shown in Figure 3d and Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Morgan and USRCAT53 fingerprints are used to represent 2D and 3D molecule structures and t-SNE54 is used for dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The figures suggest that the overall distribution matches well between the generated (blue) and test set (grey) molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' On the other hand, there are regions enriched with randomly assembled molecules (shown as dashed boxes in Figure 3d), which likely represent locations in chemical space featuring low drug-likeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To offer a more quantitative evaluation, we use maximum mean discrepancy (MMD55) to measure the overlap in the chemical space for generated and test-set molecules, as done in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='28,33 MMD is a metric used to determine the dissimilarity between two prob- ability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Here, we also use Morgan and USRCAT fingerprints as representations for MMD calculation in 2D and 3D chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Results are detailed in Table S4 and Figure 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We can see the MMD value is lower for molecules generated by the model com- pared with those randomly assembled from the building blocks, indicating higher similarity to the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We can conclude that after training, the network can enrich the output to the drug-like portion of the chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3 The quality of generated 3D structures An important aspect of the model’s performance is its ability to generate valid 3D structures of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For evaluation, we first measure its ability to correctly model the distribution of torsion angles in different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The environments are described using torsion SMARTS patterns by Schärfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='56 We compare the torsion distribution between the generated and test-set molecules for each environment using MMD, and rank the value from lowest to highest, as shown in Figure 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Since the exact value of MMD lacks interpretability, we visually inspect the torsion distribution for the environment with the highest and lowest MMD values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Results show a good match between the distribution even for the case with the highest MMD value, indicating that the model can correctly construct the local geometries of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Next, we evaluate the global conformation quality using RMSD calculated after relax- ing the generated molecules with the MMFF94s forcefield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The average RMSD values are reported in Table S4 and Figure 3f, while the distribution is shown in Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Confor- mations produced by the model generally have low RMSD values, with an average of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='69Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' As a reference, the RMSD value after relaxation for conformations generated by ETKDG using the same set of topological structures is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='14Å, a significantly higher value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' ETKDG is a conformation generation method based on distance geometry and the empirical distri- bution of torsion angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This method is initially developed to provide a faster alternative to MMFF94s optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 Case study: designing inhibitors targeting BTK’s ATP-binding pocket As mentioned previously, DeepLigBuilder+ can perform pocket-based 3D molecule design by combining a pharmacophore and shape-conditioned transformer network with MCTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To demonstrate its performance, we use DeepLigBuilder+ to design inhibitors that bind to the 13 Figure 4: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The structure of BTK kinase domain in complex with GDC-0853 (Fenebruti- nib).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The interaction between GDC-0853 and the ATP-binding pocket of BTK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The pharmacophore and shape condition extracted based on the interaction pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' As well as the seed atom used for molecule growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' d-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The distribution of: d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' the similarity with the given pharmacophore, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' the similarity with the given shape, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Smina docking scores inside the ATP-binding pocket of BTK, among generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The best rewards among all generated molecules at each step of MCTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' h-j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Several generated molecules with high predicted binding affinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The conformation generated by the model is shown in grey, and that produced by redocking the molecule is shown in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 14 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Hydrophobicity LYS 430 Seed aton ASP 539 MET 477 HBD O HBA : Shape Hydrophobic d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 12 Conditional MCTS - rollout eward 10 Conditioned on shape and 8 pharmacophore Unonditional rollout Unconditional - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='75 12 10 10 100 Pharmacophore Number of MCTS steps Shape similarity Smina score (kcal/mol) similarity (Aign-it) (log scale) h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' HO Structure 2 Structure 3 Structure 4 Morgan: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='057 Morgan: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='087 Morgan: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='071 Pharmacophore: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='361 Pharmacophore: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='274 Pharmacophore: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='211 Shape: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='836 Shape: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='831 Shape: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='644 Smina score: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='8 Smina score: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3 Smina score: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 LYS 430 LYS 430 LYS 430 ASP 539 ASP 539 MET 477 ASP 539 MET 477 MET 477ATP-binding pocket of BTK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Bruton’s tyrosine kinase (BTK) plays important roles in the signal transmission in B-cells57 and is related to a series of related diseases, including B-cell malignancies58 and autoimmune diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='59 Due to its importance, a variety of compounds have been developed to inhibit BTK, mostly targeting its ATP-binding pocket, with several of them approved for clinical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='60 However, all currently approved inhibitors bind to BTK covalently via Cys481, which may cause off-target effects by binding with other kinases with Cys481-like residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='61 Resistant mutations on Cys481 can also reduce their clinical effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='62 Non-covalent binders can in theory avoid those disadvantages, and in this section, we attempt to apply DeepLigBuilder+ for the design of non-covalent BTK inhibitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Several non-covalent BTK inhibitors are currently under clinical investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Here, we extract pharmacophore features and attempt to use DeepLigBuilder+ to generate new molecules with novel structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure 4a shows the structure of the kinase domain of BTK (PDB ID: 5vfi) complexed with GDC-0853 (Fenebrutinib), a potent BTK inhibitor (Ki=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='91nM) currently under clinical trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='63 GDC-0853 utilizes several important inter- actions inside the ATP-binding site such as hydrogen bonding with Met477, Lys430, and Asp539, as well as the hydrophobic interactions in the hinge region and the selectivity pocket (H3), as shown in Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Based on the information, we construct the shape and phar- macophore information shown in Figure 4c, and use it as a condition for DeepLigBuilder+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Note that GDC-0853 also occupies a solvent-exposed region as seen in the left part of Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This region is not included as input because we want to constrain the output molecule to a smaller size so that it can be more “lead-like” and easier to be further optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The oxygen atom of the carbonyl group is used as seed for molecule growth due to its interaction with Met477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' After the shape and pharmacophore-based inputs are determined, we combine the condi- tional generative model and MCTS to perform structure-based molecule design, with details shown in Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' First, we investigate whether the model can enrich molecules based on the given condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure 4d-e shows that the conditional model can generate molecules 15 with a better match in pharmacophore and shape compared with unconditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Next, we evaluate the benefit of using the conditional rollout inside MCTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure 4g shows the best reward among all generated molecules at each step of MCTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' It is shown that the conditional rollout can help to speed-up MCTS search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We also evaluate the benefit of using MCTS compared with direct sampling from the conditional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure 4f shows that MCTS can help to improve the docking score of generated molecules, with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='40 kcal/mol improvement in mean values (comparing the first row against the second row in Figure 4f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' It also offers more enrichment in the range of high binding affinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 13% of molecules gen- erated using MCTS have a smina score < -10 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The value is reduced to 5% if the molecules are sampled directly from the conditional network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To better demonstrate the molecule optimization process, we visualize the search tree used in MCTS in Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Due to space constraints, the tree only contains states of the first synthon, and nodes with a visit count less than 25 are dismissed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' It is shown that the model prioritizes the visits to states with higher Q-values, as those states are expanded more often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' States with lower Q-values (the nodes on the left side of the tree) are less favored, due to reasons such as bad 3D positions of the synthon anchors, as shown in Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Several generated molecules with high predicted binding affinity are shown in Figure 4h- j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A search in PubChem reveals no highly similar molecules (Tanimoto similarity > 95%), indicating that those are indeed novel structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Additionally, a search in the ChEMBL database does not reveal any structurally related molecules (Tanimoto similarity > 70%), indicating that no topologically similar molecules have been evaluated against BTK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The topological similarity with the seed molecule extracted from GDC-0853 (Figure 4c) is also low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In contrast, in terms of pharmacophore and shape, most input pharmacophore features are covered inside these generated molecules, including hydrogen bonding with Lys430 and Met477 and hydrophobic interactions at the two ends of each molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Synthesis paths are also proposed by DeepLigBuilder+ for each generated molecule, making retrosynthetic analysis of generated molecules easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For example, Figure 6a shows the proposed synthetic 16 path for Structure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The path for Structure 2 and 3 are shown in Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In summary, by combining MCTS with the conditional generative model, DeepLigBuilder+ can enrich molecules with high binding affinity based on the pharmacophore constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3 Case study: designing inhibitors targeting the NAD+ pocket of PHGDH Targeting cancer metabolism represents an important strategy for cancer drug develop- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='64 Human phosphoglycerate dehydrogenase (PHGDH), a key enzyme in the serine biosynthesis pathway, has been demonstrated to have crucial roles in tumorigenesis,65 mak- ing it a promising cancer-related target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' One strategy for targeting PHGDH is to design inhibitors that bind to its NAD+ pocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Multiple such inhibitors have been reported in pre- vious works, with most of them containing an indole-based scaffold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this case study, we use DeepLigBuilder+ to design potential binders for the NAD+ pocket with novel structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure 5a shows the structure of PHGDH (PDB ID: 6plg) together with compound 15, a potent inhibitor of the target developed by Mullarky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='66 Figure 5b demonstrates the interaction between the ligand and the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The nitrogen atom in the amide group in compound 15 acts as a hydrogen bond donor and interacts with Asp175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The carboxyl group in compound 15 can form hydrogen bonds with backbone nitrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' It also forms charge-charge interaction with Arg155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The indole structure of compound 15 resides inside a hydrophobic region, as shown by the orange arrow in Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A pharmacophore model is constructed based on those interactions, as shown in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We use the amide structure as the seed for molecule growth, as shown in Figure 5c and Figure S13b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The shape of compound 15 is also used as an input feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Next, we use DeepLigBuilder+ to generate molecules based on the pharmacophore and shape information and the target structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Similar to the previous section, we first evaluate whether the conditional model offers more enriched results based on the provided informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Indeed, Figure 5d-e shows that molecules sampled from the conditional transformer 17 Figure 5: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The structure of PHGDH with compound 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The interaction between compound 15 with the NAD+ binding pocket of PHGDH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The pharmacophore and shape condition extracted based on the interaction pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' As well as the seed atom used for molecule growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' d-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The distribution of: d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' the similarity with the given pharmacophore, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' the similarity with the given shape, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Smina docking scores inside the NAD+ binding pocket of PHGDH, among generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The best rewards among all generated molecules at each step of MCTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' h-j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Several generated molecules with high predicted binding affinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The conformation generated by the model is shown in grey, and that produced by redocking the molecule is shown in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 18 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Hydrophobicity ILE 156 ASP 175 O HBD HBA ○ Hydrophobic ARG 155 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 10 - MCTS Conditional 8 - rollout Reward Conditioned on 6 shape and pharmacophore Unonditional Unconditional rollout 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6 8 6 10 100 Pharmacophore Number of MCTS steps Shape similarity Smina score (kcal/mol) similarity (Aign-it) (log scale) h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Structure 5 Structure 6 Structure 7 Morgan: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='184 Morgan: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='165 Morgan: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='180 Pharmacophore: Pharmacophore: Pharmacophore: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='294 0 :0 Shape: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='770 Shape: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='658 Shape: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='704 HN Smina score: -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 HN Smina score: -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='56 Smina score: -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='74 HN 0 ILE 156 ILE 156 ILE 156 ASP 175 ASP 175 ASP 175 ARG 155 ARG 155 ARG 155match better to the input pharmacophore(Figure 5d) and shape(Figure 5e) compared with the unconditional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure 5g shows that introduction conditions help to accelerate the MCTS search, as demonstrated by the blue curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' When evaluating the benefit of the MCTS module, we found that MCTS search helps the model to generate molecules with bet- ter pharmacophore and shape matches (Figure 5d-e), and also helps to improve the docking score of the result, with an average improvement of Smina score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='53 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 11% of molecules generated using MCTS have a Smina score < -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 kcal/mol, compared to the value of 2% for those directly sampled from the conditional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure 5h-j shows several molecules generated by DeepLigBuilder+ with high predicted binding affinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The reaction paths generated by the model are shown in Figure 6b and Figure S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Those molecules have low topological similarities with compound 15, but share pharmacophore features such as hydrogen bond donors that interact with Asp175 and accep- tors that interact with Ile156 or Arg155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A search in PubChem does not reveal results with high topological similarity with those molecules (Tanimoto similarity >95%), indicating that those are indeed novel structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Also, the ChEMBL dataset does not contain topologically related compound records (Tanimoto similarity >70%), which means that similar molecules are not yet evaluated against PHGDH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Interestingly, those molecules contain cyclobutane structures that are similar to the oxetane structure in compound 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This structural motif helps to form a turn in the molecule shape for better accommodation with the pocket, and also creates a hydrophobic interaction with Ile177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4 Ablation studies and the effects of different hyperparameters It is important to understand how different architectural and hyperparameter choices affect the performance of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this section, we demonstrate the impact of several important network features and hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Details about the configurations explored are shown in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The performances of the model under different configurations are shown in Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 19 Figure 6: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The synthetic path of Structure 4 proposed by DeepLigBuilder+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' It involves a two-step process, which first connects the amide bond using the Schotten-Baumann reaction, and then forms the double bond using the Wittig reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Enamine IDs of reactants are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Note that the Schotten-Baumann reaction requires an additional activation step that transforms the carboxyl group into the acyl chloride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Also, the Wittig reaction requires the formation of the ylide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The synthetic path of Structure 5 proposed by DeepLigBuilder+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The first two reactants are connected using the Grignard reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The third reactant is connected by forming a urea structure using the amine group and the isocyanate group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Note that before the first step, reactant 2 needs to be transformed into the Grignard reagent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Additionally, the amine group in reactant 2 (marked grey) needs to be protected before carrying out other reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 20 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Schotten-Baumann reaction EN300-27296 Structure 4 Wittig reaction F3C EN300-88479 EN300-729603 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Grignard reaction EN300-82407 Structure 5 Urea formation EN300-313611 EN300-3789413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 The effect of changing the set of accessible building blocks A major feature of DeepLigBuilder+ is its capability to suggest synthetic paths with accessi- ble building blocks along with its generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' However, the accessibility of building blocks is a constantly changing factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' On one hand, due to technical advances, the number of synthesizable building blocks is rapidly growing over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' On the other hand, in- stock supply of such building blocks may vary between times and locations, and most require on-demand synthesis, increasing the cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Ideally, DeepLigBuilder+ should allow the user to choose a building block set that fits their need, without the need to perform re-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Here, we simulate such scenarios and report how the choice of building blocks impacts the quality of generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To simulate the lack of in-stock availability for certain building blocks, we restrict the building blocks to the EU stock, which is a smaller set with 81,235 compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In terms of the quality of topological structures, although Figure 3e shows that there is an increase in the MMD value after the restriction, Table S2 confirms that the generated molecules still maintain a high drug-likeness, with an average QED value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='60, close to the value before changing the building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In terms of the quality of 3D conformation, Table S4 and Figure 3f indicates an increase of RMSD from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='69Å to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='72Å, but still much lower than 1Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' From the results, we believe that although using a smaller building block may indeed impact the performance of the network, such an effect should be minor and still allows for regular application of DeepLigBuilder+ in drug design tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Then, to demonstrate how increasing the building block set may affect the generation result, we expand to include the comprehensive catalog, which contains more than 1 mil- lion (1,162,033) compounds, some may be synthesized on-demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure 3e-f shows that this change induces little impact on the performance of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Only 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1% of molecules generated have used the newly added building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In conclusion, we believe that DeepLig- Builder+ still offers promising performance when the building block set is changed, but a re-training may be required if we want to fully utilize newly added building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 The effect of different ways to encode 3D structural information Besides using the relative 3D positional encoding module to incorporate 3D information, DeepLigBuilder+ also uses invariant point attention (IPA), which offers a geometrically- aware way to pass information between atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To understand the benefit of including IPA, we disable the two modules consecutively and investigate the impact of those changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The results are shown in Figure 3e-f and Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We found that removing IPA has little impact on the quality of 3D conformation, as measured by RMSD, showing that IPA is not essential for maintaining 3D structure quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' However, the quality of 2D structure, as measured by 2D MMD, is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' If we remove the 3D positional embedding and keep IPA, we observe a significant improvement in 2D structural quality, but the RMSD value increased to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='718Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The results above demonstrated a trade-off between 2D and 3D structural quality, and that the two ways of including 3D information, IPA and relative positional encoding, have different emphases on the two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' One way to understand the result is to view IPA as a more regularized way of encoding geometric information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Our previous work has demon- strated that the model can not reliably generate correct 2D structures if it overly relies on accurate 3D information since it reduces the model’s ability to recover from errors during generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='33 The highly structured way to communicate 3D information in IPA can act as a form of regularization on how the model uses the 3D information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' On the other hand, there is no limitation on how 3D relative positional embedding will be processed by the net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Therefore, although the model can still generate accurate 3D structures without IPA, a lack of regularization will reduce the quality of 2D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Using both modules acts as a compromise, with an improved 3D conformation quality and a balanced 2D structure quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 22 4 Conclusion We have developed a new de novo drug design tool, DeepLigBuilder+, that generates synthesis-driven 3D molecules for a given target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' DeepLigBuilder+ uses geometric trans- former combined with an MCTS-based reinforcement learning module to navigate the space of synthesizable 3D molecules to identify potential bioactive compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This method aims to address two major challenges faced by deep molecule generative models: (1) the design of 3D molecules based on 3D constraints, and (2) the design of molecules with high synthetic accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' DeepLigBuilder+ has shown promising performances in overcoming these chal- lenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' DeepLigBuilder+ is capable of generating 3D molecules with high drug-likeness and geometric quality under the synthetic accessibility constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In the case study related to BTK and PHGDH, DeepLigBuilder+ significantly enriches molecules with high docking scores and favorable interaction patterns with the target pocket, using the 3D information provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For each generated molecule, DeepLigBuilder+ proposes a synthetic route with explicit reactions and building blocks that can be directly queried from the supplier, making retrosynthetic analysis much easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' DeepLigBuilder+ takes advantage of recent developments in 3D generative networks33 and the idea of synthetically aware de novo design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='37 To restrict the model to the chemical space of synthesizable molecules, we develop a method that calculates a stepwise constraint of the generation trajectory to ensure that it leads to purchasable building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Compared to other approaches that require a fragment-based generation scheme, our method can in theory be applied to various atom-based molecule generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In addition, it offers better scalability to large building block datasets by organizing them into a tree-based structure and avoids full database scans at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To achieve structure-based generation, we constructed a new dataset of pharmacophore-ligand pairs using large-scale 3D alignment of molecules, and then use it to develop a novel SE(3)-equivariant transformer conditioned on 3D information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This network is then combined with MCTS as the rollout policy, and it is demonstrated that the combination results in a significant improvement in the search speed 23 of MCTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' DeepLigBuilder+ could be improved in the following aspects in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' First, the present molecule assembling process relies on simple SMARTS rules, which may be limited in precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We are planning to include a more dedicated model for yield and selectivity pre- diction so that we can further improve the synthesizability of the assembled molecules by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Second, we are planning to update the reaction set to include broader, more modern reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In addition, the current version of DeepLigBuilder+ requires user-provided seed structures for molecule growth, and we are planning to develop methods to automate the seed selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Finally, we are planning to build a transformer model that is directly conditioned on the 3D pocket structure, without the need for pharmacophore extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In summary, due to its unique capability of generating highly synthesizable molecules with 3D structures, DeepLigBuilder+ provides a powerful tool to generate bioactive molecules and to accelerate the process of structure-based drug design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 5 Acknowledgements This work has been supported in part by the National Natural Science Foundation of China (22033001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We would like to thank Yuhao Ren and Kangjie Lin for their kind advice on the synthetic accessbility of generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We also appreciate Alibaba Cloud for providing the EFLOPS computation platform for the GPU-based network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' References (1) Paul, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Feng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Cheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Lin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Gao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Tian, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A Retrospective Overview of PHGDH and Its Inhibitors for Regulating Cancer Metabolism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' European Journal of Medicinal Chemistry 2021, 217, 113379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (66) Mullarky, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Inhibition of 3-Phosphoglycerate Dehydrogenase (PHGDH) by Indole Amides Abrogates de novo Serine Synthesis in Cancer Cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Bioorganic & Medicinal Chemistry Letters 2019, 29, 2503–2510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 31 Supporting Information: Synthesis-driven design of 3D molecules for structure-based drug discovery using geometric transformers Yibo Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='† Jianfeng Pei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='‡ and Luhua Lai∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='¶ † Center for Life Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Academy for Advanced Interdisciplinary Studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' China ‡ Center for Quantitative Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Academy for Advanced Interdisciplinary Studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' China ¶ BNLMS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' College of Chemistry and Molecular Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' China E-mail: jfpei@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' lhlai@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='cn 1 Supplementary Methods 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Constructing the synthon dataset We use the building block sets provided by Enamine as the set of purchasable reactants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The global stock, which contains 238,980 compounds at the time of access (July 2022), is used to assemble the training set molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To investigate the impact of changing available reactants on the model’s performance, we also downloaded the EU stock and comprehensive catalog, which contains 81,235 and 1,162,033 molecules respectively (by Oct 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='00167v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='QM] 31 Dec 2022 For reactions, we use the 58 SMARTS rules collected by Hartenfeller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=',S1 which represents a set of robust chemical reactions relevant to drug design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Based on the reaction set, we constructed a set of SMARTS rules to convert reactants to synthons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The conversion is performed using the following procedure: (1) For a given building block and reaction rule, we predict the product structure using RDKit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' If the reaction involves two reactants, the other reactant is set to be a minimum structure with the required function group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) The substructure in the product molecule that corresponds to the building block is ex- tracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The open valences resulting from the bond break in the extracted substructure are labeled with the reaction type and its role in the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (3) If the substructure extraction results in multiple valences, this indicates that new rings are formed after the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this case, we either include the new ring structure inside this synthon or leave the ring to the synthon corresponding to the other reactant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A visual demonstration of this process is given in Figure S1 and Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To simplify the generative model, we require that each reaction will result in at most one open valence in synthon structures, and such valence must correspond to a single bond in the product molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Most reactions satisfy this requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' As a result, we kept 52 reactions and constructed 108 SMARTS rules for converting reactants to synthons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The Enamine building blocks are then converted to synthons using those rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For each building block, we enumerate all possible synthon structures by iterating through the reaction rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' At most two reactions are allowed to happen in one reactant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To ensure that the generated synthons are fragment-like and relevent to drug discovery, we apply the following rules to filter the results: (1) Element types of atoms inside each molecule are restricted to the set {C, O, N, P, S, F, Cl, Br, I} and bond types are restricted to single, double, triple, and aromatic bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-2 (2) Synthons are required to be “fragment-like” based on the rule of two (Ro2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S2 (3) Each fragment can have at most 4 rings, and the size of each ring should not be larger than 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (4) Since we are focusing on designing non-covalent binders, we filter structures with the potential of forming covalent bonds with the protein, using a set of SMARTS rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S3 The filtering results in a dataset containing 241,310 synthons from the global stock, 103,385 synthons from the EU stock, and 783,195 synthons from the comprehensive catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The synthons from the global stock are later used to assemble the training set molecules (See Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 Performing molecule generation DeepLigBuilder+ generates 3D molecules using a synthon-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Each molecule is composed of 3 synthon structures, which is equivalent to combining three building blocks with two reaction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' When generating each synthon, DeepLigBuilder+ uses a graph-based approach similar to our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S4 Specifically, the model generates 3D synthon struc- tures by producing molecular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We write the output graph as G = (V, E, A, B, X), where V and E are the set of nodes (atoms) and edges (bonds), A = {av}v∈V and B = {buv}{u,v}∈E are labels representing the type of each atom and bond, and X = {xv}v∈V are the 3D coordinates of each atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' When generating each synthon, the model starts with an empty graph G0 = (, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', ), iteratively updates its structure Gt = at(Gt−1), and outputs the graph as a new synthon fragment when it is ready.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' During generating, additional information is attached to each node in the graph to record the generation history, which includes: (1) The currently focused node, denoted as v∗ t , where t represents the step ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The defini- tion of a “focused” node is similar to that in our previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S4 Briefly, all edits to S-3 the molecular graph, whether to add new atoms or new bonds, happen on the focused node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) The parent of each node Pt = {pv}v∈Vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A node pv is called the “parent” of another node v if pv is the focused node when v is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Note that the parent-child relationship induces a spanning tree of the molecular graph Gt, which can be used to calculate tree-based distances between atoms (or nodes) in the graph as input features to the transformer network (as detailed in Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We denote the state of the molecular graph at step t with the additional information as G′ t = (Vt, Et, At, Bt, Xt, v∗ t , Pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The graph structure is iteratively modified based on actions at sampled from the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The following types of actions are allowed during generation: (1) Initialization, which adds the first atom to the molecular graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) Append, which attaches a new atom to the focused atom using a new bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For this action, the model needs to decide the type of the new atom and bond, as well as the 3D position of the new atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' When generating the position, the model uses a spherical coordinate frame attached to the focused atom, following our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S4 Besides the element type of the new atom, the model also needs to decide whether there will be branches on this atom and whether the atom will be a target of future ring closure, similar to Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S5 (3) Backtracking, which requires the model to move the focused atom to its closest ancestor that allows branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The generation terminates if the focused node has no parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (4) Search loop target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This action indicates that a new ring will be formed, and the model should examine the closest ancestor of the focused node to see whether it is a suitable target during the ring closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The network can generate this action multiple times until a suitable target is found for ring closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-4 (5) Start loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This action happens after a “search loop target” action when the appropriate target of ring closure is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this action, the model determines the size of the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' After this action, a series of “append” actions should be issued by the model to complete the ring formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (6) Close loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This action happens after all ring atoms have been generated, and the model is ready to connect the ring to the target atom determined in the “search loop target” step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The model decides the type of bond used to close the ring during this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The process of molecule generation can be represented using a finite state machine, as shown in Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A full path for generating an example molecule is shown in Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A major difference between the generation scheme proposed in this work compared with the previous version of DeepLigBuilderS4 is its emphasis on ring generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Before gener- ating explicit ring structures, DeepLigBuilder+ will first determine the size of the ring, as well as the location the ring will be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This information will help to guide the process of ring generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' It will also help to avoid problems when the user changes the synthon dataset (to be discussed in the next section, also see Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3 Constraining the model to generate structures inside the syn- thon database To ensure synthetic accessibility, the synthons generated by DeepLigBuilder+ must be re- stricted to the synthon dataset derived from purchasable building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To achieve this goal, most previous methods use reactants as basic units for molecule generation and ap- ply neural networks to parametrize scoring functions that filter the reactant dataset for appropriate candidates at each generation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this work, we propose a radically dif- ferent approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Instead of adopting a generation scheme based on reactants, we still use atoms as basic generation units, building on the foundation of previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S4,S6 To en- S-5 force constraints on the chemical space, we apply masks on the action space at each step of generation, so that we can ensure that the output topological structure can be found in the synthon dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Next, we show how such action masks can be calculated at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' First, for each synthon in the synthon database s ∈ S, we represent it as a list of actions that can be used to generate its molecular graph, which is indicated as (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', aT) → s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The definition of the action space and the generation process follows Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Since we only need to constrain the topological structures, 3D action information, such as the position of new atoms, is removed from the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For convenience, we refer to such action sequences as trajectories and write them as τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this way, we convert the synthon dataset into a collection of trajectories T (S) = {τ|∃s ∈ S s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' τ → s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To ensure synthons generated by the model lie in S, we need to ensure that the generation trajectories lie in T (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' At each step t, given the generation history τt = τ[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='.t − 1] = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', at−1), in order to ensure that the final trajectory will lie inside T (S), we need to make sure that the next action at is inside the following set: at ∈ A(τt, S) = {a|∃τ ′ ∈ T (S) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' τ ′[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='.t] = τt · a} It is easy to prove that as long as this requirement holds at each step, we can guarantee that the resulting synthon will be inside S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In order to efficiently calculate A(τt, S), the trajectories in T (S) are organized into a prefix tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Inside the tree, each node represents a prefix of some trajectory in T (S), and each edge represents an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' During retrieval, we descend the tree to find the node that equals τt, and then collect all its outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' It can be seen that those edges form the set A(τt, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this way, we can construct action masks at each step to ensure that the resulting synthon can be found in the synthon dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' As a method to constrain the chemical space of the generative model, our method has some advantages compared with previous methods: S-6 (1) The complexity of generating each synthon does not scale with the size of the synthon dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Searching inside the prefix tree has an average cost of O(L), where L is the average number of steps required to generate a synthon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Other approaches generally require a full scan of the reactant dataset, which has more limited scalability for larger synthon databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Constructing the prefix tree will cost O(LN), where N is the size of the synthon, but we only need to construct the tree once, and it can then be reused for subsequent generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) Our method only constrains the 2D (topological) structure of molecules, while the 3D coordinates are generated by the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This eliminates the need of build- ing a 3D fragment database, as done in several previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S7,S8 Such approaches may cause some technical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' First, enumerating 3D conformers will significantly increase the size of the fragment dataset, especially when the dataset contains large flexible structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Second, the conformation of a fragment depends on its environ- ment, and enumerating its conformation in isolation may result in inaccurate results when the fragment is attached to another molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Some practical issues need to be considered when applying this method: (1) There are generally multiple ways to generate a molecular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To reduce com- plexity and computational cost, we follow the approach in previous works,S4,S6 which uses a depth-first, canonically ordered way to generate molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this way, a molec- ular graph will correspond to exactly one trajectory, when the starting atom for gen- eration is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To make the model more flexible, we allow multiple starting points for the generation as in our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S4 (2) When using the proposed method to constrain the chemical space, there may be issues related to ring conformation when the synthon dataset is changed without model re- training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' An illustrative example is given in Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To alleviate this problem, we S-7 adopt a more refined ring-generation scheme, which allocates the size and location of the ring before its structures are generated, as discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4 Network architecture In this section, we give a detailed description of the neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We first describe the inputs required by the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Then, we show how the inputs are embedded before feeding to the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Next, we detail the architecture of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Finally, we demonstrate how the output features from the transformer are used to generate the action at each step with a MADE-based policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Input features The model receives previously generated synthons as inputs, as well as shape and pharma- cophore information for structure-based generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Graph inputs All graph-related inputs, including previously generated synthons and the intermediate synthon structures, are represented as their generation trajectories τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This acts as a sequence-based representation of molecular graphs, similar to the concept of a “sentence” in NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Each action in the sequence corresponds to a “token” or “word” in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The following information is included in each token: (1) The current step id (t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) Action performed at this step, including the action type (actt) and the type of new bonds added (nbtt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (3) Information of the focused node after the action is applied, such as its index (idt, or- dered based on the step each atom is generated), element type (elt), formal charge(fct), and the number of explicit hydrogens (neht) attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Information about whether the S-8 node allows for branching (bt) or whether the node can act as a target for ring closure is also included(rt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (4) If a ring is being generated, we also input the expected size of the ring (rst) and the expected target for ring closure (rtt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In addition to the information above, each token is attached with a 3D coordinate frame (ot, Rt) using the method developed in our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S4 Those frames have several functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (1) The coordinates of newly generated atoms are defined under those frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The spheri- cal coordinate values of the local frames correspond to bond lengths, bond angles, and torsion angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) Those frames are used in the IPA modules to communicate 3D information between the tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (3) Those frames are used to define the relative 3D positional embeddings between tokens (to be discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Besides input features for each token, we also include features for each action pair as relative embedding to increase the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Those pair features include: (1) The topological distance between focused atoms at each step (toptt′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) The distance between the focused atoms in the spanning tree induced by the generation trajectory (treett′, recall descriptions in Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (3) The relative 3D positions between coordinate frames attached to each action (∆xtt′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Specifically, for the action pair (at, at′), we first calculate the displacement between the origins of each frame and then transform the coordinate values to the local coordinate frame attached to ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-9 Pharmacophore inputs The input pharmacophore model can be represented as a se- quence of individual pharmacophore features, with definitions adopted from Align-it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S9 Each pharmacophore p contains information about its type (ptp) and radius (prp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We use the Euclidean distances (pdpp′) between pharmacophore pairs (p, p′) as relative positional em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In the decoder, we need to communicate between the pharmacophore model and the synthon structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Therefore, we feed the model with the following information about the 3D relationship between each pharmacophore-action pair: (1) The position of each pharmacophore in the local coordinate system attached to each action (∆xpt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) The direction for each pharmacophore (only HBDs and HBAs) in the local coordinate systems attached to each action (ˆnpt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Shape inputs The shape input can originate from known active ligands or directly from the target pocket using programs such as PANTHER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S10 In both cases, we represent the input shape as a set of 3D spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Most previous models use 3D-CNN to encode shape information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S11 This method is not equivariant and induces additional computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In comparison, DeepLigBuilder+ uses a more compact, SE(3) equivariant representation for 3D shapes based on 3D Zernike coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For an input shape composed of 3D spheres, we can represent it using a 3D scalar function following Grant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' :S12 f(x) = N � i=1 pi exp(−αi|x − xi|2) Where xi is the location of each sphere, and pi and αi are parameters related to the radius of each sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We define the function in the coordinate frame placed in the center of the S-10 spheres xc = 1 N �N i=1 xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The function is then decomposed as: f(x) = � nlm cm nlZm nl(x) Where Zm nl(x) are 3D Zernike polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S13 In this work, we use a truncated series with n ≤ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Those functions are generalizations of Zernike polynomials defined in 2D space and act as the orthogonal basis for 3D functions (defined in a ball with radius 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Additionally, those coefficients c = {cm nl}nlm changes in an equivariant manner when a rotation R ∈ SO(3) is applied to the function f: f ′(x) = f(R−1x) = � nlm � m′ Rl mm′cm′ nl Zm nl(x) Where Rl mm′ are the matrices that can be used to “rotate” the coefficients of the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Those matrices can be efficiently calculated using the methods proposed by Ivanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S14 The representation of the shape can then be written as (xc, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' When feeding into the model, we rotate it into the local coordinate frames of each action (xc t, ct), and concatenate it with other action features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In other words, the shape information is used by the transformer similar to a positional embedding for each token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 Embedding layers Several types of embedding layers are used for the input information discussed above, in- cluding: (1) Lookup tables with trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This type of layer is used to embed atom, bond, and pharmacophore types, as well as topological distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The full list of inputs includes actt, nbtt, elt, fct, neht, bt, rt, rst, toptt′, treett′, and ptp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) Positional embedding using sine and cosine functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S15 This type of embedding is widely used in transformer models to embed position-related data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this work, we S-11 use it to embed time, position, and distance-related information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The full list includes t, idxt, rtt, ∆xtt′, pdpp′, ∆xtp and xc t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (3) Some inputs with continuous representations are input to the model as-is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This in- cludes: prp, ˆnpt, and ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' After the inputs are embedded, for each token (action or pharmacophore) and token pair, we concatenate all input information into a vector and use a linear layer to project the inputs to a predefined dimension, which is then used as transformer inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We write the size of the dimension as F for each token and F ′ for each token pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this work, we have F ′ = F 2 and two values {512, 256} are experimented for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3 The transformer architecture The transformer is responsible for processing the input features to generate a state embedding at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Later, this state embedding will be used by the policy network for action sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The transformer consists of multiple encoders responsible for processing different inputs, and a decoder used to generate the state embedder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The decoder and encoders are composed of transformer layers, each containing one or more attention layers and a tokenwise dense layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The attention and dense layers are wrapped inside residue blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Attention layers Two types of attention layers are used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The first is the widely used scaled dot-product attention(SDPA),S15 with additional relative positional bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Given the features of the source sequence {hs i}ls i=1 , the target sequence {ht i}lt i=1, and the source-target token pairs {hp ij}j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=',ls i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=',lt , the attention layer performs the following operations to calculate the output feature {h′i}lt i=1 for each target token: [kh j , vh j ]H h=1 = Linearkv(hs j), [qh i ]H h=1 = Linearq(ht i), [bh ij, zh ij]H h=1 = Linearp(hp ij) S-12 ah ij = softmaxj( 1 √ d qh i · kh j + bh ij) h′ i = Linearout([ ls � j=1 ah ijvh j , ls � j=1 ah ijzh ij]H h=1) where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', ls;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', H H denotes the number of attention heads, which is set to be 16 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' d is the dimension of each query, key, or value vector, which is set to be d = F H , the [·] operator represents concatenation or unpacking respectively when it appears in the right or left side of the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The second type of attention is invariant point attention (IPA), initially proposed in Al- phaFold2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S16 Similar to SDPA, we first calculate vector-based queries for the target sequence, as well as the keys and values for the source sequence: [kh j , vh j ]H h=1 = Linearkv(hs j), [qh i ]H h=1 = Linearq(ht i), [bh ij, zh ij]H h=1 = Linearp(hp ij) where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', ls Different from SDPA, IPA also calculates keys, queries, and values based on 3D points: [⃗kph i ,⃗vph i ]h=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=',H p=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=',P = Linear3D kv (hs i), [⃗qph j ]h=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=',H p=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=',P = Linear3D q (ht j) where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', ls Where P is the number of points for each attention head and is set to be 8 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Those points are defined on the local coordinate system attached to each token and can be transformed into the global coordinate system as: ⃗k′ph j = Rs j⃗kph j + os j, ⃗v′ph j = Rs j⃗vph j + os j, ⃗q′ph i = Rt i⃗qph i + ot i S-13 where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', ls;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', P Where {(ot i, Rt i)}lt i=1 and {(os i, Rs i)}ls i=1 are coordinate frames attached to each source and target tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Attention maps are then calculated as: ah ij = softmaxj(wL( 1 √ d qh i · kh j + bh ij − γhwC 2 P � p=1 |⃗q′ph i − ⃗k′ph j |2)) where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', ls;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', H In which wL = � 1 3 and wC = � 2 9P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The values are then used to calculate the output features: f h i = ls � j=1 ah ijvh j ˜f h i = ls � j=1 ah ijzh ij ⃗f hp i = (Rt i)−1( ls � j=1 ah ij⃗vph j − ot i) h′ i = Linearout([f h i ,˜f h i , [⃗f ph i ]P p=1]H h=1) where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', lt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', P Note that compared with the original implementation, we do not include the norm of ⃗f hp i in the input of the linear projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' SDPA and IPA are used in different situations in the transformer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The encoder for previous synthons uses IPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In the decoder, self- attention layers and outer-attention layers with previous synthons use IPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The encoder for pharmacophores and the decoder outer-attention layers with pharmacophores use SDPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The token-wise dense layers (MLP layers) MLP layers consist of two dense layers, each with a normalization-activation-linear architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this work, we use layer nor- malizationS17 and ELUS18 as activation units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The number of hidden features is set to be S-14 2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Transformer layers The attention layers and MLP layers are composed of transformer layers that are later stacked into the encoder and decoder networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Each transformer layer in the encoder consists of one attention layer and one MLP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Each transformer layer in the decoder consists of two attention layers, one for self-attention and the other for outer attention, and an MLP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The attention and MLP layers are all wrapped inside residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Encoders, decoders, and the transformer network Multiple transformer layers are stacked to form the encoder and decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For unconditional generation tasks, we use 6 blocks for the encoder of previous synthons and 6 blocks for the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For structure-based generation tasks, the model also receives shape and pharmacophore-based information, which uses 3 more encoder and decoder layers for processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A shallow configuration is also exper- imented with in unconditional generation tasks, as a way to demonstrate the performance using different network scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this configuration, the encoder and the decoder each uses 3 blocks of transformer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Two versions of geometric transformers are developed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' An unconditional transformer is used to access the ability of this method to generate drug-like, geometrically valid molecules with high synthesizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A conditional one, which receives user-provided pharmacophores and shapes as extract inputs, is used as the rollout policy to accelerate MCTS in SBDD problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4 The policy network Using the state embedding generated by the transformer network, a policy network is then applied to generate the action for the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Before specifying the architecture of this network, we need to first define the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Two types of decisions need to be made at each step of the generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The first one relates to the topological structure of the molecule, S-15 including the type of action to be carried out, the type of the new atom and bond, the size of the new ring, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The second one relates to the 3D molecular structure, that is the position of the new atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For topological decisions, we iterate through the synthon dataset to collect the actions that are needed to produce all the synthon structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This result in the topological action space Atopo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For 3D actions, we write the location of the new atom added to each step in the local spherical coordinate system attached to the focused node (r, θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We then discretize r, θ and φ, each using two integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Take the φ coordinate for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We first split its domain (−π, π] into N1 equal-sized intervals (−π+ 2π N1i, −π+ 2π N1(i+1)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', N1, and find the one containing the coordinate value φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The interval found is named φcrude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To achieve further precision, φcrude is then divided into N2 smaller chunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We find the one containing φ, and name it φrefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Similar procedures are applied for r and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Following the definition above, we can now write the action as: a = (atopo, rcrude, rrefined, θcrude, θrefined, φcrude, φrefined) where atopo ∈ Atopo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' rcrude, θcrude, φcrude ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', N1} rrefined, θrefined, φrefined ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=', N2} In this work, N1 is set to be 30 and N2 to be 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The task of the policy network is to parametrize the distribution of a using the neural network: pη(a|h), where η is the parameter of the network, and h is the state embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To efficiently model the joint distribution of discrete variables in a, we factorize pη autoregressively and use MADE (masked autoencoder for density estimation) as the model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The network contains 3 layers and 630 hidden units for each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The general idea of the policy network is similar to the previous version of DeepLig- S-16 Builder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S4 The major difference is that we now use a discretized action space for the 3D positioning of new atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Previously, we found that modeling the continuous distribution of atom positions faces numerical issues, and proposed SoftMADE to address those issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' However, SoftMADE works by adding noise to the 3D coordinates, which reduces the ac- curacy of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Here in DeepLigBuilder+, we use a two-step discretization process, which ensures the precision of the distribution and can also avoid numerical instabilities in continuous distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 Dataset and network training 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Training the unconditional model We use a dataset containing drug-like molecules randomly assembled using the building blocks in the Enamine global stock as the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The assembling process follows a step-wise procedure, which is initialized with a random building block sampled from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' At each step, the possible reactions that can happen to the molecule are enumer- ated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The reaction type is then randomly selected from the results, and the next reactant is sampled from the building block set based on the selected reaction type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Molecules are assembled with three reactants combined using two reaction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Several filters are applied to obtain drug-like molecules, including (1) Lipinski’s rule-of-5 (Ro5),S19 (2) Veber’s rule,S20 (3) PAINS patterns,S21 and (4) a QEDS22 threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' After this process, we obtained approximately 1 million (974,917) molecules, with 4/5 of which used as the training set, and the rest used for validation and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The 3D conformers of those molecules are generated using RDKit, by first using ETKDG to embed the molecules into 3D space, and then opti- mizing them using MMFF94s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To create more stable conformers, at most 10 conformers are generated for each molecule, and the one with the lowest energy is used for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Finally, those molecular structures are converted to synthons and subsequently transformed into generation trajectories to train the unconditional transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The network is implemented using PyTorch, and Adam is used for model optimization,S23 S-17 with a linear learning rate warm-up of one epoch to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='001, followed by an exponential learning rate decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The decay rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='01, and several decay frequencies are experimented with (see Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The batch size is set to 1024, and the model is trained for 100 epochs using 4 A100 GPUs, which may take 1-2 days to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To train the shape and pharmacophore-conditioned model, a dataset of input-output pairs is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' First, we extracted a set of ligand-based pharmacophore models and shapes from the 3D ligands in the PDBBind 2020 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S24 We use PDBBind as the data source due to its ligand diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' It not only contains drug-like ligands, but also metabolites, peptides, fragments, and other types of ligands that lack drug-likeness but are still frequently used in pharmacophore extraction and interaction analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this way, we can increase the diversity of the pharmacophore and shape inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Note that we do not use the protein structure and bioactivity values inside the PDBBind dataset, therefore the extracted information is fully unlabeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Future research may also consider that information to improve the quality of the extracted pharmacophores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' An overall 12,456 pharmacophores and shapes are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Next, we align the assem- bled molecules to the extracted pharmacophores and filter those with a good match to form the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This process requires 974, 917 × 12, 456 3D alignment operations, and due to its time cost, we utilize an approach based on sequential filtering: (1) First, we perform similarity calculation based on USRCAT fingerprints, and filter the top 10,000 most similar molecules to each pharmacophore and shape query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Since USRCAT similarity does not involve 3D alignment, it can be carried out with high efficiency inside GPU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) Next, we perform 3D alignments between the 10, 000 × 12, 456 pairs of molecules and queries based on 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The shapes are represented as a combination of 3D Gaus- sian functions, as described previously,S12 each “colored” with a pharmacophore type assigned using a set of SMARTS patterns defined in RDKit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' PCA is then performed on the point sets to obtain the principle axes, and those axes are aligned to form the initial S-18 pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 4 candidate poses are created by rotating 180◦ around each axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A gradient- based optimization process is then used to tune the rotation and translation to achieve the best overlap between two shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The alignment process is accomplished using an in-house PyTorch program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Shape similarity is computed using the aligned pose, and the top 100 most similar molecules are retained for each pharmacophore-shape query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (3) After shape-based alignment and filtering, a more refined pharmacophore-based filter- ing is used to further enrich molecule-query pairs with a good match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' At this step, we retain the top 10 most similar molecules for each pharmacophore and shape-based query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The filtering process described above creates a dataset containing 10×12, 456 = 124, 560 input-output pairs for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' When training the conditional model, we use the pre- trained unconditional model as the base model and add pharmacophore and shape-related layers at the tail of the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To avoid overtraining, only the parameters of the newly added layers are allowed to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The training is performed for 160 epochs and the learning rate decay is performed for every 30 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Other hyperparameters for training the conditional model are similar to that used to train its unconditional counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6 Monte Carlo tree search Monte Carlo tree search is a widely used technique in reinforcement learning which finds promising solutions for a given problem by strategically expanding the search tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this work, we combine MCTS with the pharmacophore and shape-conditioned transformer for the design of synthesizable 3D molecules inside a given pocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In order to search for promising molecular structures, MCTS maintains a look-ahead tree T and iteratively builds T using four steps: selection, expansion, simulation, and backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' DeepLigBuilder+ uses a variant of MCTS that includes several modifications to better suit it to the 3D molecule generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this section, we first describe the data structure of the look-ahead tree S-19 T , then discuss how the tree is updated at each step, and finally specify the details of the hyperparameters used during MCTS runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The look-ahead tree in MCTS is used to store the history of previous visits, with each node representing an intermediate state during molecule generation, and each edge representing an action carried out at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In DeepLigBuilder+, we introduced several custom modifications in the data structure of nodes and edges: (1) Edges in the tree contain topological and 3D actions applied to the molecular graph at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The 3D action is discretized using the method introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Note that the edge only stores the value of the φ coordinate, or torsion angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This is because the bond lengths r and bond angles θ are largely determined by the bond types and ring sizes, which are already stored in the topological actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) The torsion angles of new atoms are stored in a coarse-grained form, that is φcrude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this way, nodes in T now represent sets of molecule structures with similar 3D conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This has a similar effect of clustering intermediate states using the torsion fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' At each step of MCTS, the following operation are carried out consecutively: (1) The selection operation, which chooses a promising state from the tree based on its estimated value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We follow MENTSS25 and use E2W (Empirical Exponential Weight) to generate the selection policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' As mentioned that each node represents a cluster of states with the same topological structure and similar 3D conformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We randomly select one of the states from the cluster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) The expansion operation, which enumerates all possible actions that can be carried out given a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In practice, we found that although the allowed action space for the 3D generative model is large, generally only a small subset of actions will be selected by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Based on this observation, we first perform multiple independent sampling S-20 of actions given the state selected using the transformer, and cluster the actions based on their topological action and torsion angle, as described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This procedure is similar to pruning branches in the search tree with a small probability of being chosen by the transformer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (3) The simulation operation, in which a full rollout is performed based on the selected state, and the results are evaluated using the Smina scoring function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Note that all new states created in the expansion operation are used to perform the rollout, which acts as a form of leaf-level parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S26 Additionally, all subsequent actions generated at each rollout are added to the tree, which may help to reduce the instability during the tree search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Compared with the previous version of DeepLigBuilder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' during the rollout,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' we uses a shape and pharmacophore-conditioned generative model When evaluating the generated outcome,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' DeepLigBuilder+ uses a soft version of the Smina score as the reward function: R(m) = softplus(−S(m)) + �3 i=1 softplus(−S(si)) 2 Where m is generated molecule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 3} are the synthons fragments composed of the molecule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S(·) is the Smina score (evaluated directly without minimization),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' and softplus(x) = ln(1+exp(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Adding softplus to the equation helps to reduce large penalties from the clashes with the pocket, making the reward function softer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (4) The backtracking operation, in which the calculated rewards are used to update the Q value estimates for each edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Following MENTS, we use the soft-bellman backup as the operator to update the Q values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The number of rollout steps for each case study is set to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To fully utilize GPU resources, tree-level, root-level, and leaf-level parallelism are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S26 During selection, S-21 an overall of 16 nodes is selected at once for simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Also, 20 trees are constructed independently for each case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' During expansion, 32 actions are sampled from the full action space for clustering, and those actions are all used to form updated states for rollout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' There are two major parameters controlling the balance between exploration and exploitation in MENTS,S25 the temperature parameter τ and the exploration parameter ϵ in E2W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' In this work, we have τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='25 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The MCTS program uses one NVIDIA 3070 graphics card with 1 CPU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='7 Model evaluation Several evaluations are performed to examine the performance of DeepLigBuilder+ in dif- ferent aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 The unconditional generation tasks For the unconditional model, we investigate whether it can generate drug-like molecules with high-quality 3D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To this end, a dataset containing molecules randomly assembled from the building blocks without drug-likeness filters is constructed as the target of comparison, and the following evaluation metrics are employed: (1) Metrics related to the molecular properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For each molecule, a series of 2D or 3D properties are calculated, including molecular weight, LogP, QED, the number of hydrogen bond donors and acceptors, the number of rotatable bonds, total and po- lar solvent accessible surface areas, and the radius of gyration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The distributions of those properties are then compared between the generated molecules, the assembled molecules, and the test set molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To make the comparison, the mean and stan- dard deviation values are calculated for each property in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We also use a metric calculated using the RMSD between mean and standard deviation values for a quantitative measurement: W = � (µ1 − µ2)2 + (σ1 − σ2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Mathematically, this S-22 metric is equivalent to the Wasserstein distance between Gaussian approximations of two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) For a more quantitative measurement of whether the model correctly constructed the drug-like chemical space of synthesizable molecules, we evaluate the MMDS27 between generated molecules and test set molecules using 2D (morgan) and 3D (USRCAT) fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' MMD, which stands for the maximum mean discrepancy, is a widely used technique for evaluating the differences between distributions, and are applied in several previous works for the evaluation of molecule generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S4,S28 MMD can be calculated as follows: MMD2 u(X, Y ) = 1 m(m − 1) � i̸=j k(xi, xj) + 1 n(n − 1) � i̸=j k(yi, yj) − 2 mn � ij k(xi, yj) Where X = {xi}m i=1 and Y = {yi}n i=1 are two datasets to be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' k is the kernel function based on either the Morgan or USRCAT fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (3) To evaluate the quality of 3D structures, we first examine the quality of local geometries of generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Similar to the previous work,S4 we compare the distribution of torsion angles between generated molecules and the test set molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The en- vironments are described using torsion SMARTS patterns by Schärfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='S29 The difference between torsion angle distributions is quantized using MMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We use the cosine values of torsion angle difference as kernel function when calculating the MMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (4) To further access the quality of generated conformers by the model, we optimize each generated molecule using the MMFF94s force field and calculate the RMSD value between conformers before and after the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To provide a context of the model’s performance, we also perform this evaluation on conformers generated using the ETKDGS30 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This method is initially proposed as a faster alternative for forcefield-based conformation optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 The structure-based generation tasks When evaluating the performance of DeepLigBuilder+ in structure-based generation tasks, we mainly focus on answering the following two questions: (1) Can MCTS help enrich molecules with high docking scores?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) Can shape-based and pharmacophore-based conditional rollout policy help MCTS to discover better results faster?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A series of ablation studies are performed to answer those two questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' First, in terms of the benefit of MCTS-based sampling, we compare the distribution of Smina docking scores for molecules generated with or without MCTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' When calculating the Smina scores, we first move the generated molecules out of the protein pocket, perform local relaxation for each molecule using the MMFF94s forcefield, and then re-dock the relaxed conformers back into the target pocket using Smina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To access the benefit of the conditional rollout policy, the following studies are performed: (1) We determine whether the conditional model can achieve enrichment in pharmacophore and shape compared with its unconditional counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' This question can be answered by examining the distribution of shape and pharmacophore similarity between gener- ated molecules and input queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' (2) We investigate whether the extra conditional inputs can help MCTS to achieve faster search speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' To this end, ablation studies are performed by enabling and then disabling the conditional inputs for the rollout policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' At each MCTS step, we record the reward value for the best molecule found so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We then compare whether the conditional rollout policy can help MCTS reach solutions with higher rewards faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 2 Supplementary Figures S-24 Figure S1: The process of converting a reactant in the building block dataset into a synthon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-25 Reactant: Reaction Schotten-Baumann 0H EN300-729603 Convert to the reaction product 个 CH,CH,NH, NHCH,CH3 Extract substrcture as synthon Anchor label Reaction: Schotten-Baumann NHCH,CH, Role: carboxylic acid SynthonFigure S2: Special treatment is needed when converting reactants to synthons when the reaction involves ring formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-26 Huisgen reaction OH H3CH2C- NN EN300-107725 H3CH2C H3CH2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' H3CH2C N orFigure S3: A representation of the molecule generation process using a finite state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' From the figure, we can see that the model moves back and forth between ring generation (represented as the “searching ring target” state and the “generating ring” state) and chain generation (represented as the “generating chains” state and the “backtracking” state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-27 START END Initialize Terminate Backtrack Append Generating chains Backtracking Backtrack Append CloseLoop SearchLoopTarget Searching ring target StartLoop Generating rings SearchLoopTarget AppendFigure S4: The process of generating a 3D molecule using DeepLigBuilder+ S-28 APPEND APPEND APPEND Search StartLoop APPEND INIT (C,-,C,X。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=') (C,=,C,X) (C,-,r,x2) LoopTarget size=5 (C,ar,C,X,) 2/5 3/5 4/5 5/5 APPEND APPEND APPEND APPEND APPEND CloseLoop Backtrack (C,ar,C,X4) (C,ar,b,x,) (C,ar,C,X。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=') (C,ar,c,X,) (C,-,b,Xg) b b APPEND APPEND Backtrack Backtrack Backtrack Backtrack TER (N,-,C,Xg) (O,=,C,X1) Carbon Focused atom Loop target Single bond Nitrogen Branching atom Loop count down Double bond 0/5 Aromatic bond Oxygen Potential loop targetFigure S5: Issues related to ring generation when the synthon dataset is changed, and how the refined ring-based generation scheme can alleviate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The original synthon dataset, and the structures of generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The six-membered ring is removed from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Without retraining, the model will not acknowledge this change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' As a result, the model will still attempt to generate six-membered rings, but the generation will be terminated in advance due to the constraints on available synthons, resulting in problematic structures shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' However, if the ring size is determined before generating its structures, the model will acknowledge the change in the synthon dataset, since the action of generating a six-membered ring is masked due to the imposed synthon constraints, resulting in molecule structures with higher quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-29 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' DeepLigBuilder+ Synthon dataset Generated structure b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Removed DeepLigBuilder+ Synthon dataset Generated structure Removed Blocked DeepLigBuilder+ Synthon dataset Generated structureFigure S6: Examples of several generated 3D molecules using the unconditional transformer, along with the building blocks and proposed reactions to synthesize the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-30 Structure S1 Structure S2 Structure S3 Topological structure HC Building HaN blocks HO N NH2 N (1) (2) (3) (1) (2) (3) (1) (2) (3) (1) + (2): (1) + (2) & (1-2) + (3): (1) + (2): Grignard reactiont Reactions Oxadiazole formation Schotten-Baumann reaction (1-2) + (3): using hydroxylamine (1-2) + (3): Wittig reaction Schotten-Baumann reaction Generated structuresFigure S7: The distribution of 2D and 3D molecular properties of the generated (blue),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' test set (grey),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' and randomly assembled (red) molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 2D properties includes: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Molecular weight, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' LogP, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' QED, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' the number of hydrogen bond acceptors (HBA), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' the number of hydrogen bond donors, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The number of rotatable bonds (ROT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' 3D properties includes: g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The total amount of solvent accessible surface area (SASA), h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' polar solvent accessible surface areas (PolarSASA), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' the radius of gyration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-31 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Density Generated Assembled Test set 300 400 500 600 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='0 Molecular Weight LogP QED f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Density Generated Assembled Test set 0 5 10 0 2 4 6 5 10 0 15 HBA HBD ROTB h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Density Generated Assembled Test set 500 600 700 800 900 100 200 300 4003 4 5 6 7 SASA Polar SASA Radius of gyrationFigure S8: A t-SNE visualization of the distribution of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Morgan fingerprint and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' US- RCAT fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Blue dots represent generated molecules, grey dots represent test set molecules, and red dots represents molecules randomly assembled from the building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure S9: The distribution of RMSD values after relaxation with MMFF94s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Blue: molecules with conformations generated by the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Grey: molecules with conforma- tions generated by the ETKDG method provided by RDKit S-32 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Generated Assembled Test set1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='0- Probability Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='0 - ETKDG Generated 0 2 3 4 5 RMSD (A)Figure S10: The detailed structure of the search tree for the first synthon in the BTK’s ATP-binding pocket (only containing nodes with visit count larger than 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Some gener- ated synthon structures are shown below the tree, with important pharmacophore features highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The leftmost example shows a state with a low estimated Q-value, which largely resulted from the unfavorable anchor position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' As the result, this state is not as frequently visited as other states with higher Q-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-33 Seed Initial state Q-values: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='0 Anchor atom Hydrophobic Unfavorable anchor position HBD HBAFigure S11: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The topological structure of GDC-0853 (Fenebrutinib).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The part of GDC- 0853 used extract pharmacophore, shape, and seed for molecule generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Note that some part of GDC-0853 inside the solvent-exposed region is not considered for pharmacophore and shape extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure S12: The proposed synthetic path for Structure 2 (a) and Structure 3 (b) by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Functional groups marked in grey needs to be protected before the reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-34 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' GDC-0853 (Fenebrutinib) Kd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='91 nM Smina score: -14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The sub-structure used to generate the shape and pharmacophore conditions, as well as the seed Solvent structure for molecule growth exposed region Smina score: -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6 O HBD HBA Hydrophobic Shape o Seed atomNegishi reaction EN300-6496494 Grignard reaction EN300-72460 EN300-50850 Structure 2 b Huisgen reaction EN300-79402 Buchwald-Hartwig reaction EN300-20284 EN300-107725 Structure 3Figure S13: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The topological structure of Compound 15, a potent inhibitor targeting the NAD pocket of PHGDH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' The pharmacophore features extracted from the binding mode of the molecule, as well as the seed location for molecule growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Figure S14: The proposed synthetic path for Structure 6 (a) and Structure 7 (b) by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Functional groups marked in grey needs to be protected before the reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' S-35 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Cl 1 Compound 15 (Mullarky et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=') OHBD HBA● Hydrophobic Kd = 15 nM Negative charge :: Shape Seed atom Smina score: -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='9Sonogashira reaction EN300-82406 Structure 6 Urea formation EN300-27699201 EN300-139617 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Grignard HN reaction EN300-27745883 Structure 7 Urea formation EN300-58858 EN300-824063 Supplementary tables Table S1: A summary of different hyperparameter configurations experimented in this work (BBs: building blocks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' LR: learning rate) Model architecture Training parameter Generation parameter Group Variant name IPA 3D pair embedding Width (F) DepthDecay frequency (#steps) Noise probability Noise scale(Å) Building block Default configuration ✓ ✓ 512 6 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Global BBs EU stock ✓ ✓ 512 6 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 EU Comprehensive catalog ✓ ✓ 512 6 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Comprehensive 3D en- coding Dropped IPA × ✓ 512 6 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Global Dropped 3D pair embedding ✓ × 512 6 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Global Scale Narrow network ✓ ✓ 256 6 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Global Shallow network ✓ ✓ 512 3 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Global LR Fast learning rate decay ✓ ✓ 512 6 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Global S-36 Model architecture Training parameter Generation parameter Slow learning rate decay ✓ ✓ 512 6 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Global Noise High noise probability ✓ ✓ 512 6 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Global Low noise probability ✓ ✓ 512 6 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 Global High noise scale ✓ ✓ 512 6 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 Global Low noise scale ✓ ✓ 512 6 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05 Global S-37 Table S2: The distribution of 2D molecular properties among molecules generated from models with different hyperparameters (see Table S1 for a detailed description of each configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We report the mean and standard deviation for each property, as well as the estimated Wasserstein distance to the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For the first row, we report the statistics of molecules randomly assembled from building blocks without any drug-likeness filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Model variant MW LogP HBA HBD ROT QED mean std wd mean std wd mean std wd mean std wd mean std wd mean std wd Randomly assembled 454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='32 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='45 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='46 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='57 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='33 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='18 Default configuration 412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='31 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='39 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='03 EU stock 416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='63 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='96 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='02 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='03 High noise probability 414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='14 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='61 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='03 S-38 Model variant MW LogP HBA HBD ROT QED Low noise probability 416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='82 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='94 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05 Large noise scale 413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='76 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='78 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='09 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='03 Small noise scale 413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='24 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='93 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='09 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='04 Validation set 419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='52 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='27 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='92 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='09 Test set 419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='72 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='13 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='11 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='92 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='09 S-39 Table S3: The distribution of 3D molecular properties among molecules generated from models with different hyperparameters (see Table S1 for a detailed description of each configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' We report the mean and standard deviation for each property, as well as the estimated Wasserstein distance to the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' For the first row, we report the statistics of molecules randomly assembled from building blocks without any drug-likeness filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Model variant SASA Polar SASA Radius of gyration mean std wd mean std wd mean std wd Randomly assembled 667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='72 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='84 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='41 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='77 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='93 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3 Default configuration 617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='86 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='27 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='07 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='93 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='96 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05 EU stock 624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='14 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='58 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='01 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='68 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='83 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='03 Comprehensive catalog 620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='45 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='08 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='91 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='53 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='83 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='04 Dropped IPA 620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='16 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='23 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='24 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='19 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='44 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='03 Dropped 3D pair embedding 621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='13 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='84 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='8 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='87 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='02 Narrow network 618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='08 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='04 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='95 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='56 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='03 Shallow network 619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='66 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='62 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='59 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='02 Fast learning rate decay 616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='74 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='97 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='34 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='7 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='63 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='04 Slow learning rate decay 619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='42 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='54 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='15 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='41 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05 High noise probability 620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='89 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='67 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='99 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='02 Low noise probability 614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='44 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='41 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='35 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='52 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='61 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='96 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='15 Large noise scale 619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='44 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='45 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='75 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='03 Small noise scale 618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='84 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='38 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='16 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='05 Validation set 627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='61 50.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='795 Large noise scale 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='000163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='000080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='685 Small noise scale 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Meier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Nieto-Oberhuber, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Altmann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Schnei- der, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Jacoby, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' Renner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQfWvcM/content/2301.00167v1.pdf'} +page_content=' A Collection of Robust Organic Synthesis 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a/ZtAyT4oBgHgl3EQfWvfe/content/tmp_files/2301.00171v1.pdf.txt b/ZtAyT4oBgHgl3EQfWvfe/content/tmp_files/2301.00171v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d949d4eeae611246660b99eb913ce0761de5601 --- /dev/null +++ b/ZtAyT4oBgHgl3EQfWvfe/content/tmp_files/2301.00171v1.pdf.txt @@ -0,0 +1,642 @@ +arXiv:2301.00171v1 [math.GR] 31 Dec 2022 +EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF +FREE PRODUCTS +JIALIN LEI AND QIANG ZHANG +ABSTRACT. For an automorphism φ of a free group Fn of rank n, Bestvina and Handel +showed that the rank rkFix(φ) of the fixed subgroup is not greater than n (the so-called +Scott conjecture). Soon after Bestvina and Handel’s announcement, their result was gen- +eralized by many authors in various directions. In this paper, we are interested in the fixed +subgroups of endomorphisms of free products, focusing on explicit bounds for their ranks. +1. INTRODUCTION +For a finitely generated group G, the rank of G denoted rk(G) is the minimal number of +generators of G. There are many researches on the rank of subgroups in finitely generated +groups. For an abelian group G, the rank rk(H) of a subgroup group H of G can not be +greater than rk(G). However, in general, the statement is false even in free groups. It is +easy to see that the free group Fn of rank n is a subgroup of F2 for all n ∈ N. Denote +the monoid of endomorphisms (resp. monomorphisms, i.e. injective endomorphisms) of +G by End(G) (resp. Mon(G)), and the group of automorphisms of G by Aut(G). For an +endomorphism φ ∈ End(G), the fixed subgroup of φ is defined to be +Fix(φ) := {g ∈ G | φ(g) = g}, +that is a subgroup of G with many special properties. +For a free group Fn of rank n, in 1975, Dyer and Scott [7] proved that for a finite or- +der automorphism φ of Fn, the rank rkFix(φ) of the fixed subgroup is not greater than +n. Moreover, Scott conjectured that rkFix(φ) ≤ n for any φ ∈ Aut(Fn), which is the +so-called Scott conjecture. Once the conjecture was put forward, it attracted a lot of re- +search, see [6] for a survey. Finally, in 1989, the conjecture was solved by Bestvina and +Handel [2]. Soon after Bestvina and Handel’s announcement, their result was generalized +by many authors in various directions (for example, [4, 9, 5, 6, 1, 12, 28, 27, 31] etc.). In +particular, by introducing an important concept of stable image (see Section 2 for more +details), Imrich and Turner [9] reduced the fixed subgroups of endomorphisms to that of +automorphisms, and then showed +Theorem 1.1 (Imrich-Turner, [9]). If φ ∈ End(Fn), then rkFix(φ) ≤ n. +For surface groups, a lot is known too. In this paper, a surface group is the fundamental +group π1(S) of a closed (orientable or not) surface S with Euler characteristic χ(S) < 0. +Nielsen [14, 15], Jaco-Shalen [10], and Zieschang [32] gave the following results. (Alter- +native proofs can also be found in [12].) +Date: January 3, 2023. +2010 Mathematics Subject Classification. 20F65, 20F34, 57M07. +Key words and phrases. Fixed subgroups, free products, Gromov hyperbolic groups, surface groups. +The second author is partially supported by NSFC (Nos. 11961131004 and 11971389). +1 + +2 +JIALIN LEI AND QIANG ZHANG +Theorem 1.2 (Nielsen [14, 15], Jaco-Shalen [10], and Zieschang [32]). Suppose G is a +surface group. Then for any endomorphism φ ∈ End(G), we have +(1) rkFix(φ) ≤ rk(G) if φ is epimorphic, with equality if and only if φ = id; +(2) rkFix(φ) ≤ 1 +2rk(G) if φ is not epimorphic. +To our knowledge, there are only a few results of this type for the fundamental group +of a 3-manifold, see [11, 13, 29, 30]. In particular, Lin and Wang [13], investigated the +fundamental group π1(M) of a hyperbolic 3-manifold M, i.e., M is compact, orientable, +and the interior of M admits a complete hyperbolic structure of finite volume (then M is +either closed or the boundary ∂M of M is a union of tori. Note that when M is closed, +π1(M) is hyperbolic in the sense of Gromov, while ∂M is a union of tori, π1(M) contains +a subgroup isomorphic to Z × Z). +Theorem 1.3 (Lin-Wang, [13]). Suppose φ is an automorphism of G = π1(M), where M +is a hyperbolic 3-manifold. Then rkFix(φ) < 2rk(G). +For a Gromov hyperbolic group G, Paulin [17] proved that the fixed subgroup Fix(f) +is finitely generated for any automorphism f ∈ Aut(G). See [21, 8] for more information +on the fixed subgroups in hyperbolic groups. +In [27, 31], the authors investigated direct products of finitely many free groups and +surface groups, and showed +Theorem 1.4 (Ventura-Wu-Zhang, [31]). Let G = ×n +i=1Gi be a direct product of surface +groups and free groups. If neither of the factors is cyclic, then for any automorphism +φ ∈ Aut(G), +rkFix(φ) ≤ rk(G). +Otherwise, if G contains a non-cyclic factor and a factor Z, then there exists f ∈ Aut(G) +such that Fix(f) is not finitely generated. +In this paper, we are mainly interested in the fixed subgroups of free products, focusing +on explicit bounds for their ranks. +For a group G, we say that G has the finitely generated fixed subgroup property of +monomorphisms (resp. automorphisms, endomorphisms), abbreviated as FGFPm (resp. +FGFPa, FGFPe), if for any f ∈ Mon(G) (resp. Aut(G), End(G)), the fixed subgroup +Fix(f) is finitely generated. Clearly, if G has FGFPe, then it has FGFPm and hence +has FGFPa. Theorem 1.1 and 1.2 imply that free groups and surface groups have FGFPe. +Furthermore, Theorem 1.4 shows that the free groups F2 and Z both have FGFPa but their +direct product don’t (see also Lemma 3.2). Then it is natural to ask +Question 1.5. If two groups G1 and G2 both have FGFPm (FGFPa or FGFPe), then, +what about their free product G1 ∗ G2? +For the cases FGFPm and FGFPa, we have an affirmative answer to Question 1.5. +Theorem 1.6. A free product ∗n +i=1Gi has FGFPm (resp. FGFPa) if and only if the factor +groups G1, G2, . . . , Gn all have FGFPm (resp. FGFPa). +Since the fundamental group π1(M) of a hyperbolic 3-manifold M is co-Hopfian (see +[26]), every φ ∈ Mon(π1(M)) is an automorphism. As a direct application of Theorem +1.3, Theorem 1.6 and Proposition 3.3, we have +Corollary 1.7. Let M = #n +i=1Mi be a connected sum of finitely many hyperbolic 3- +manifolds. Then the fundamental group π1(M) has FGFPm (and hence FGFPa). More +precisely, for any monomorphism f ∈ Mon(π1(M)), we have rkFix(f) < 2n · rkπ1(M). + +EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF FREE PRODUCTS +3 +Moreover, to quantitatively analysis the ranks of fixed subgroups of a group G, we +introduce a concept of UFP degree, denoted Duf(G) (see Definition 3.1) and show some +explicit bounds for the fixed subgroups, see Proposition 3.3 and Proposition 3.6 in Section +3. +For FGFPe, we can give affirmative answers for some special kinds of groups. +Theorem 1.8. Let G = ∗n +i=1Gi, where each factor Gi is a torsion-free hyperbolic group +with finite UFP degree Duf(Gi). For an endomorphism φ ∈ End(G), if φn(G) is hyper- +bolic for arbitrary lager n, then +rkFix(φ) ≤ 1 +4ℓ(rk(G) + 1)2, +where the number ℓ = maxn +i=1 Duf(Gi), the maximal one of Duf(Gi), i = 1, . . . , n. +Remark 1.9. (1) Note that a subgroup of a hyperbolic group may be not hyperbolic, as +conjectured by O’Neill and Turner [16], we do not know whether or not every torsion-free +hyperbolic group has the property that φn(G) is hyperbolic for arbitrary lager n for any +φ ∈ End(G). (2) Sela [19] proved that a non-elementary, torsion-free hyperbolic group +is co-Hopfian (i.e. every monomorphism is an automorphism) if and only if it is freely +indecomposable. Thus, for such groups, FGFPm is equivalent to FGFPa. +It is well-known that surface groups and free groups are torsion-free hyperbolic in the +sense of Gromov, and a subgroup of a surface group is either a surface group or a free +group. Let G = ∗n +i=1Gi be a free product with each factor Gi a free group or a surface +group. Then G is a torsion-free hyperbolic group satisfying the hypothesis of Theorem 1.8 +with the UFP degree Duf(Gi) = 1, and hence, as a corollary, we have explicit bounds for +the fixed subgroups of G. +Theorem 1.10. Let G = ∗t +i=1Gi ∗ Fs be a free product, where Fs is a free group of rank +s, and each factor Gi is a surface group. Then +(1) for any endomorphism φ ∈ End(G), the fixed subgroup has rank +rkFix(φ) ≤ 1 +4(rk(G) + 1)2. +(2) for any φ ∈ Mon(G), we have rkFix(φ) ≤ (s + t)(rk(G) − s − t + 1). In particular, +if s = 0 and all the surface groups Gi share the same rank, then +rkFix(φ) ≤ rk(G). +In [18], Rodaro, Silva and Sykiotis studied fixed subgroups of graph groups (i.e., right +angled Artin groups) and showed +Theorem 1.11. [18, Theorem 3.1] Let G be a graph group. Then the following two condi- +tions are equivalent +(1) Fix(φ) is finitely generated for every endomorphism φ ∈ End(G); +(2) G is a free product of finitely many free abelian groups of finite rank. +Now, we give explicit bounds for the fixed subgroups of free products of free abelian +groups. +Theorem 1.12. Let G = ∗n +i=1Zti be a free product of free abelian groups Zti of rank ti. +Then for any endomorphism φ ∈ End(G), we have +rkFix(φ) ≤ n(rk(G) − n + 1). +In particular, if the ranks t1 = t2 = . . . = tn, then rkFix(φ) ≤ rk(G) = �n +i=1 ti. + +4 +JIALIN LEI AND QIANG ZHANG +The paper is organized as follows. In Section 2, we review some useful facts on free +products, especially Sykiotis’ work on symmetric endomorphisms and the structure of +fixed subgroups of free products. In Section 3, we introduce some concepts on fixed sub- +groups to quantitatively analysis the ranks. At last, in Section 4, we give proofs of the main +results of this paper. +2. PRELIMINARIES +In papers [23, 24], Sykiotis studied the fixed subgroups of symmetric endomorphisms +of free products of groups. For later use, let us first review some important definitions and +facts in this section. +2.1. Stable image. To study the fixed subgroups of endomorphisms of free groups, Imrich +and Turner [9] introduced the following definition. +Definition 2.1 ([9]). For a group G and an endomorphism φ ∈ End(G), the stable image +φ∞(G) of φ is the intersection +φ∞(G) := +∞ +� +n=1 +φn(G). +Clearly, the fixed subgroup Fix(φ) = Fix(φ∞) ≤ φ∞(G), where φ∞ : φ∞(G) → +φ∞(G) is the restriction of φ to the stable image φ∞(G). +For a free group Fn, Imrich and Turner proved that for arbitrary endomorphism φ ∈ +End(Fn), φ∞ ∈ Aut(φ∞(Fn)). Here φ∞(Fn) is a subgroup of Fn, so it is a free group. +They also proved that rk(φ∞(Fn)) ≤ rk(Fn) = n. Then the Bestvina-Handel theorem +(Scott conjecture) implies that rkFix(φ) = rkFix(φ∞) ≤ rk(φ∞(Fn)) ≤ n. +2.2. Kurosh rank. Let G be a group and let H be a nontrivial subgroup of G. A famous +result of Kurosh showed that, G can be represented as a free product of freely indecompos- +able factors, G = ∗n +i=1Gi, and the set of factors (and hence the number n of the factors) +is well-defined up to isomorphism. Then n is said to be the (absolute) Kurosh rank of G, +denoted Krk(G). By the Kurosh subgroup theorem, the subgroup H is a free product +H = ∗t +j=1Hj ∗ Fs, +where Fs is a free group of rank s and every factor Hj is the intersection of H with a +conjugate of some factor Gi (either of s or t could be 0 or infinity). If the representation +of G as a free product is changed by an isomorphism, then these intersections change, but +the number s + t is invariant. We say s + t the Kurosh rank of H in G. denoted KrkG(H). +Note that Grushko’s theorem states that the rank of groups is additive under free prod- +ucts, i.e., rk(∗n +i=1Gi) = �n +i=1 rk(Gi). Thus +KrkG(H) ≤ Krk(H) ≤ rk(H). +The first equality holds if s = 0 and none of Hi can be split as a nontrivial free products, +and the second equality holds if each Hj is cyclic. +2.3. Symmetric endomorphism. Let G = ∗n +i=1Gi and K = ∗m +i=1Ki be two free products +(the factors Gi and Ki may be freely decomposable). Sykiotis [24] gave the following +definition. +Definition 2.2 ([24]). A homomorphism φ : G → K is said to be symmetric if each non- +infinite-cyclic free factor of G is mapped by φ into a conjugate of some non-infinite-cyclic +free factor of K. + +EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF FREE PRODUCTS +5 +It is easy to see that if each factor Gi is freely indecomposable, then each injective +homomorphism is symmetric. Since surface groups are freely indecomposable, we have +Lemma 2.3. Let G = ∗t +i=1Gi ∗ Fs be a free product, where Fs is a free group of rank +s, and each factor Gi is a surface group. Then every monomorphism φ ∈ Mon(G) is +symmetric. +For a free product G = ∗n +i=1Gi of freely indecomposable factors, Collins and Turner +[5] studied the fixed subgroups of an automorphism φ ∈ Aut(G), and showed that the +Kurosh rank KrkG(Fix(φ)) of the fixed subgroup can not exceed Krk(G) = n. Sykiotis +[24] extended Collins and Turner’s result to monomorphisms. Since every group can be +represented as a free product of freely indecomposable factors, we translate [24, Corollary +4] to the following. +Theorem 2.4 (Sykiotis, [24]). Let G be a group and φ ∈ Mon(G) a monomorphism. Then +the Kurosh rank +KrkG(Fix(φ)) ≤ Krk(G). +In general, the fixed subgroups of endomorphisms of a free product is more complicate +than that of monomorphisms. But for some special groups, we can get some bounds for +the fixed subgroups. For example, Sykiotis [24] gave the following bound on the fixed +subgroups of free products of nilpotent groups. +Theorem 2.5. [24, Theorem 7] Let G = ∗n +i=1Gi be a free product of finitely generated +nilpotent and finite groups. If φ ∈ End(G) is an endomorphism of G, then the fixed +subgroup Fix(φ) of φ has Kurosh rank at most n. +Sykiotis also showed the structure of the fixed subgroup of symmetric endomorphisms. +Theorem 2.6. [23, Theorem 1.2] Let G be a group which acts on a tree X with finite +quotient graph and finite edge groups. Suppose that φ is a symmetric endomorphism of G. +Then the fixed subgroup Fix(φ) of G is the fundamental group of a finite graph of groups +with the following properties. +(1) The edge groups are contained in groups of the form �n +i=1 giGeig−1 +i +, where ei are +edges of X and gi ∈ G. +(2) The vertex groups are either of the form Fix(φ|Gv) where v is a vertex of X fixed by +˜φ, or are contained in groups of the form in item (1), and hence are finite. +By using Bass-Serre theory, combining Theorem 2.4 and Theorem 2.6, we have the +following. +Theorem 2.7. Let G = ∗n +i=1Gi be a free product of freely indecomposable factors. Then +for any monomorphism φ ∈ Mon(G), +Fix(φ) = ∗t +j=1Fix(φ|gjGσ(j)g−1 +j ) ∗ Fs, +where KrkG(Fix(φ)) = s + t ≤ n, and φ|gjGσ(j)g−1 +j +: gjGσ(j)g−1 +j +→ gjGσ(j)g−1 +j +is the +restriction of φ to a conjugate of some factor Gσ(j). +Proof. Since G = ∗n +i=1Gi is a free product, G acts on a tree X with finite quotient graph +and trivial edge groups. Moreover, the stabilizer Gv of a vertex v is a conjugate of some +factor Gi. Then the conclusion of Theorem 2.7 is clear. +□ + +6 +JIALIN LEI AND QIANG ZHANG +2.4. Gromov hyperbolic group. Now we consider torsion-free hyperbolic groups. In +[16], O’Neill and Turner studied the stable image of hyperbolic groups, and gave the fol- +lowing. +Proposition 2.8. [16, Proposition 2] Let G be a torsion-free hyperbolic group. If φ : G → +G is an endomorphism with the property that φn(G) is hyperbolic for arbitrary lager n, +then φ∞(G) is a free factor of φN(G) for some N. +In [20], Sela gave the following. +Definition 2.9. [20, Definition 1.11] Let Γ be a torsion-free hyperbolic group. A finitely +generated group G is said to be a Γ-limit group, if G is isomorphic to a subgroup of Γ or if +G is a (strict) Γ-limit group. +Theorem 2.10. [20, Theorem 1.12] Let Γ be a torsion-free hyperbolic group, and let G be +a finitely generated group. Every decreasing sequence of Γ-limit groups that are quotients +of G, +R1 > R2 > R3 > · · · +terminates after finitely many steps. +We do not need the concept of strict Γ-limit group but the relation “>”. Let Γ be a +torsion-free hyperbolic group, and let G be a finitely generated group. On the set of Γ- +limit groups, we define a relation. Given two Γ-limit groups R1, R2, that are quotients of +G, with prescribed maps ηi : G → Ri, i = 1, 2 we say that R1 > R2, if there exists an +epimorphism with non-trivial kernel: τ : R1 → R2, so that η2 = τ ◦ η1. +For an endomorphism φ ∈ End(G) of a finitely generated torsion-free hyperbolic group +G, it is easy to see that φn(G) is a G-limit group. Moreover, we have an analogue of [24, +Lemma 5] as follows. +Proposition 2.11. Let G be a finitely generated torsion-free hyperbolic group, and φ ∈ +End(G) an endomorphism. Then the restriction φ∞ : φ∞(G) → φ∞(G) is an automor- +phism, and the Kurosh rank +Krkφ∞(G)(Fix(φ)) ≤ Krk(φ∞(G)). +Proof. Let φn = φ|φn(G) : φn(G) → φn+1(G) be the restriction of φ to φn(G). First, +we prove that the kernel ker(φN) is trivial for some N, i.e., φN is injective. To see this, +assume that ker(φn) is non-trivial for all n. Since the subgroups φn(G) of G are also +quotients of G, we have a decreasing sequence of G-limit groups as in Theorem 2.10 but +with infinitely many terms, +G > φ(G) > φ2(G) > φ3(G) > · · · , +we get a contradiction. +Below we show that φ∞ is an automorphism. Note that for any +g ∈ φ∞(G) = +∞ +� +n=N +φn(G), +there exists gn ∈ G such that φn(gn) = g for every n ≥ N. Let cn = φn(gn+1) ∈ φn(G). +Then φ(cn) = g. Since φN is injective, we have cN = cn ∈ φn(G) for all n ≥ N and +hence cN ∈ φ∞(G). This gives surjectivity of φ∞, and hence φ∞ is an automorphism. +Note that Fix(φ) = Fix(φ∞) ≤ φ∞(G), by using Theorem 2.4, we have +Krkφ∞(G)(Fix(φ)) = Krkφ∞(G)(Fix(φ∞)) ≤ Krk(φ∞(G)). + +EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF FREE PRODUCTS +7 +The proof is completed. +□ +Remark 2.12. Although many hyperbolic groups have been showed to be residually fi- +nite, Gromov’s famous question remains open: Is every hyperbolic group residually fi- +nite? Therefore, Proposition 2.11 is not a direct corollary of [24, Lemma 5]: Let φ be +an endomorphism of a finitely generated residually finite group G. Then the restriction +φ∞ : φ∞(G) → φ∞(G) of φ to φ∞(G) is a monomorphism. +3. FIXED SUBGROUPS OF MONOMORPHISMS +In this section, we first introduce two properties on fixed subgroups, and then show +some results on the fixed subgroups of monomorphisms of free products. +3.1. FGFP property. Recall that a group G has FGFPm (resp. FGFPa, FGFPe), if for +any f ∈ Mon(G) (resp. Aut(G), End(G)), the fixed subgroup Fix(f) is finitely gener- +ated. Clearly, if G has FGFPe, then it has FGFPm and hence has FGFPa. Moreover, +the property FGFPa is not heritable, for example, according to Theorem 1.4, the group +Fn × Fm for m, n > 1 has FGFPa but its subgroup F2 × Z does not. To quantitatively +analysis the ranks of the fixed subgroups, we have +Definition 3.1. Let G be a finitely generated group, and let Mon(G) denote the monoid of +monomorphisms of G. +(1) G is said to have k-FGFP, if for any φ ∈ Mon(G), rkFix(φ) ≤ k · rk(G). The +minimal number k satisfying the above equation is said to be the FP degree for the +group G, denoted Df(G). Namely, +Df(G) := sup{rkFix(φ) +rk(G) +| φ ∈ Mon(G)} ∈ [1, +∞]. +(2) G is said to have k-UFGFP (“U” for uniformly), if for every finitely generated sub- +group H ≤ G and any φ ∈ Mon(H), rkFix(φ) ≤ k · rk(H). The minimal number k +satisfying the above equation is said to be the UFP degree for the group G, denoted +Duf(G). Namely, +Duf(G) := sup{rkFix(φ) +rk(H) +| H ≤ G, φ ∈ Mon(H)} ∈ [1, +∞]. +Note that in the above definitions, we only consider monomorphisms, and omit the +cases of endomorphisms and automorphisms because that are similar. Clearly, k-UFGFP +implies k-FGFP, and many kinds of groups have k-FGFP. Furthermore, the property +k-UFGFP is heritable. In precisely, +Lemma 3.2. Let G be a finitely generated group and let H ≤ G be a finitely generated +subgroup. +(1) 1 ≤ Df(G) ≤ Duf(G) ≤ ∞; +(2) If G has k-UFGFP, then its subgroup H also has k-UFGFP and hence has k-FGFP, +i.e., +Df(H) ≤ Duf(H) ≤ Duf(G) ≤ k. +(3) Df(G) = Duf(G) = 1 if G is one of the following, +(a) free abelian groups Zn, +(b) free groups Fn, +(c) surfaces groups. +(4) Df(F2×Z) = ∞, and hence Duf(G) = ∞ if G contains a subgroup that is isomorphic +to F2 × Z. + +8 +JIALIN LEI AND QIANG ZHANG +Proof. Items (1) and (2) are trivial. Item (3) follows from Theorem 1.1 and Theorem 1.2 +clearly. To prove item (4), it suffices to show that Fix(f) is not finitely generated for some +f ∈ Mon(F2 × Z). Indeed, let +F2 × Z = ⟨a, b, t | [a, t], [b, t]⟩, +and let f : F2 ×Z → F2 ×Z such that a �→ at, b �→ b, t �→ t. Then an element u ∈ Fix(f) +if and only if it has zero exponent sum in a. So Fix(f) is isomorphic to F∞ × Z generated +by the infinite set {t, aiba−i|i ∈ Z}. +□ +3.2. FGFP in free products. Now we have the following key proposition. +Proposition 3.3. Let G = ∗n +i=1Gi be a free product, where each factor Gi is a freely +indecomposable group satisfying rkFix(f) ≤ ki · rk(Gi) for any f ∈ Mon(Gi). Then for +any monomorphism φ ∈ Mon(G), we have +rkFix(φ) ≤ n( +n +max +i=1 ki)(rk(G) − n + 1), +in particular, if all the factors Gi have the same rank, then +rkFix(φ) ≤ ( +n +max +i=1 ki) · rk(G). +Before give the proof, we have a direct corollary on the FP degree of fixed subgroups +as follows. +Corollary 3.4. Let G = ∗n +i=1Gi, where each Gi is a freely indecomposable group with +finite FP degree Df(Gi). Then +Df(G) ≤ n · +n +max +i=1 Df(Gi), +in particular, if all the factors Gi have the same rank, then Df(G) ≤ maxn +i=1 Df(Gi). +Proof of Proposition 3.3. According to Theorem 2.7, Fix(φ) = ∗s +i=1Hi∗Ft, where s+t ≤ +n and each Hi = Fix(φ|giGσ(i)g−1 +i ) is the fixed subgroup of a conjugate of some Gσ(i). +Since Gi’s have the property rkFix(f) ≤ ki · rk(Gi) for any f ∈ Mon(Gi), we have +rk(Hi) ≤ kσ(i) · rk(giGσ(i)g−1 +i +) = kσ(i) · rk(Gσ(i)) ≤ kσ(i) · (rk(G) − n + 1). +It follows +rkFix(φ) += +t + +s +� +i=1 +rk(Hi) ≤ t + +s +� +i=1 +kσ(i) · rk(Gσ(i)) +≤ +n( +n +max +i=1 ki)(rk(G) − n + 1). +(3.1) +In particular, if all the factors Gi have the same rank rk(G1) = · · · = rk(Gn), then +rkFix(φ) ≤ t + +s +� +i=1 +kσ(i) · rk(Gσ(i)) ≤ ( +n +max +i=1 ki) · rk(G). +□ +Remark 3.5. Note that in the previous proof, Gσ(i) may equal to Gσ(j) for distinct i ̸= j, +see Section 4 for an example. + +EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF FREE PRODUCTS +9 +Proposition 3.6. Let G = ∗n +i=1Gi be a free product, where each factor is a (not necessarily +freely indecomposable) group Gi with finite UFP degree Duf(Gi). Then for any subgroup +H ≤ G and any monomorphism φ ∈ Mon(H), +rkFix(φ) ≤ 1 +4ℓ(rk(H) + 1)2, +where ℓ = maxn +i=1 Duf(Gi). +Proof. Let H ≤ G be a finitely generated subgroup and φ ∈ Mon(H) a monomorphism. +Split H as a free product of freely indecomposable factors, H = ∗s +j=1Hj, where the +absolute Kurosh rank (other than the Kurosh rank in G) Krk(H) = s ≤ rk(H), and each +factor Hj is either a freely indecomposable group contained in a conjugate of a factor Gij +or the free cyclic group Z (with Duf(Z) = 1 clearly). Then Lemma 3.2 implies +Duf(Hj) ≤ Duf(Gij ) ≤ +n +max +i=1 Duf(Gi) = ℓ, +and hence rkFix(f) ≤ ℓ · rk(Hj) for any f ∈ Mon(Hj). Applying Proposition 3.3 to the +monomorphism +φ : H → H = ∗s +i=jHj, +we have rkFix(φ) ≤ ℓs(rk(H) − s + 1) ≤ 1 +4ℓ(rk(H) + 1)2. +□ +4. MAIN RESULTS AND SOME EXAMPLES +In this section, we first prove the main theorems, and then give some examples. +Theorem 1.6. A free product ∗n +i=1Gi has FGFPm (resp. FGFPa) if and only if the factor +groups G1, G2, . . . , Gn all have FGFPm (resp. FGFPa). +Proof. Since the proofs of the two cases FGFPm and FGFPa are parallel, we only con- +sider the case FGFPm, and leave the other case FGFPa for the reader. +Ar first, we prove the “only if” part. Without loss of generality, suppose G1 dose not +have FGFPm, i.e., there is a monomorphism f : G1 → G1 such that Fix(f) is not finitely +generated. Let idi ∈ Mon(Gi) be the identity of Gi. Then the free product f ∗ id2 ∗ · · · ∗ +idn ∈ Mon(G). Clearly, the fixed subgroup +Fix(f ∗ id2 ∗ · · · ∗ idn) = Fix(f) ∗ G2 ∗ · · · ∗ Gn +is not finitely generated, contradicting the hypothesis that G has FGFPm. +Now let us consider the “if” part. There are two cases: +(i) All the factors Gi are freely indecomposable. Then any f ∈ Mon(G) maps each +factor Gi to a conjugate of some Gj, as in the proof of Proposition 3.3, we have Fix(f) = +∗s +i=1Hi ∗ Ft, where s + t ≤ n and each Hi = Fix(f|giGσ(i)g−1 +i ) is the fixed subgroup of a +conjugate of some Gσ(i). Since Gi all have FGFPm, the Hi are all finitely generated and +hence Fix(f) is also finitely generated. +(ii) Some factors Gi are freely decomposable. Then each Gi can be decomposed into +Gi = G′ +i1 ∗ · · · ∗ G′ +ij, where each factor G′ +ik is freely indecomposable. Since Gi has +FGFPm, the factors G′ +ik all have FGFPm by the “only if” part. Therefore, we have +reduced case (ii) to case (i). +□ +Proof of Corollary 1.7. Since each Mi is a hyperbolic 3-manifold, the fundamental group +π1(Mi) is co-Hopfian by [3], i.e., every φ ∈ Mon(π1(Mi)) is an automorphism. Then +Theorem 1.3 implies that π1(Mi) all have FGFPm. Note that the fundamental group + +10 +JIALIN LEI AND QIANG ZHANG +π1(M) = ∗n +i=1π1(Mi), and each π1(Mi) is freely indecomposable because Mi is a hyper- +bolic 3-manifold, then Theorem 1.6 and Proposition 3.3 imply that π1(M) has FGFPm +and +rkFix(f) < 2n · rkπ1(M) +for any f ∈ Mon(π1(M)). +□ +Theorem 1.8. Let G = ∗n +i=1Gi, where each factor Gi is a torsion-free hyperbolic group +with finite UFP degree Duf(Gi). For an endomorphism φ ∈ End(G), if φn(G) is hyper- +bolic for arbitrary lager n, then +rkFix(φ) ≤ 1 +4ℓ(rk(G) + 1)2, +where the number ℓ = maxn +i=1 Duf(Gi), the maximal one of Duf(Gi), i = 1, . . . , n. +Proof. Since G is a free product of torsion-free hyperbolic groups, G itself is also hyper- +bolic and torsion-free. Then Proposition 2.11 implies that the restriction φ∞ = φ|φ∞(G) : +φ∞(G) → φ∞(G) is an automorphism. Furthermore, since G satisfies the condition that +φn(G) is hyperbolic for arbitrary lager n, by Proposition 2.8, the stable image φ∞(G) is a +free factor of φN(G) for some N. So +rk(φ∞(G)) ≤ rk(φN(G)) ≤ rk(G), +the last inequality holds because φN(G)) is a quotient of G. Note that Fix(φ) = Fix(φ∞) +and φ∞(G) ≤ G, applying Proposition 3.6 to the automorphism φ∞ : φ∞(G) → φ∞(G), +we have +rkFix(φ) = rkFix(φ∞) ≤ 1 +4ℓ · (rk(φ∞(G)) + 1)2 ≤ 1 +4ℓ(rk(G) + 1)2, +where ℓ = maxn +i=1 Duf(Gi). +□ +Proof of Theorem 1.10. According to Lemma 3.2, surface groups and free groups have +UFP degree Duf = 1. Moreover, for arbitrary lager n, by the Kurosh subgroup theorem, +φn(G) is a free product of finitely many free and surface groups, and hence it is hyperbolic +with rank rk(φn(G)) ≤ rk(G). Then item (1) follows from Theorem 1.8, and item (2) +follows from Proposition 3.3 clearly. +□ +A free abelian group is not hyperbolic in general, but it is nilpotent. By Theorem 2.5, +we have +Theorem 1.12. Let G = ∗n +i=1Zti be a free product of free abelian groups Zti of rank ti. +Then for any endomorphism φ ∈ End(G), we have +rkFix(φ) ≤ n(rk(G) − n + 1). +In particular, if the ranks t1 = t2 = . . . = tn, then rkFix(φ) ≤ rk(G) = �n +i=1 ti. +Proof. Without loss of generality, let G = ∗n +i=1Zti with t1 ≥ t2 ≥ · · · ≥ tn. Then For +any φ ∈ End(G), according to Theorem 2.5, the fixed subgroup has the form Fix(φ) = +∗m +j=1Hj with KrkG(Fix(φ)) = m ≤ n, where each Hj is either Z or a subgroup of some +conjugate of Zti and hence rk(Hj) ≤ rk(Zti) ≤ t1. Thus +rkFix(φ) ≤ nt1 ≤ n(rk(G) − n + 1). +If t1 = . . . = tn, then the conclusion holds clearly. +□ +The following example shows that the bound in Theorem 1.12 is sharp. + +EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF FREE PRODUCTS +11 +Example 4.1. Let G = Z ∗ Z2 = ⟨t⟩ ∗ ⟨a, b|[a, b]⟩, and ϕ ∈ Aut(G) such that +ϕ(t) = ta, +ϕ(a) = a, +ϕ(b) = b. +Then Fix(ϕ) = Z2 ∗ tZ2t−1, and hence rkFix(φ) = 2(rk(G) − 1). +At last, let us give an example for the fixed subgroup of a free product of surface and +free groups. +Example 4.2. Let H = π1(S3) = ⟨a1, b1, a2, b2, a3, b|[a1, b1][a2, b2][a3, b]⟩, the funda- +mental group of an orientable surface of genus 3, and the commutator [a, b] = aba−1b−1. +Let ψ : H → H as follows, +ψ(ai) = ai, i = 1, 2, 3; +ψ(b1) = b1, ψ(b2) = b2, ψ(b) = ba3. +Then id ̸= ψ ∈ Aut(H), and hence, by Theorem 1.2, rkFix(ψ) < rk(H) = 6. Indeed, it +is clear that Fix(ψ) = ⟨a1, b1, a2, b2, a3⟩ ∼= F5. Let Z ∗ H = ⟨t⟩ ∗ H, and +ϕ : Z ∗ H → Z ∗ H, +ϕ(t) = tbψ(b−1), +ϕ|H = ψ. +Then ϕ ∈ Aut(Z ∗ H). Note that for any h ∈ H, +ϕ(tht−1) = tbψ(b−1) · ψ(h) · (tbψ(b−1))−1 = t[ib ◦ ψ ◦ ib−1(h)]t−1 ∈ tHt−1, +where ib : H → H, h �→ bhb−1 is the inner automorphism induced by b. Therefore, +ϕ|tHt−1 : tHt−1 → tHt−1 is an automorphism, and +tht−1 ∈ Fix(ϕ|tHt−1) ⇐⇒ h ∈ Fix(ib ◦ ψ ◦ ib−1) ∼= Fix(ψ). +It implies that Fix(ϕ|tHt−1) ∼= Fix(ψ) ∼= F5. Note that KrkG(Fix(ϕ)) ≤ Krk(Z∗H) = 2 +according to Theorem 2.7, so +Fix(ϕ) = Fix(ψ) ∗ Fix(ϕ|tHt−1) ∼= F10, +and hence +rk(Z ∗ H) = 7 < rkFix(ϕ) = 10 < 2(rk(Z ∗ H) − 1) = 12. +REFERENCES +[1] G. Bergman, Supports of derivarions, free factorizations and ranks of fixed subgroups on free groups, Trans. +Amer. Math. Soc. 351 (1999), 1531-1550. +[2] M. Bestvina and M. Handel, Train tracks and automorphisms of free groups, Ann. of Math., 135 (1992), +1-51. +[3] M. Bridson, D. Groves, J. Hillman and G. Martin, Cofinitely Hopfian groups, open mappings and knot +complements, Groups Geom. Dyn. 4 (2010), 693–707 +[4] D. Collins and E. Turner, Free product fixed points, J. London Math. Soc., 38(2)(1988), 67-76. +[5] D. Collins and E. Turner, Efficient representatives for automorphisms of free products, Michigan Math. 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Wang, Fixed subgroups of automorphisms of hyperbolic 3-manifold groups, Topology Appl., +173 (2014), 175–187. +[14] J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flachen, Acta Math. 50 (1927), +189–358. +[15] J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flachen. II., Acta Math. 53 +(1929), 1–76. +[16] J. O’Neill and E. Turner, Test elements and the retract theorem in hyperbolic groups, New York J. Math. 6 +(2000), 107–117. +[17] F. Paulin, Points fixes des automorphismes de groupe hyperbolique, Ann. Inst. Fourier (Grenoble) +39(3)(1989), 651–662. +[18] E. Rodaro, P. Silva and M. Sykiotis, Fixed points of endomorphisms of graph groups, J. group Theory 16 +(2013), 573–583. +[19] Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II, +Geom. Funct. Anal. 7 (3)(1997), 561–593. +[20] Z. Sela, Diophantine geometry over groups VII: The elementary theorey of a hyperbolic group, Proc. Lon- +don. Math. Soc., 99 (3) (2009) 217-273. +[21] J. Shor, On fixed subgroups of automorphisms in hyperbolic groups, PhD thesis, Columbia University, 1999. +[22] J. Stallings, Group Theory and Three-Dimensional Manifolds, Yale University Press, 1971. +[23] M. Sykiotis, Fixed points of symmetric endomorphisms of groups, Internat. J. Algebra Comput. 12 (5)(2002), +737–745. +[24] M. Sykiotis, Fixed subgroups of endomorphisms of free products, J. Algebra, 312 (2007), 274–278. +[25] E. Ventura, Fixed subgroups in free groups: A survey, Contemp. Math., 296 (2002), 231–255. +[26] W. Wang and Y. Wu, Covering invariants and co-Hopficity of 3-manifold groups, Proc. London Math. Soc., +68 (3) (1994), 203–224. +[27] J. Wu, E. Ventura and Q. Zhang, Fixed subgroups in direct products of surface groups of Euclidean type, +Commun. Algebra, 48(7)(2020), 3003–3010, +[28] J. Wu and Q. Zhang, The group fixed by a family of endomorphisms of a surface group, J. Algebra, 417 +(2014), 412–432. +[29] Q. Zhang, Bounds for fixed points on Seifert manifolds, Topology Appl., 159 (2012), 3263–3273. +[30] Q. Zhang, The fixed subgroups of homeomorphisms of Seifert manifolds, Acta Math. Sin. (Engl. Ser.), 31 +(2015), 797–810. +[31] Q. Zhang, E. Ventura and J. Wu, Fixed subgroups are compressed in surface groups, Internat. J. Algebra +Comput., 25 (2015), 865–887. +[32] H. Zieschang, ¨Uber einfache Kurven auf Vollbrezeln, Abh. Math. Sem. Univ. Hamburg, 25 (1961/1962), +231–250. +SCHOOL OF MATHEMATICS AND STATISTICS, XI’AN JIAOTONG UNIVERSITY, XI’AN 710049, CHINA +Email address: leijialin0218@stu.xjtu.edu.cn +SCHOOL OF MATHEMATICS AND STATISTICS, XI’AN JIAOTONG UNIVERSITY, XI’AN 710049, CHINA +Email address: zhangq.math@mail.xjtu.edu.cn + diff --git a/ZtAyT4oBgHgl3EQfWvfe/content/tmp_files/load_file.txt b/ZtAyT4oBgHgl3EQfWvfe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..280cb4994e52dd1a1c30ef8f2f5bf35c2ab19a70 --- /dev/null +++ b/ZtAyT4oBgHgl3EQfWvfe/content/tmp_files/load_file.txt @@ -0,0 +1,615 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf,len=614 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='00171v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='GR] 31 Dec 2022 EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF FREE PRODUCTS JIALIN LEI AND QIANG ZHANG ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For an automorphism φ of a free group Fn of rank n, Bestvina and Handel showed that the rank rkFix(φ) of the fixed subgroup is not greater than n (the so-called Scott conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Soon after Bestvina and Handel’s announcement, their result was gen- eralized by many authors in various directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In this paper, we are interested in the fixed subgroups of endomorphisms of free products, focusing on explicit bounds for their ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' INTRODUCTION For a finitely generated group G, the rank of G denoted rk(G) is the minimal number of generators of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' There are many researches on the rank of subgroups in finitely generated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For an abelian group G, the rank rk(H) of a subgroup group H of G can not be greater than rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' However, in general, the statement is false even in free groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' It is easy to see that the free group Fn of rank n is a subgroup of F2 for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Denote the monoid of endomorphisms (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' monomorphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' injective endomorphisms) of G by End(G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Mon(G)), and the group of automorphisms of G by Aut(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For an endomorphism φ ∈ End(G), the fixed subgroup of φ is defined to be Fix(φ) := {g ∈ G | φ(g) = g}, that is a subgroup of G with many special properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For a free group Fn of rank n, in 1975, Dyer and Scott [7] proved that for a finite or- der automorphism φ of Fn, the rank rkFix(φ) of the fixed subgroup is not greater than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Moreover, Scott conjectured that rkFix(φ) ≤ n for any φ ∈ Aut(Fn), which is the so-called Scott conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Once the conjecture was put forward, it attracted a lot of re- search, see [6] for a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Finally, in 1989, the conjecture was solved by Bestvina and Handel [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Soon after Bestvina and Handel’s announcement, their result was generalized by many authors in various directions (for example, [4, 9, 5, 6, 1, 12, 28, 27, 31] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In particular, by introducing an important concept of stable image (see Section 2 for more details), Imrich and Turner [9] reduced the fixed subgroups of endomorphisms to that of automorphisms, and then showed Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='1 (Imrich-Turner, [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' If φ ∈ End(Fn), then rkFix(φ) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For surface groups, a lot is known too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In this paper, a surface group is the fundamental group π1(S) of a closed (orientable or not) surface S with Euler characteristic χ(S) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Nielsen [14, 15], Jaco-Shalen [10], and Zieschang [32] gave the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (Alter- native proofs can also be found in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=') Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 20F65, 20F34, 57M07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Fixed subgroups, free products, Gromov hyperbolic groups, surface groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' The second author is partially supported by NSFC (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 11961131004 and 11971389).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 1 2 JIALIN LEI AND QIANG ZHANG Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2 (Nielsen [14, 15], Jaco-Shalen [10], and Zieschang [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Suppose G is a surface group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then for any endomorphism φ ∈ End(G), we have (1) rkFix(φ) ≤ rk(G) if φ is epimorphic, with equality if and only if φ = id;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (2) rkFix(φ) ≤ 1 2rk(G) if φ is not epimorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' To our knowledge, there are only a few results of this type for the fundamental group of a 3-manifold, see [11, 13, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In particular, Lin and Wang [13], investigated the fundamental group π1(M) of a hyperbolic 3-manifold M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=', M is compact, orientable, and the interior of M admits a complete hyperbolic structure of finite volume (then M is either closed or the boundary ∂M of M is a union of tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Note that when M is closed, π1(M) is hyperbolic in the sense of Gromov, while ∂M is a union of tori, π1(M) contains a subgroup isomorphic to Z × Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3 (Lin-Wang, [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Suppose φ is an automorphism of G = π1(M), where M is a hyperbolic 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then rkFix(φ) < 2rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For a Gromov hyperbolic group G, Paulin [17] proved that the fixed subgroup Fix(f) is finitely generated for any automorphism f ∈ Aut(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' See [21, 8] for more information on the fixed subgroups in hyperbolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In [27, 31], the authors investigated direct products of finitely many free groups and surface groups, and showed Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='4 (Ventura-Wu-Zhang, [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ×n i=1Gi be a direct product of surface groups and free groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' If neither of the factors is cyclic, then for any automorphism φ ∈ Aut(G), rkFix(φ) ≤ rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Otherwise, if G contains a non-cyclic factor and a factor Z, then there exists f ∈ Aut(G) such that Fix(f) is not finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In this paper, we are mainly interested in the fixed subgroups of free products, focusing on explicit bounds for their ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For a group G, we say that G has the finitely generated fixed subgroup property of monomorphisms (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' automorphisms, endomorphisms), abbreviated as FGFPm (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' FGFPa, FGFPe), if for any f ∈ Mon(G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Aut(G), End(G)), the fixed subgroup Fix(f) is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Clearly, if G has FGFPe, then it has FGFPm and hence has FGFPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2 imply that free groups and surface groups have FGFPe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Furthermore, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='4 shows that the free groups F2 and Z both have FGFPa but their direct product don’t (see also Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then it is natural to ask Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' If two groups G1 and G2 both have FGFPm (FGFPa or FGFPe), then, what about their free product G1 ∗ G2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For the cases FGFPm and FGFPa, we have an affirmative answer to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' A free product ∗n i=1Gi has FGFPm (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' FGFPa) if and only if the factor groups G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' , Gn all have FGFPm (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' FGFPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since the fundamental group π1(M) of a hyperbolic 3-manifold M is co-Hopfian (see [26]), every φ ∈ Mon(π1(M)) is an automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' As a direct application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='6 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3, we have Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let M = #n i=1Mi be a connected sum of finitely many hyperbolic 3- manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then the fundamental group π1(M) has FGFPm (and hence FGFPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' More precisely, for any monomorphism f ∈ Mon(π1(M)), we have rkFix(f) < 2n · rkπ1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF FREE PRODUCTS 3 Moreover, to quantitatively analysis the ranks of fixed subgroups of a group G, we introduce a concept of UFP degree, denoted Duf(G) (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='1) and show some explicit bounds for the fixed subgroups, see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='6 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For FGFPe, we can give affirmative answers for some special kinds of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗n i=1Gi, where each factor Gi is a torsion-free hyperbolic group with finite UFP degree Duf(Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For an endomorphism φ ∈ End(G), if φn(G) is hyper- bolic for arbitrary lager n, then rkFix(φ) ≤ 1 4ℓ(rk(G) + 1)2, where the number ℓ = maxn i=1 Duf(Gi), the maximal one of Duf(Gi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (1) Note that a subgroup of a hyperbolic group may be not hyperbolic, as conjectured by O’Neill and Turner [16], we do not know whether or not every torsion-free hyperbolic group has the property that φn(G) is hyperbolic for arbitrary lager n for any φ ∈ End(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (2) Sela [19] proved that a non-elementary, torsion-free hyperbolic group is co-Hopfian (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' every monomorphism is an automorphism) if and only if it is freely indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Thus, for such groups, FGFPm is equivalent to FGFPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' It is well-known that surface groups and free groups are torsion-free hyperbolic in the sense of Gromov, and a subgroup of a surface group is either a surface group or a free group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗n i=1Gi be a free product with each factor Gi a free group or a surface group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then G is a torsion-free hyperbolic group satisfying the hypothesis of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='8 with the UFP degree Duf(Gi) = 1, and hence, as a corollary, we have explicit bounds for the fixed subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗t i=1Gi ∗ Fs be a free product, where Fs is a free group of rank s, and each factor Gi is a surface group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then (1) for any endomorphism φ ∈ End(G), the fixed subgroup has rank rkFix(φ) ≤ 1 4(rk(G) + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (2) for any φ ∈ Mon(G), we have rkFix(φ) ≤ (s + t)(rk(G) − s − t + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In particular, if s = 0 and all the surface groups Gi share the same rank, then rkFix(φ) ≤ rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In [18], Rodaro, Silva and Sykiotis studied fixed subgroups of graph groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=', right angled Artin groups) and showed Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' [18, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='1] Let G be a graph group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then the following two condi- tions are equivalent (1) Fix(φ) is finitely generated for every endomorphism φ ∈ End(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (2) G is a free product of finitely many free abelian groups of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Now, we give explicit bounds for the fixed subgroups of free products of free abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗n i=1Zti be a free product of free abelian groups Zti of rank ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then for any endomorphism φ ∈ End(G), we have rkFix(φ) ≤ n(rk(G) − n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In particular, if the ranks t1 = t2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' = tn, then rkFix(φ) ≤ rk(G) = �n i=1 ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 4 JIALIN LEI AND QIANG ZHANG The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In Section 2, we review some useful facts on free products, especially Sykiotis’ work on symmetric endomorphisms and the structure of fixed subgroups of free products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In Section 3, we introduce some concepts on fixed sub- groups to quantitatively analysis the ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' At last, in Section 4, we give proofs of the main results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' PRELIMINARIES In papers [23, 24], Sykiotis studied the fixed subgroups of symmetric endomorphisms of free products of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For later use, let us first review some important definitions and facts in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Stable image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' To study the fixed subgroups of endomorphisms of free groups, Imrich and Turner [9] introduced the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='1 ([9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For a group G and an endomorphism φ ∈ End(G), the stable image φ∞(G) of φ is the intersection φ∞(G) := ∞ � n=1 φn(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Clearly, the fixed subgroup Fix(φ) = Fix(φ∞) ≤ φ∞(G), where φ∞ : φ∞(G) → φ∞(G) is the restriction of φ to the stable image φ∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For a free group Fn, Imrich and Turner proved that for arbitrary endomorphism φ ∈ End(Fn), φ∞ ∈ Aut(φ∞(Fn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Here φ∞(Fn) is a subgroup of Fn, so it is a free group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' They also proved that rk(φ∞(Fn)) ≤ rk(Fn) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then the Bestvina-Handel theorem (Scott conjecture) implies that rkFix(φ) = rkFix(φ∞) ≤ rk(φ∞(Fn)) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Kurosh rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G be a group and let H be a nontrivial subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' A famous result of Kurosh showed that, G can be represented as a free product of freely indecompos- able factors, G = ∗n i=1Gi, and the set of factors (and hence the number n of the factors) is well-defined up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then n is said to be the (absolute) Kurosh rank of G, denoted Krk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' By the Kurosh subgroup theorem, the subgroup H is a free product H = ∗t j=1Hj ∗ Fs, where Fs is a free group of rank s and every factor Hj is the intersection of H with a conjugate of some factor Gi (either of s or t could be 0 or infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' If the representation of G as a free product is changed by an isomorphism, then these intersections change, but the number s + t is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' We say s + t the Kurosh rank of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' denoted KrkG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Note that Grushko’s theorem states that the rank of groups is additive under free prod- ucts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=', rk(∗n i=1Gi) = �n i=1 rk(Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Thus KrkG(H) ≤ Krk(H) ≤ rk(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' The first equality holds if s = 0 and none of Hi can be split as a nontrivial free products, and the second equality holds if each Hj is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Symmetric endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗n i=1Gi and K = ∗m i=1Ki be two free products (the factors Gi and Ki may be freely decomposable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Sykiotis [24] gave the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2 ([24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' A homomorphism φ : G → K is said to be symmetric if each non- infinite-cyclic free factor of G is mapped by φ into a conjugate of some non-infinite-cyclic free factor of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF FREE PRODUCTS 5 It is easy to see that if each factor Gi is freely indecomposable, then each injective homomorphism is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since surface groups are freely indecomposable, we have Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗t i=1Gi ∗ Fs be a free product, where Fs is a free group of rank s, and each factor Gi is a surface group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then every monomorphism φ ∈ Mon(G) is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For a free product G = ∗n i=1Gi of freely indecomposable factors, Collins and Turner [5] studied the fixed subgroups of an automorphism φ ∈ Aut(G), and showed that the Kurosh rank KrkG(Fix(φ)) of the fixed subgroup can not exceed Krk(G) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Sykiotis [24] extended Collins and Turner’s result to monomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since every group can be represented as a free product of freely indecomposable factors, we translate [24, Corollary 4] to the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='4 (Sykiotis, [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G be a group and φ ∈ Mon(G) a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then the Kurosh rank KrkG(Fix(φ)) ≤ Krk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In general, the fixed subgroups of endomorphisms of a free product is more complicate than that of monomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' But for some special groups, we can get some bounds for the fixed subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For example, Sykiotis [24] gave the following bound on the fixed subgroups of free products of nilpotent groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' [24, Theorem 7] Let G = ∗n i=1Gi be a free product of finitely generated nilpotent and finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' If φ ∈ End(G) is an endomorphism of G, then the fixed subgroup Fix(φ) of φ has Kurosh rank at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Sykiotis also showed the structure of the fixed subgroup of symmetric endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' [23, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2] Let G be a group which acts on a tree X with finite quotient graph and finite edge groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Suppose that φ is a symmetric endomorphism of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then the fixed subgroup Fix(φ) of G is the fundamental group of a finite graph of groups with the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (1) The edge groups are contained in groups of the form �n i=1 giGeig−1 i , where ei are edges of X and gi ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (2) The vertex groups are either of the form Fix(φ|Gv) where v is a vertex of X fixed by ˜φ, or are contained in groups of the form in item (1), and hence are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' By using Bass-Serre theory, combining Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='4 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='6, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗n i=1Gi be a free product of freely indecomposable factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then for any monomorphism φ ∈ Mon(G), Fix(φ) = ∗t j=1Fix(φ|gjGσ(j)g−1 j ) ∗ Fs, where KrkG(Fix(φ)) = s + t ≤ n, and φ|gjGσ(j)g−1 j : gjGσ(j)g−1 j → gjGσ(j)g−1 j is the restriction of φ to a conjugate of some factor Gσ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since G = ∗n i=1Gi is a free product, G acts on a tree X with finite quotient graph and trivial edge groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Moreover, the stabilizer Gv of a vertex v is a conjugate of some factor Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then the conclusion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='7 is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' □ 6 JIALIN LEI AND QIANG ZHANG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Gromov hyperbolic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Now we consider torsion-free hyperbolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In [16], O’Neill and Turner studied the stable image of hyperbolic groups, and gave the fol- lowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' [16, Proposition 2] Let G be a torsion-free hyperbolic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' If φ : G → G is an endomorphism with the property that φn(G) is hyperbolic for arbitrary lager n, then φ∞(G) is a free factor of φN(G) for some N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In [20], Sela gave the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' [20, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='11] Let Γ be a torsion-free hyperbolic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' A finitely generated group G is said to be a Γ-limit group, if G is isomorphic to a subgroup of Γ or if G is a (strict) Γ-limit group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' [20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='12] Let Γ be a torsion-free hyperbolic group, and let G be a finitely generated group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Every decreasing sequence of Γ-limit groups that are quotients of G, R1 > R2 > R3 > · · · terminates after finitely many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' We do not need the concept of strict Γ-limit group but the relation “>”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let Γ be a torsion-free hyperbolic group, and let G be a finitely generated group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' On the set of Γ- limit groups, we define a relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Given two Γ-limit groups R1, R2, that are quotients of G, with prescribed maps ηi : G → Ri, i = 1, 2 we say that R1 > R2, if there exists an epimorphism with non-trivial kernel: τ : R1 → R2, so that η2 = τ ◦ η1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For an endomorphism φ ∈ End(G) of a finitely generated torsion-free hyperbolic group G, it is easy to see that φn(G) is a G-limit group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Moreover, we have an analogue of [24, Lemma 5] as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G be a finitely generated torsion-free hyperbolic group, and φ ∈ End(G) an endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then the restriction φ∞ : φ∞(G) → φ∞(G) is an automor- phism, and the Kurosh rank Krkφ∞(G)(Fix(φ)) ≤ Krk(φ∞(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let φn = φ|φn(G) : φn(G) → φn+1(G) be the restriction of φ to φn(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' First, we prove that the kernel ker(φN) is trivial for some N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=', φN is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' To see this, assume that ker(φn) is non-trivial for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since the subgroups φn(G) of G are also quotients of G, we have a decreasing sequence of G-limit groups as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='10 but with infinitely many terms, G > φ(G) > φ2(G) > φ3(G) > · · · , we get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Below we show that φ∞ is an automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Note that for any g ∈ φ∞(G) = ∞ � n=N φn(G), there exists gn ∈ G such that φn(gn) = g for every n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let cn = φn(gn+1) ∈ φn(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then φ(cn) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since φN is injective, we have cN = cn ∈ φn(G) for all n ≥ N and hence cN ∈ φ∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' This gives surjectivity of φ∞, and hence φ∞ is an automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Note that Fix(φ) = Fix(φ∞) ≤ φ∞(G), by using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='4, we have Krkφ∞(G)(Fix(φ)) = Krkφ∞(G)(Fix(φ∞)) ≤ Krk(φ∞(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF FREE PRODUCTS 7 The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Although many hyperbolic groups have been showed to be residually fi- nite, Gromov’s famous question remains open: Is every hyperbolic group residually fi- nite?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Therefore, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='11 is not a direct corollary of [24, Lemma 5]: Let φ be an endomorphism of a finitely generated residually finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then the restriction φ∞ : φ∞(G) → φ∞(G) of φ to φ∞(G) is a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' FIXED SUBGROUPS OF MONOMORPHISMS In this section, we first introduce two properties on fixed subgroups, and then show some results on the fixed subgroups of monomorphisms of free products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' FGFP property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Recall that a group G has FGFPm (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' FGFPa, FGFPe), if for any f ∈ Mon(G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Aut(G), End(G)), the fixed subgroup Fix(f) is finitely gener- ated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Clearly, if G has FGFPe, then it has FGFPm and hence has FGFPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Moreover, the property FGFPa is not heritable, for example, according to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='4, the group Fn × Fm for m, n > 1 has FGFPa but its subgroup F2 × Z does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' To quantitatively analysis the ranks of the fixed subgroups, we have Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G be a finitely generated group, and let Mon(G) denote the monoid of monomorphisms of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (1) G is said to have k-FGFP, if for any φ ∈ Mon(G), rkFix(φ) ≤ k · rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' The minimal number k satisfying the above equation is said to be the FP degree for the group G, denoted Df(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Namely, Df(G) := sup{rkFix(φ) rk(G) | φ ∈ Mon(G)} ∈ [1, +∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (2) G is said to have k-UFGFP (“U” for uniformly), if for every finitely generated sub- group H ≤ G and any φ ∈ Mon(H), rkFix(φ) ≤ k · rk(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' The minimal number k satisfying the above equation is said to be the UFP degree for the group G, denoted Duf(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Namely, Duf(G) := sup{rkFix(φ) rk(H) | H ≤ G, φ ∈ Mon(H)} ∈ [1, +∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Note that in the above definitions, we only consider monomorphisms, and omit the cases of endomorphisms and automorphisms because that are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Clearly, k-UFGFP implies k-FGFP, and many kinds of groups have k-FGFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Furthermore, the property k-UFGFP is heritable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In precisely, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G be a finitely generated group and let H ≤ G be a finitely generated subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (1) 1 ≤ Df(G) ≤ Duf(G) ≤ ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (2) If G has k-UFGFP, then its subgroup H also has k-UFGFP and hence has k-FGFP, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=', Df(H) ≤ Duf(H) ≤ Duf(G) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (3) Df(G) = Duf(G) = 1 if G is one of the following, (a) free abelian groups Zn, (b) free groups Fn, (c) surfaces groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (4) Df(F2×Z) = ∞, and hence Duf(G) = ∞ if G contains a subgroup that is isomorphic to F2 × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' 8 JIALIN LEI AND QIANG ZHANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Items (1) and (2) are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Item (3) follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2 clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' To prove item (4), it suffices to show that Fix(f) is not finitely generated for some f ∈ Mon(F2 × Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Indeed, let F2 × Z = ⟨a, b, t | [a, t], [b, t]⟩, and let f : F2 ×Z → F2 ×Z such that a �→ at, b �→ b, t �→ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then an element u ∈ Fix(f) if and only if it has zero exponent sum in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' So Fix(f) is isomorphic to F∞ × Z generated by the infinite set {t, aiba−i|i ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' FGFP in free products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Now we have the following key proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗n i=1Gi be a free product, where each factor Gi is a freely indecomposable group satisfying rkFix(f) ≤ ki · rk(Gi) for any f ∈ Mon(Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then for any monomorphism φ ∈ Mon(G), we have rkFix(φ) ≤ n( n max i=1 ki)(rk(G) − n + 1), in particular, if all the factors Gi have the same rank, then rkFix(φ) ≤ ( n max i=1 ki) · rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Before give the proof, we have a direct corollary on the FP degree of fixed subgroups as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗n i=1Gi, where each Gi is a freely indecomposable group with finite FP degree Df(Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then Df(G) ≤ n · n max i=1 Df(Gi), in particular, if all the factors Gi have the same rank, then Df(G) ≤ maxn i=1 Df(Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' According to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='7, Fix(φ) = ∗s i=1Hi∗Ft, where s+t ≤ n and each Hi = Fix(φ|giGσ(i)g−1 i ) is the fixed subgroup of a conjugate of some Gσ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since Gi’s have the property rkFix(f) ≤ ki · rk(Gi) for any f ∈ Mon(Gi), we have rk(Hi) ≤ kσ(i) · rk(giGσ(i)g−1 i ) = kσ(i) · rk(Gσ(i)) ≤ kσ(i) · (rk(G) − n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' It follows rkFix(φ) = t + s � i=1 rk(Hi) ≤ t + s � i=1 kσ(i) · rk(Gσ(i)) ≤ n( n max i=1 ki)(rk(G) − n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='1) In particular, if all the factors Gi have the same rank rk(G1) = · · · = rk(Gn), then rkFix(φ) ≤ t + s � i=1 kσ(i) · rk(Gσ(i)) ≤ ( n max i=1 ki) · rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Note that in the previous proof, Gσ(i) may equal to Gσ(j) for distinct i ̸= j, see Section 4 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF FREE PRODUCTS 9 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗n i=1Gi be a free product, where each factor is a (not necessarily freely indecomposable) group Gi with finite UFP degree Duf(Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then for any subgroup H ≤ G and any monomorphism φ ∈ Mon(H), rkFix(φ) ≤ 1 4ℓ(rk(H) + 1)2, where ℓ = maxn i=1 Duf(Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let H ≤ G be a finitely generated subgroup and φ ∈ Mon(H) a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Split H as a free product of freely indecomposable factors, H = ∗s j=1Hj, where the absolute Kurosh rank (other than the Kurosh rank in G) Krk(H) = s ≤ rk(H), and each factor Hj is either a freely indecomposable group contained in a conjugate of a factor Gij or the free cyclic group Z (with Duf(Z) = 1 clearly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2 implies Duf(Hj) ≤ Duf(Gij ) ≤ n max i=1 Duf(Gi) = ℓ, and hence rkFix(f) ≤ ℓ · rk(Hj) for any f ∈ Mon(Hj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3 to the monomorphism φ : H → H = ∗s i=jHj, we have rkFix(φ) ≤ ℓs(rk(H) − s + 1) ≤ 1 4ℓ(rk(H) + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' MAIN RESULTS AND SOME EXAMPLES In this section, we first prove the main theorems, and then give some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' A free product ∗n i=1Gi has FGFPm (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' FGFPa) if and only if the factor groups G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' , Gn all have FGFPm (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' FGFPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since the proofs of the two cases FGFPm and FGFPa are parallel, we only con- sider the case FGFPm, and leave the other case FGFPa for the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Ar first, we prove the “only if” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Without loss of generality, suppose G1 dose not have FGFPm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=', there is a monomorphism f : G1 → G1 such that Fix(f) is not finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let idi ∈ Mon(Gi) be the identity of Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then the free product f ∗ id2 ∗ · · · ∗ idn ∈ Mon(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Clearly, the fixed subgroup Fix(f ∗ id2 ∗ · · · ∗ idn) = Fix(f) ∗ G2 ∗ · · · ∗ Gn is not finitely generated, contradicting the hypothesis that G has FGFPm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Now let us consider the “if” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' There are two cases: (i) All the factors Gi are freely indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then any f ∈ Mon(G) maps each factor Gi to a conjugate of some Gj, as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3, we have Fix(f) = ∗s i=1Hi ∗ Ft, where s + t ≤ n and each Hi = Fix(f|giGσ(i)g−1 i ) is the fixed subgroup of a conjugate of some Gσ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since Gi all have FGFPm, the Hi are all finitely generated and hence Fix(f) is also finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' (ii) Some factors Gi are freely decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then each Gi can be decomposed into Gi = G′ i1 ∗ · · · ∗ G′ ij, where each factor G′ ik is freely indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since Gi has FGFPm, the factors G′ ik all have FGFPm by the “only if” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Therefore, we have reduced case (ii) to case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' □ Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since each Mi is a hyperbolic 3-manifold, the fundamental group π1(Mi) is co-Hopfian by [3], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=', every φ ∈ Mon(π1(Mi)) is an automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3 implies that π1(Mi) all have FGFPm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Note that the fundamental group 10 JIALIN LEI AND QIANG ZHANG π1(M) = ∗n i=1π1(Mi), and each π1(Mi) is freely indecomposable because Mi is a hyper- bolic 3-manifold, then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='6 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3 imply that π1(M) has FGFPm and rkFix(f) < 2n · rkπ1(M) for any f ∈ Mon(π1(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' □ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗n i=1Gi, where each factor Gi is a torsion-free hyperbolic group with finite UFP degree Duf(Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' For an endomorphism φ ∈ End(G), if φn(G) is hyper- bolic for arbitrary lager n, then rkFix(φ) ≤ 1 4ℓ(rk(G) + 1)2, where the number ℓ = maxn i=1 Duf(Gi), the maximal one of Duf(Gi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Since G is a free product of torsion-free hyperbolic groups, G itself is also hyper- bolic and torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='11 implies that the restriction φ∞ = φ|φ∞(G) : φ∞(G) → φ∞(G) is an automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Furthermore, since G satisfies the condition that φn(G) is hyperbolic for arbitrary lager n, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='8, the stable image φ∞(G) is a free factor of φN(G) for some N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' So rk(φ∞(G)) ≤ rk(φN(G)) ≤ rk(G), the last inequality holds because φN(G)) is a quotient of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Note that Fix(φ) = Fix(φ∞) and φ∞(G) ≤ G, applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='6 to the automorphism φ∞ : φ∞(G) → φ∞(G), we have rkFix(φ) = rkFix(φ∞) ≤ 1 4ℓ · (rk(φ∞(G)) + 1)2 ≤ 1 4ℓ(rk(G) + 1)2, where ℓ = maxn i=1 Duf(Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' According to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2, surface groups and free groups have UFP degree Duf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Moreover, for arbitrary lager n, by the Kurosh subgroup theorem, φn(G) is a free product of finitely many free and surface groups, and hence it is hyperbolic with rank rk(φn(G)) ≤ rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then item (1) follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='8, and item (2) follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='3 clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' □ A free abelian group is not hyperbolic in general, but it is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='5, we have Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = ∗n i=1Zti be a free product of free abelian groups Zti of rank ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then for any endomorphism φ ∈ End(G), we have rkFix(φ) ≤ n(rk(G) − n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' In particular, if the ranks t1 = t2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' = tn, then rkFix(φ) ≤ rk(G) = �n i=1 ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Without loss of generality, let G = ∗n i=1Zti with t1 ≥ t2 ≥ · · · ≥ tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then For any φ ∈ End(G), according to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='5, the fixed subgroup has the form Fix(φ) = ∗m j=1Hj with KrkG(Fix(φ)) = m ≤ n, where each Hj is either Z or a subgroup of some conjugate of Zti and hence rk(Hj) ≤ rk(Zti) ≤ t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Thus rkFix(φ) ≤ nt1 ≤ n(rk(G) − n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' If t1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' = tn, then the conclusion holds clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' □ The following example shows that the bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='12 is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' EXPLICIT BOUNDS FOR FIXED SUBGROUPS OF ENDOMORPHISMS OF FREE PRODUCTS 11 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let G = Z ∗ Z2 = ⟨t⟩ ∗ ⟨a, b|[a, b]⟩, and ϕ ∈ Aut(G) such that ϕ(t) = ta, ϕ(a) = a, ϕ(b) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then Fix(ϕ) = Z2 ∗ tZ2t−1, and hence rkFix(φ) = 2(rk(G) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' At last, let us give an example for the fixed subgroup of a free product of surface and free groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let H = π1(S3) = ⟨a1, b1, a2, b2, a3, b|[a1, b1][a2, b2][a3, b]⟩, the funda- mental group of an orientable surface of genus 3, and the commutator [a, b] = aba−1b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let ψ : H → H as follows, ψ(ai) = ai, i = 1, 2, 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' ψ(b1) = b1, ψ(b2) = b2, ψ(b) = ba3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then id ̸= ψ ∈ Aut(H), and hence, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='2, rkFix(ψ) < rk(H) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Indeed, it is clear that Fix(ψ) = ⟨a1, b1, a2, b2, a3⟩ ∼= F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Let Z ∗ H = ⟨t⟩ ∗ H, and ϕ : Z ∗ H → Z ∗ H, ϕ(t) = tbψ(b−1), ϕ|H = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Then ϕ ∈ Aut(Z ∗ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Note that for any h ∈ H, ϕ(tht−1) = tbψ(b−1) · ψ(h) · (tbψ(b−1))−1 = t[ib ◦ ψ ◦ ib−1(h)]t−1 ∈ tHt−1, where ib : H → H, h �→ bhb−1 is the inner automorphism induced by b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Therefore, ϕ|tHt−1 : tHt−1 → tHt−1 is an automorphism, and tht−1 ∈ Fix(ϕ|tHt−1) ⇐⇒ h ∈ Fix(ib ◦ ψ ◦ ib−1) ∼= Fix(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' It implies that Fix(ϕ|tHt−1) ∼= Fix(ψ) ∼= F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Note that KrkG(Fix(ϕ)) ≤ Krk(Z∗H) = 2 according to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='7, so Fix(ϕ) = Fix(ψ) ∗ Fix(ϕ|tHt−1) ∼= F10, and hence rk(Z ∗ H) = 7 < rkFix(ϕ) = 10 < 2(rk(Z ∗ H) − 1) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' REFERENCES [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' Bergman, Supports of derivarions, free factorizations and ranks of fixed subgroups on free groups, Trans.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content=' SCHOOL OF MATHEMATICS AND STATISTICS, XI’AN JIAOTONG UNIVERSITY, XI’AN 710049, CHINA Email address: leijialin0218@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='cn SCHOOL OF MATHEMATICS AND STATISTICS, XI’AN JIAOTONG UNIVERSITY, XI’AN 710049, CHINA Email address: zhangq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfWvfe/content/2301.00171v1.pdf'} +page_content='math@mail.' metadata={'source': 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a/ZtE4T4oBgHgl3EQfOgxT/content/tmp_files/2301.04965v1.pdf.txt b/ZtE4T4oBgHgl3EQfOgxT/content/tmp_files/2301.04965v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fee5f86aaf81b9acbc672788fda7de53b574754 --- /dev/null +++ b/ZtE4T4oBgHgl3EQfOgxT/content/tmp_files/2301.04965v1.pdf.txt @@ -0,0 +1,653 @@ +arXiv:2301.04965v1 [math.AP] 12 Jan 2023 +ON POSITIVITY SETS FOR HELMHOLTZ SOLUTIONS +PU-ZHAO KOW, MIKKO SALO, AND HENRIK SHAHGHOLIAN +Dedicated to Carlos E. Kenig on the occasion of his 70th birthday +Abstract. We address the question of finding global solutions of the Helmholtz equation +that are positive in a given set. This question arises in inverse scattering for penetrable +obstacles. In particular, we show that there are solutions that are positive on the boundary +of a bounded Lipschitz domain. +1. Introduction +The objective in this short note is to consider the following problem. +Question 1.1. Let k > 0 and let E be a subset of Rn (n ≥ 2). Does there exist a solution +of (∆ + k2)u = 0 in Rn with u|E > 0? +Note that any solution of the Helmholtz equation (∆ + k2)u = 0 is C∞, and thus the +condition u|E > 0 can be understood pointwise. There is a substantial literature on zero +sets of solutions of elliptic equations and eigenfunctions, as discussed in the review [LM20]. +In our setting, any real valued solution of (∆ + k2)u = 0 in Rn must have a zero in any +closed ball of radius j n−2 +2 ,1k−1 where j n−2 +2 +,1 is the first zero of the Bessel function J n−2 +2 +(see +e.g. [SS21, Lemma 3.1]). Question 1.1 above is related to producing a global solution whose +zero set avoids a given set E. +Our motivation comes from inverse scattering theory and the works [CV21, SS21, KLSS22]. +In these works, one considers a bounded open set D ⊂ Rn (penetrable obstacle) together with +a coefficient h ∈ L∞(Rn) with |h| ≥ c > 0 a.e. near ∂D (contrast), and asks whether it is +possible to find a solution u0 ̸≡ 0 of (∆ + k2)u0 = 0 in Rn (incident wave) such that the +obstacle D with contrast h does not produce any scattering response. The last condition can +be precisely formulated as the existence of a function u solving +(∆ + k2 + hχD)u = 0 in Rn, +u = u0 outside some ball. +If this happens for some contrast h, then the obstacle D is called a non-scattering domain +and it will be invisible with respect to probing with the incident wave u0. +It was proved in [SS21, Theorem 2.1] that if D has real-analytic boundary and if there is +an incident wave u0 with u0|∂D > 0, then D is a non-scattering domain. Similarly, the work +[KLSS22] introduced the notion of quadrature domains for the Helmholtz operator ∆ + k2 +and proved that if D is such a domain, and if there is an incident wave u0 with u0|∂D > 0, +then D is a non-scattering domain. On the other hand, the works [CV21, SS21] show that +under a nonvanishing condition for u0 on ∂D, the boundary of a non-scattering domain can +2020 Mathematics Subject Classification. 35J05; 35J15; 35J20; 35R30; 35R35. +Key words and phrases. Helmholtz equation; acoustic equation; Lipschitz domain; inverse scattering +problem. +1 + +2 +PU-ZHAO KOW, MIKKO SALO, AND HENRIK SHAHGHOLIAN +be interpreted as a free boundary in an obstacle-type problem and hence such a domain must +be either regular or have thin complement near any boundary point. +It was also proved in [SS21] that one may be able to find incident waves that are positive on +the boundary of a bounded C1 domain (Lipschitz if n = 2, 3). Our first main result extends +this to Lipschitz domains in any dimension. +Theorem 1.1. Let D ⊂ Rn (n ≥ 2) be a bounded Lipschitz domain such that Rn \ D is +connected. Suppose that k2 > 0 is not a Dirichlet eigenvalue of −∆ in D. Then there exists +a Herglotz wave function u0 (see Definition 2.1) satisfying +(∆ + k2)u0 = 0 in Rn and u0|∂D > 0. +The proof of Theorem 1.1 is done in two steps. +One first constructs a solution v of +(∆ + k2)v = 0 in D with v|∂D > 0 by solving a Dirichlet problem. Then one approximates +v in D by a suitable Herglotz wave u0 in Rn via a Runge approximation argument. This +approximation needs to be done in a suitable norm to obtain the pointwise condition u0|∂D > +0, but since D only has Lipschitz boundary the solution v is not very regular and this limits +the choice of possible norms. We will work with fractional Sobolev spaces Hs,p and invoke +the theory of boundary value problems in Lipschitz domains.1 +We remark that the assumption in Theorem 1.1 that k2 is not an eigenvalue is necessary, +at least when D is a ball (see Example 2.5). For the first eigenvalue this was pointed out in +[SS21, Remark 3.2]. +Another instance of subsets E ⊂ Rn where one can arrange u0|E > 0 is given in the +following result. +Theorem 1.2. Let k > 0, and let D ⊂ Rn (n ≥ 2) be a bounded Lipschitz domain such that +Rn \ D is connected and |D| ≤ |Br| where r = j n−2 +2 ,1k−1. If E ⊂ D is compact, then there +exists a Herglotz wave function u0 (see Definition 2.1) satisfying +(∆ + k2)u0 = 0 in Rn and u0|E > 0. +The proof is similar to that of Theorem 1.1, except that in the first step we use the +Faber-Krahn inequality to produce a solution v that is positive near E. +2. Solutions satisfying the positivity condition +In this section we will prove Theorems 1.1 and 1.2. We begin with some preparations. +2.1. Fractional Sobolev spaces. For each s ∈ R and 1 < p < ∞, the fractional Sobolev +space Hs,p(Rn) is the Banach space equipped with the norm +∥u∥Hs,p(Rn) := ∥⟨D⟩su∥Lp(Rn), +where ⟨D⟩s is the the Bessel potential of order s, i.e. the Fourier multiplier corresponding +to ⟨ξ⟩s = (1 + |ξ|2) +s +2. In particular when s = k ≥ 1 is an integer, we also have Hk,p(Rn) = +W k,p(Rn), where +W k,p(Rn) = +� u ∈ Lp(Rn) +Dαu ∈ Lp(Rn) for all multi-indices α with |α| ≤ k � +. +From [BL76, Corollary 6.2.8], we have the duality statement +(2.1) +(Hs,p(Rn))∗ = H−s,p′(Rn) +for all s ∈ R and 1 < p < ∞, +1This is one of the areas where Carlos Kenig has made pioneering contributions. + +ON POSITIVITY SETS FOR HELMHOLTZ SOLUTIONS +3 +where (p′)−1 + p−1 = 1. We also recall the Sobolev embedding ([BL76, Theorem 6.5.1]): +(2.2) +Hs,p(Rn) ⊂ Hs1,p1(Rn) +whenever 1 < p ≤ p1 < ∞, −∞ < s1 ≤ s < ∞, and s − n +p = s1 − n +p1. +Let D be an open set in Rn. We define +Hs,p(D) := +� u|D +u ∈ Hs,p(Rn) � +for all s ∈ R and 1 < p < ∞. +This is a Banach space equipped with the quotient norm +∥v∥Hs,p(D) := inf +� ∥u∥Hs,p(Rn) +u|D = v � +. +When D is a bounded Lipschitz domain, from [JK95, Theorem 2.3] we know that there exists +a bounded linear extension operator +(2.3) +E : Hs,p(D) → Hs,p(Rn) +with Eu = u in D for all u ∈ Hs,p(D). +If F ⊂ Rn is closed, we define +Hs,p +F (Rn) := +� u ∈ Hs,p(Rn) +supp(u) ⊂ F � +. +If D is a bounded Lipschitz domain, the following result can be found in [JK95, Remark 2.7]: +(2.4) +C∞ +c (D) is dense in Hs,p +D (Rn) for each s ∈ R and 1 < p < ∞. +2.2. Runge-Herglotz approximation. The next objective is to prove a result stating that +solutions in Hs,p(D) can be approximated in D by Herglotz waves. We first give a definition. +Definition 2.1. Let k > 0 and consider the operator Pk : C∞(Sn−1) → C∞(Rn) defined by +(Pkf)(x) := +� +Sn−1 eikx·ˆzf(ˆz) dˆz, +x ∈ Rn. +The functions u = Pkf with f ∈ C∞(Sn−1) are called Herglotz waves, and they are particular +solutions of (∆ + k2)u = 0 in Rn. +Proposition 2.2. Let k > 0, 0 < s ≤ 1, 1 < p < ∞, and let D ⊂ Rn (n ≥ 2) be a bounded +Lipschitz domain such that Rn \ D is connected. Given any v ∈ Hs,p(D) with (∆ + k2)v = 0 +in D, there exist Herglotz waves uj ∈ C∞(Rn) such that +∥uj − v∥Hs,p(D) → 0 +as j → ∞. +If v is real-valued, then so are uj. +The proof of Proposition 2.2 is very similar to [SS21, Proposition 3.4] that considered +approximation in W 1,p(D). Here we need to work with fractional Sobolev spaces instead. +Proof. In view of the Hahn-Banach theorem, it is enough to prove that any bounded linear +functional ℓ : Hs,p(D) → C that vanishes on +� Pkf|D +f ∈ C∞(Sn−1) � +must also vanish +on +� v ∈ Hs,p(D) +−(∆ + k2)v = 0 in D � +. Let ℓ be such a linear functional, and define a +bounded linear functional ℓ1 : Hs,p(Rn) → C by ℓ1(u) := ℓ(u|D). By duality (2.1), there +exists a unique µ ∈ H−s,p′(Rn) such that +ℓ1(u) = (u, µ) +for all u ∈ Hs,p(Rn), +where (·, ·) is the sesquilinear distributional pairing in Rn. It is easy to see that µ = 0 in +Rn \ D, and the condition ℓ(Pkf|D) = 0 for all f ∈ C∞(Sn−1) implies that +(2.5) +(Pkf, µ) = 0 +for all f ∈ C∞(Sn−1). + +4 +PU-ZHAO KOW, MIKKO SALO, AND HENRIK SHAHGHOLIAN +We now define the distribution w := Φk ∗ µ, where +(2.6) +Φk(x) = +ik +n−2 +2 +4(2π) +n−2 +2 |x|− n−2 +2 H(1) +n−2 +2 (k|x|) +is the outgoing fundamental solution of the Helmholtz operator −(∆ + k2) and H(1) +α +is the +Hankel function (see [Yaf10, §1.2.3]). Then w is a distributional solution of +(2.7) +− (∆ + k2)w = µ +in Rn. +Elliptic regularity yields w ∈ H2−s,p′ +loc +(Rn), and since supp(µ) ⊂ D we also have that w is C∞ +in Rn \ D. +Given any f ∈ C∞(Sn−1), we write u = Pkf ∈ C∞(Rn). Using (2.5) and the fact that µ +has compact support, we have +(2.8) +0 = (u, µ) = lim +r→∞(u, µ)Br, +where (·, ·)Br is the sesquilinear distributional pairing in the ball Br. We now consider a +cut-off function χ ∈ C∞ +c (Rn) satisfying 0 ≤ χ ≤ 1 and χ = 1 near D. Using (2.7), we can +write (2.8) as +0 = lim +r→∞ +� +(χu, (∆ + k2)w)Br + ((1 − χ)u, (∆ + k2)w)Br +� += lim +r→∞ +� +((∆ + k2)(χu), w)Br + ((∆ + k2)((1 − χ)u), w)Br ++ +� +∂Br +(u∂|x|w − (∂|x|u)w) dS +� += lim +r→∞ +� +∂Br +(u∂|x|w − (∂|x|u)w) dS, +(2.9) +where ∂|x| = ˆx·∇ denotes the radial derivative. Here we also used the fact that (∆+k2)u = 0 +in Rn. +Using [Mel95, Lemma 1.2 and equation (1.18)], we know that the Herglotz function u = Pkf +has the following asymptotics as |x| → ∞: +u(x) = c′ +n,k|x|− n−1 +2 +� +eik|x|f(ˆx) + in−1e−ik|x|f(−ˆx) +� ++ O(|x|− n+1 +2 ), +(2.10a) +∂|x|u(x) = c′ +n,k|x|− n−1 +2 ik +� +eik|x|f(ˆx) − in−1e−ik|x|f(−ˆx) +� ++ O(|x|− n+1 +2 ), +(2.10b) +where c′ +n,k = k +n−1 +2 e +π(n−1)i +4 +(2π)− n−1 +2 . On the other hand, from [Yaf10, equation (2.27)], we +know that w has the asymptotics +w(x) = c′′ +n,k|x|− n−1 +2 eik|x|ˆµ(kˆx) + O(|x|− n+1 +2 ) +as |x| → ∞, +(2.10c) +∂|x|w(x) = c′′ +n,k|x|− n−1 +2 ikeik|x|ˆµ(kˆx) + O(|x|− n+1 +2 ) +as |x| → ∞, +(2.10d) + +ON POSITIVITY SETS FOR HELMHOLTZ SOLUTIONS +5 +where c′′ +n,k = 2−1e− π(n−3)i +4 +(2π)− n−1 +2 k +n−3 +2 +and ˆµ ∈ C∞(Rn) is the Fourier transform of the +compactly supported distribution µ. +Combining (2.9) with (2.10a)–(2.10d), we obtain +� +Sn−1 f(ˆx)ˆµ(kˆx) dˆx = 0. +By the fact that f ∈ C∞(Sn−1) was arbitrary, we conclude ˆµ(kˆx) = 0 for all ˆx ∈ Sn−1. +Consequently, (2.10c) becomes +w(x) = O(|x|− n+1 +2 ) +as |x| → ∞. +In other words, the far-field pattern of w is vanishing. By the Rellich uniqueness theorem +[CK19, Hör73], the unique continuation principle and the connectedness of Rn \ D, we +conclude that +(2.11a) +w = 0 +in Rn \ D. +Since w ∈ H2−s,p′ +loc +(Rn), we also conclude that w ∈ H2−s,p′ +D +(Rn). +Now let v ∈ Hs,p(D) be any solution of (∆ + k2)v = 0 in D, and let ˜v ∈ Hs,p(Rn) be such +that ˜v|D = v. We see that +ℓ(v) = ℓ1(˜v|D) = (˜v, µ) = (˜v, (∆ + k2)w). +From (2.4), we know that there are wj ∈ C∞ +c (D) with wj → w in H2−s,p′(Rn). +Since +(∆ + k2)˜v = 0 in D, we finally conclude that +ℓ(v) = lim +j→∞(˜v, (∆ + k2)wj) = lim +j→∞((∆ + k2)˜v, wj) = 0, +which is our desired result. +□ +2.3. Proof of the main result. Theorem 1.1 is an immediate consequence of the following +result: +Theorem 2.3. Let D be a bounded Lipschitz domain in Rn (n ≥ 2) such that Rn \ D is +connected. +Suppose that k2 > 0 is not a Dirichlet eigenvalue of −∆ in D. +Given any +constant c0 ∈ R, there exist Herglotz wave functions uj ∈ C∞(Rn) solving (∆ + k2)uj = 0 in +Rn such that +lim +j→∞ ∥uj − c0∥L∞(∂D) = 0. +Before we prove Theorem 2.3 we need the following result, which is a special case of [JK95, +Theorems 1.1 & 1.3]. +Proposition 2.4. Let D be a bounded Lipschitz domain in Rn (n ≥ 2). If 2 ≤ p < ∞ and +f ∈ Hs−2,p(D) where +1 +p < s < 3 +p, +then there exists a unique u ∈ Hs,p(D) satisfying −∆u = f in D and u = 0 on ∂D. +Proof. We first consider the case when n ≥ 3. Let p0 be as in [JK95, Theorem 1.1] (with +Ω = D). If p′ +0 ≤ p < ∞, the result follows from [JK95, Theorem 1.1(c)]. On the other hand, +if 2 ≤ p < p′ +0, the result follows from [JK95, Theorem 1.1(a)] since s < 3 +p ≤ 1 + 1 +p. The +case when n = 2 can be proved using identical reasoning using [JK95, Theorem 1.3] and the +observation 3 +p ≤ 2 +p + 1 +2. +□ + +6 +PU-ZHAO KOW, MIKKO SALO, AND HENRIK SHAHGHOLIAN +Proof of Theorem 2.3. Since k2 is not a Dirichlet eigenvalue in D, there exists a unique +solution v ∈ H1,2(D) such that +(∆ + k2)v = 0 in D +and +v = c0 on ∂D. +If v ∈ Hs,p(D) for some 0 < s ≤ 1 and p > n/s, using Proposition 2.2, we know that there +exist Herglotz waves uj ∈ C∞(Rn) such that +∥uj − c0∥L∞(∂D) = ∥uj − v∥L∞(∂D) ≤ ∥uj − v∥C(D) ≤ C∥uj − v∥Hs,p(D) → 0, +where we used the Sobolev embedding. +It remains to show that v ∈ Hs,p(D) for some s, p with s > n/p, and this follows from a +standard bootstrap argument based on Proposition 2.4. We claim that +(2.12) +v ∈ H +2 +pj ,pj(D) for 0 ≤ j < n − 2 +4 +, +where +1 +pj += 1 +2 − j +2 +n − 2. +The case j = 0 follows since v ∈ H1,2(D). We argue by induction and assume that this holds +for j. Define w := v − c0 and note that w solves +−∆w = k2v ∈ H +2 +pj ,pj(D), +w|∂D = 0. +We next use the Sobolev embedding H +2 +pj ,pj(D) ⊂ H +2 +q −2,q(D) where +2 +pj > 2 +q − 2 and +2 +pj +− n +pj += 2 +q − 2 − n +q . +It follows that q = pj+1 and then indeed +2 +pj > 2 +q − 2. In particular −∆w ∈ H +2 +pj+1 −2,pj+1(D) +with w|∂D = 0, and we may use Proposition 2.4 to conclude that w ∈ H +2 +pj+1 ,pj+1(D). This +completes the induction step and proves (2.12). +We have proved that v ∈ H +2 +pj ,pj(D) where j is the largest integer < +n−2 +4 . +Using the +above notation, we have ∆w ∈ H +2 +pj ,pj(D) and w|∂D = 0. By Sobolev embedding we have +∆w ∈ Hs−2,p(D) whenever p ≥ pj and +2 +pj +− n +pj += s − 2 − n +p . +The last condition implies that +s − n +p = 2 + 2 − n +pj += 2 + 2 − n +2 ++ 2j ≥ 0 +since j ≥ n−2 +4 +− 1. If j > n−2 +4 +− 1, using Proposition 2.4 once again we obtain that w and +hence v is in Hs,p for some s > n/p. On the other hand, if j = +n−2 +4 +− 1 we iterate the +argument once more to get v ∈ Hs,p for some s > n/p. This concludes the proof. +□ +The next simple example shows that the condition that k2 is not an eigenvalue is necessary +at least for balls. + +ON POSITIVITY SETS FOR HELMHOLTZ SOLUTIONS +7 +Example 2.5. Let v(x) := |x| +2−n +2 J n−2 +2 (|x|). We see that v ∈ C∞(Rn) and (∆ + 1)v = 0 in +Rn. Suppose that u1 is a real-valued function satisfying (∆ + 1)u1 = 0 in Rn. Since +v(x) = 0 +when |x| = j n−2 +2 +,m for any m ≥ 1, +where j n−2 +2 ,m denotes the mth positive zero of J n−2 +2 , we have +� +|x|=j n−2 +2 +,m +u1 +∂v +∂r dS = +� +|x| 0 +when |x| = j n−2 +2 +,m, +it follows that u1 must change sign on |x| = j n−2 +2 +,m. +Similarly, if R > 0 and if u0 solves (∆ + k2 +m)u0 = 0 in Rn where km = R−1j n−2 +2 ,m, define u1 +via the rescaling +u0(x) = u1(R−1j n−2 +2 +,mx) +for x ∈ Rn. +We see that (∆ + 1)u1 = 0 in Rn. The above discussion shows that u0 must change sign on +∂BR. +The following strong maximum principle can be found in [KLSS22, Appendix A]. However, +for readers’ convenience, here we exhibit the statement as well as its proof. +Lemma 2.6 (Strong maximum principle). Let D be a bounded Lipschitz domain in Rn +(n ≥ 2), and let k2 < λ1(D), where λ1(D) > 0 denotes the smallest H1 +0(D)-eigenvalue of +−∆. If the solution u ∈ H1(D) satisfies +(∆ + k2)u = 0 in D, +u ≥ 0 on ∂D, +then for each open component G of D we have either u ≡ 0 in G or u > 0 in G (note that +u ∈ C∞(G) by elliptic regularity). +Proof. It is easy to see that for each component G of D we have k2 < λ1(G) and +(∆ + k2)u = 0 in G, +u ≥ 0 on ∂G. +Testing the equation above by u− ∈ H1 +0(G) and using Poincaré inequality, we have +� +G +|u−|2 dx ≤ +1 +λ1(G) +� +G +|∇u−|2 dx = +k2 +λ1(G) +� +G +|u−|2 dx. +Since +k2 +λ1(G) < 1, then u− ≡ 0 in G, that is, +(2.13) +u ≥ 0 in G. +Let x0 ∈ G such that u(x0) = 0. The mean value theorem for Helmholtz equation (see e.g. +[KLSS22, Appendix A]) gives that +(2.14) +� +Bǫ(x0) +u(x) dx = 0 +for all sufficiently small ǫ > 0 so that Bǫ(x0) ⊂ G. Since u is continuous in G, combining +(2.13) and (2.14) we know that u = 0 in Bǫ(x0), and this shows that +� x ∈ G +u(x) = 0 � +is +both open and closed in G. Since G is connected, then we have either +� x ∈ G +u(x) = 0 � += G +or +� x ∈ G +u(x) = 0 � += ∅, + +8 +PU-ZHAO KOW, MIKKO SALO, AND HENRIK SHAHGHOLIAN +which concludes our lemma. +□ +Finally, we give the proof of Theorem 1.2. +Proof of Theorem 1.2. Since |D| ≤ |Br| where r = j n−2 +2 ,1k−1, the Faber-Krahn inequality (see +e.g. [Cha01, Theorem III.3.1]) implies that each connected component G of D satisfies +λ1(G) ≥ λ1(Br) = k2. +Case 1. +If λ1(G) = k2, we choose v to be an eigenfunction corresponding to the first +eigenvalue with v > 0 in G, i.e. v solves (∆ + k2)v = 0 in G with v ∈ H1 +0(G), see e.g. [Eva10, +Theorem 2(ii) in Section 6.5.1]. +Case 2. If λ1(G) > k2, then there exists a unique solution v ∈ H1(G) such that +(∆ + k2)v = 0 in G, +v = 1 on ∂G. +Using the strong maximum principle in Lemma 2.6, we know that v > 0 in G. +Next we choose a bounded Lipschitz domain D1 that satisfies E ⊂ D1, D1 ⊂ D, and +Rn \ D1 is connected. The function v|D1 is in H1,p(D1) for any p > n and satisfies v|D1 > 0. +The approximation result in Proposition 2.2 yields a sequence of Herglotz waves uj satisfying +∥uj|D1 − v∥H1,p(D1) → 0 as j → ∞. +If j is sufficiently large, the Sobolev embedding ensures that uj|E > 0. +□ +Acknowledgments +This project was finalized while the authors stayed at Institute Mittag Leffler (Sweden), +during the program Geometric aspects of nonlinear PDE. Kow and Salo were partly supported +by the Academy of Finland (Centre of Excellence in Inverse Modelling and Imaging, 312121) +and by the European Research Council under Horizon 2020 (ERC CoG 770924). Shahgholian +was supported by Swedish Research Council. +Declarations +Data availability statement: All data needed are contained in the manuscript. +Funding and/or Conflicts of interests/Competing interests: The authors declare +that there are no financial, competing or conflict of interests. +References +[BL76] +J. +Bergh +and +J. +Löfström. +Interpolation +spaces, +volume +223 +of +Grundlehren +der +Mathematischen +Wissenschaften. +Springer-Verlag, +Berlin +Heidelberg, +1976. +MR0482275, +doi:10.1007/978-3-642-66451-9. +[CV21] +F. Cakoni and M. S. Vogelius. Singularities almost always scatter: +Regulari results for non- +scattering inhomogeneities. arXiv preprint, 2021. arXiv:2104.05058. +[Cha01] +I. Chavel. Isoperimetric inequalities, volume 145 of Cambridge Tracts in Mathematics. Cambridge +University Press, Cambridge, 2001. MR1849187. +[CK19] +D. Colton and R. Kress. Inverse acoustic and electromagnetic scattering theory, volume 93 +of +Applied +Mathematical +Sciences. +Springer, +Cham, +fourth +edition, +2019. +MR3971246, +doi:10.1007/978-3-030-30351-8. +[Eva10] +L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. +American +Mathematical +Society, +Providence, +RI, +second +edition, +2010. +MR1625845, +doi:10.1090/gsm/019. + +ON POSITIVITY SETS FOR HELMHOLTZ SOLUTIONS +9 +[Hör73] +L. Hörmander. Lower bounds at infinity for solutions of differential equations with constant +coefficients. Israel J. Math., 16:103–116, 1973. MR0340793, doi:10.1007/BF02761975. +[JK95] +D. Jerison and C. E. Kenig. The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. +Anal., 130(1):161–219, 1995. MR1331981, doi:10.1006/jfan.1995.1067. +[KLSS22] P.-Z. Kow, S. Larson, M. Salo, and H. Shahgholian. Quadrature and non-scattering domains for +the helmholtz equation. arXiv preprint, 2022. arXiv:2204.13934. +[LM20] +A. Logunov and E. Malinnikova. Review of Yau’s conjecture on zero sets of Laplace eigenfunctions. +In Current developments in mathematics, volume 2018, pages 179–212. Int. Press, Somerville, MA, +2020. MR4363378, doi:10.4310/CDM.2018.v2018.n1.a4, arXiv:1908.01639. +[Mel95] +R. B. Melrose. Geometric scattering theory. Stanford Lectures. Cambridge University Press, +Cambridge, 1995. MR1350074. +[SS21] +M. Salo and H. Shahgholian. Free boundary methods and non-scattering phenomena. Res. Math. +Sci., 8(4):Paper No. 58, 2021. MR4323345, doi:10.1007/s40687-021-00294-z, arXiv:2106.15154. +[Yaf10] +D. R. Yafaev. Mathematical scattering theory. Analytic theory, volume 158 of Mathematical +Surveys and Monographs. American Mathematical Society, Providence, RI, 2010. MR2598115, +doi:10.1090/surv/158. +Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of +Jyväskylä, Finland. +Email address: pu-zhao.pz.kow@jyu.fi +Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of +Jyväskylä, Finland. +Email address: mikko.j.salo@jyu.fi +Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, +Sweden. +Email address: henriksh@kth.se + diff --git a/ZtE4T4oBgHgl3EQfOgxT/content/tmp_files/load_file.txt b/ZtE4T4oBgHgl3EQfOgxT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..137cb2b3b100ec5f90c8983ca264c19974231a6f --- /dev/null +++ b/ZtE4T4oBgHgl3EQfOgxT/content/tmp_files/load_file.txt @@ -0,0 +1,404 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf,len=403 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='04965v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='AP] 12 Jan 2023 ON POSITIVITY SETS FOR HELMHOLTZ SOLUTIONS PU-ZHAO KOW, MIKKO SALO, AND HENRIK SHAHGHOLIAN Dedicated to Carlos E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Kenig on the occasion of his 70th birthday Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We address the question of finding global solutions of the Helmholtz equation that are positive in a given set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' This question arises in inverse scattering for penetrable obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' In particular, we show that there are solutions that are positive on the boundary of a bounded Lipschitz domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Introduction The objective in this short note is to consider the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Let k > 0 and let E be a subset of Rn (n ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Does there exist a solution of (∆ + k2)u = 0 in Rn with u|E > 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Note that any solution of the Helmholtz equation (∆ + k2)u = 0 is C∞, and thus the condition u|E > 0 can be understood pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' There is a substantial literature on zero sets of solutions of elliptic equations and eigenfunctions, as discussed in the review [LM20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' In our setting, any real valued solution of (∆ + k2)u = 0 in Rn must have a zero in any closed ball of radius j n−2 2 ,1k−1 where j n−2 2 ,1 is the first zero of the Bessel function J n−2 2 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' [SS21, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1 above is related to producing a global solution whose zero set avoids a given set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Our motivation comes from inverse scattering theory and the works [CV21, SS21, KLSS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' In these works, one considers a bounded open set D ⊂ Rn (penetrable obstacle) together with a coefficient h ∈ L∞(Rn) with |h| ≥ c > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' near ∂D (contrast), and asks whether it is possible to find a solution u0 ̸≡ 0 of (∆ + k2)u0 = 0 in Rn (incident wave) such that the obstacle D with contrast h does not produce any scattering response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' The last condition can be precisely formulated as the existence of a function u solving (∆ + k2 + hχD)u = 0 in Rn, u = u0 outside some ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' If this happens for some contrast h, then the obstacle D is called a non-scattering domain and it will be invisible with respect to probing with the incident wave u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' It was proved in [SS21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1] that if D has real-analytic boundary and if there is an incident wave u0 with u0|∂D > 0, then D is a non-scattering domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Similarly, the work [KLSS22] introduced the notion of quadrature domains for the Helmholtz operator ∆ + k2 and proved that if D is such a domain, and if there is an incident wave u0 with u0|∂D > 0, then D is a non-scattering domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' On the other hand, the works [CV21, SS21] show that under a nonvanishing condition for u0 on ∂D, the boundary of a non-scattering domain can 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' 35J05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' 35J15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' 35J20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' 35R30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' 35R35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Helmholtz equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' acoustic equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Lipschitz domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' inverse scattering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' 1 2 PU-ZHAO KOW, MIKKO SALO, AND HENRIK SHAHGHOLIAN be interpreted as a free boundary in an obstacle-type problem and hence such a domain must be either regular or have thin complement near any boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' It was also proved in [SS21] that one may be able to find incident waves that are positive on the boundary of a bounded C1 domain (Lipschitz if n = 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Our first main result extends this to Lipschitz domains in any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Let D ⊂ Rn (n ≥ 2) be a bounded Lipschitz domain such that Rn \\ D is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Suppose that k2 > 0 is not a Dirichlet eigenvalue of −∆ in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Then there exists a Herglotz wave function u0 (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1) satisfying (∆ + k2)u0 = 0 in Rn and u0|∂D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1 is done in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' One first constructs a solution v of (∆ + k2)v = 0 in D with v|∂D > 0 by solving a Dirichlet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Then one approximates v in D by a suitable Herglotz wave u0 in Rn via a Runge approximation argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' This approximation needs to be done in a suitable norm to obtain the pointwise condition u0|∂D > 0, but since D only has Lipschitz boundary the solution v is not very regular and this limits the choice of possible norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We will work with fractional Sobolev spaces Hs,p and invoke the theory of boundary value problems in Lipschitz domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1 We remark that the assumption in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1 that k2 is not an eigenvalue is necessary, at least when D is a ball (see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' For the first eigenvalue this was pointed out in [SS21, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Another instance of subsets E ⊂ Rn where one can arrange u0|E > 0 is given in the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Let k > 0, and let D ⊂ Rn (n ≥ 2) be a bounded Lipschitz domain such that Rn \\ D is connected and |D| ≤ |Br| where r = j n−2 2 ,1k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' If E ⊂ D is compact, then there exists a Herglotz wave function u0 (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1) satisfying (∆ + k2)u0 = 0 in Rn and u0|E > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' The proof is similar to that of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1, except that in the first step we use the Faber-Krahn inequality to produce a solution v that is positive near E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Solutions satisfying the positivity condition In this section we will prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We begin with some preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Fractional Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' For each s ∈ R and 1 < p < ∞, the fractional Sobolev space Hs,p(Rn) is the Banach space equipped with the norm ∥u∥Hs,p(Rn) := ∥⟨D⟩su∥Lp(Rn), where ⟨D⟩s is the the Bessel potential of order s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' the Fourier multiplier corresponding to ⟨ξ⟩s = (1 + |ξ|2) s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' In particular when s = k ≥ 1 is an integer, we also have Hk,p(Rn) = W k,p(Rn), where W k,p(Rn) = � u ∈ Lp(Rn) Dαu ∈ Lp(Rn) for all multi-indices α with |α| ≤ k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' From [BL76, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='8], we have the duality statement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1) (Hs,p(Rn))∗ = H−s,p′(Rn) for all s ∈ R and 1 < p < ∞, 1This is one of the areas where Carlos Kenig has made pioneering contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' ON POSITIVITY SETS FOR HELMHOLTZ SOLUTIONS 3 where (p′)−1 + p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We also recall the Sobolev embedding ([BL76, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1]): (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='2) Hs,p(Rn) ⊂ Hs1,p1(Rn) whenever 1 < p ≤ p1 < ∞, −∞ < s1 ≤ s < ∞, and s − n p = s1 − n p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Let D be an open set in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We define Hs,p(D) := � u|D u ∈ Hs,p(Rn) � for all s ∈ R and 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' This is a Banach space equipped with the quotient norm ∥v∥Hs,p(D) := inf � ∥u∥Hs,p(Rn) u|D = v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' When D is a bounded Lipschitz domain, from [JK95, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='3] we know that there exists a bounded linear extension operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='3) E : Hs,p(D) → Hs,p(Rn) with Eu = u in D for all u ∈ Hs,p(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' If F ⊂ Rn is closed, we define Hs,p F (Rn) := � u ∈ Hs,p(Rn) supp(u) ⊂ F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' If D is a bounded Lipschitz domain, the following result can be found in [JK95, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='7]: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='4) C∞ c (D) is dense in Hs,p D (Rn) for each s ∈ R and 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Runge-Herglotz approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' The next objective is to prove a result stating that solutions in Hs,p(D) can be approximated in D by Herglotz waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We first give a definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Let k > 0 and consider the operator Pk : C∞(Sn−1) → C∞(Rn) defined by (Pkf)(x) := � Sn−1 eikx·ˆzf(ˆz) dˆz, x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' The functions u = Pkf with f ∈ C∞(Sn−1) are called Herglotz waves, and they are particular solutions of (∆ + k2)u = 0 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Let k > 0, 0 < s ≤ 1, 1 < p < ∞, and let D ⊂ Rn (n ≥ 2) be a bounded Lipschitz domain such that Rn \\ D is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Given any v ∈ Hs,p(D) with (∆ + k2)v = 0 in D, there exist Herglotz waves uj ∈ C∞(Rn) such that ∥uj − v∥Hs,p(D) → 0 as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' If v is real-valued, then so are uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='2 is very similar to [SS21, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='4] that considered approximation in W 1,p(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Here we need to work with fractional Sobolev spaces instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' In view of the Hahn-Banach theorem, it is enough to prove that any bounded linear functional ℓ : Hs,p(D) → C that vanishes on � Pkf|D f ∈ C∞(Sn−1) � must also vanish on � v ∈ Hs,p(D) −(∆ + k2)v = 0 in D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Let ℓ be such a linear functional, and define a bounded linear functional ℓ1 : Hs,p(Rn) → C by ℓ1(u) := ℓ(u|D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' By duality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1), there exists a unique µ ∈ H−s,p′(Rn) such that ℓ1(u) = (u, µ) for all u ∈ Hs,p(Rn), where (·, ·) is the sesquilinear distributional pairing in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' It is easy to see that µ = 0 in Rn \\ D, and the condition ℓ(Pkf|D) = 0 for all f ∈ C∞(Sn−1) implies that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='5) (Pkf, µ) = 0 for all f ∈ C∞(Sn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' 4 PU-ZHAO KOW, MIKKO SALO, AND HENRIK SHAHGHOLIAN We now define the distribution w := Φk ∗ µ, where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='6) Φk(x) = ik n−2 2 4(2π) n−2 2 |x|− n−2 2 H(1) n−2 2 (k|x|) is the outgoing fundamental solution of the Helmholtz operator −(∆ + k2) and H(1) α is the Hankel function (see [Yaf10, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Then w is a distributional solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='7) − (∆ + k2)w = µ in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Elliptic regularity yields w ∈ H2−s,p′ loc (Rn), and since supp(µ) ⊂ D we also have that w is C∞ in Rn \\ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Given any f ∈ C∞(Sn−1), we write u = Pkf ∈ C∞(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='5) and the fact that µ has compact support, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='8) 0 = (u, µ) = lim r→∞(u, µ)Br, where (·, ·)Br is the sesquilinear distributional pairing in the ball Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We now consider a cut-off function χ ∈ C∞ c (Rn) satisfying 0 ≤ χ ≤ 1 and χ = 1 near D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='7), we can write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='8) as 0 = lim r→∞ � (χu, (∆ + k2)w)Br + ((1 − χ)u, (∆ + k2)w)Br � = lim r→∞ � ((∆ + k2)(χu), w)Br + ((∆ + k2)((1 − χ)u), w)Br + � ∂Br (u∂|x|w − (∂|x|u)w) dS � = lim r→∞ � ∂Br (u∂|x|w − (∂|x|u)w) dS, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='9) where ∂|x| = ˆx·∇ denotes the radial derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Here we also used the fact that (∆+k2)u = 0 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Using [Mel95, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='2 and equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='18)], we know that the Herglotz function u = Pkf has the following asymptotics as |x| → ∞: u(x) = c′ n,k|x|− n−1 2 � eik|x|f(ˆx) + in−1e−ik|x|f(−ˆx) � + O(|x|− n+1 2 ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='10a) ∂|x|u(x) = c′ n,k|x|− n−1 2 ik � eik|x|f(ˆx) − in−1e−ik|x|f(−ˆx) � + O(|x|− n+1 2 ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='10b) where c′ n,k = k n−1 2 e π(n−1)i 4 (2π)− n−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' On the other hand, from [Yaf10, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='27)], we know that w has the asymptotics w(x) = c′′ n,k|x|− n−1 2 eik|x|ˆµ(kˆx) + O(|x|− n+1 2 ) as |x| → ∞, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='10c) ∂|x|w(x) = c′′ n,k|x|− n−1 2 ikeik|x|ˆµ(kˆx) + O(|x|− n+1 2 ) as |x| → ∞, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='10d) ON POSITIVITY SETS FOR HELMHOLTZ SOLUTIONS 5 where c′′ n,k = 2−1e− π(n−3)i 4 (2π)− n−1 2 k n−3 2 and ˆµ ∈ C∞(Rn) is the Fourier transform of the compactly supported distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='9) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='10a)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='10d), we obtain � Sn−1 f(ˆx)ˆµ(kˆx) dˆx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' By the fact that f ∈ C∞(Sn−1) was arbitrary, we conclude ˆµ(kˆx) = 0 for all ˆx ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Consequently, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='10c) becomes w(x) = O(|x|− n+1 2 ) as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' In other words, the far-field pattern of w is vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' By the Rellich uniqueness theorem [CK19, Hör73], the unique continuation principle and the connectedness of Rn \\ D, we conclude that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='11a) w = 0 in Rn \\ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Since w ∈ H2−s,p′ loc (Rn), we also conclude that w ∈ H2−s,p′ D (Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Now let v ∈ Hs,p(D) be any solution of (∆ + k2)v = 0 in D, and let ˜v ∈ Hs,p(Rn) be such that ˜v|D = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We see that ℓ(v) = ℓ1(˜v|D) = (˜v, µ) = (˜v, (∆ + k2)w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='4), we know that there are wj ∈ C∞ c (D) with wj → w in H2−s,p′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Since (∆ + k2)˜v = 0 in D, we finally conclude that ℓ(v) = lim j→∞(˜v, (∆ + k2)wj) = lim j→∞((∆ + k2)˜v, wj) = 0, which is our desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Proof of the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1 is an immediate consequence of the following result: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Let D be a bounded Lipschitz domain in Rn (n ≥ 2) such that Rn \\ D is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Suppose that k2 > 0 is not a Dirichlet eigenvalue of −∆ in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Given any constant c0 ∈ R, there exist Herglotz wave functions uj ∈ C∞(Rn) solving (∆ + k2)uj = 0 in Rn such that lim j→∞ ∥uj − c0∥L∞(∂D) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Before we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='3 we need the following result, which is a special case of [JK95, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1 & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Let D be a bounded Lipschitz domain in Rn (n ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' If 2 ≤ p < ∞ and f ∈ Hs−2,p(D) where 1 p < s < 3 p, then there exists a unique u ∈ Hs,p(D) satisfying −∆u = f in D and u = 0 on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We first consider the case when n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Let p0 be as in [JK95, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1] (with Ω = D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' If p′ 0 ≤ p < ∞, the result follows from [JK95, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' On the other hand, if 2 ≤ p < p′ 0, the result follows from [JK95, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='1(a)] since s < 3 p ≤ 1 + 1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' The case when n = 2 can be proved using identical reasoning using [JK95, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='3] and the observation 3 p ≤ 2 p + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' □ 6 PU-ZHAO KOW, MIKKO SALO, AND HENRIK SHAHGHOLIAN Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Since k2 is not a Dirichlet eigenvalue in D, there exists a unique solution v ∈ H1,2(D) such that (∆ + k2)v = 0 in D and v = c0 on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' If v ∈ Hs,p(D) for some 0 < s ≤ 1 and p > n/s, using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='2, we know that there exist Herglotz waves uj ∈ C∞(Rn) such that ∥uj − c0∥L∞(∂D) = ∥uj − v∥L∞(∂D) ≤ ∥uj − v∥C(D) ≤ C∥uj − v∥Hs,p(D) → 0, where we used the Sobolev embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' It remains to show that v ∈ Hs,p(D) for some s, p with s > n/p, and this follows from a standard bootstrap argument based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We claim that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='12) v ∈ H 2 pj ,pj(D) for 0 ≤ j < n − 2 4 , where 1 pj = 1 2 − j 2 n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' The case j = 0 follows since v ∈ H1,2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We argue by induction and assume that this holds for j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Define w := v − c0 and note that w solves −∆w = k2v ∈ H 2 pj ,pj(D), w|∂D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We next use the Sobolev embedding H 2 pj ,pj(D) ⊂ H 2 q −2,q(D) where 2 pj > 2 q − 2 and 2 pj − n pj = 2 q − 2 − n q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' It follows that q = pj+1 and then indeed 2 pj > 2 q − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' In particular −∆w ∈ H 2 pj+1 −2,pj+1(D) with w|∂D = 0, and we may use Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='4 to conclude that w ∈ H 2 pj+1 ,pj+1(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' This completes the induction step and proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We have proved that v ∈ H 2 pj ,pj(D) where j is the largest integer < n−2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Using the above notation, we have ∆w ∈ H 2 pj ,pj(D) and w|∂D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' By Sobolev embedding we have ∆w ∈ Hs−2,p(D) whenever p ≥ pj and 2 pj − n pj = s − 2 − n p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' The last condition implies that s − n p = 2 + 2 − n pj = 2 + 2 − n 2 + 2j ≥ 0 since j ≥ n−2 4 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' If j > n−2 4 − 1, using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='4 once again we obtain that w and hence v is in Hs,p for some s > n/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' On the other hand, if j = n−2 4 − 1 we iterate the argument once more to get v ∈ Hs,p for some s > n/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' □ The next simple example shows that the condition that k2 is not an eigenvalue is necessary at least for balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' ON POSITIVITY SETS FOR HELMHOLTZ SOLUTIONS 7 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Let v(x) := |x| 2−n 2 J n−2 2 (|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' We see that v ∈ C∞(Rn) and (∆ + 1)v = 0 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Suppose that u1 is a real-valued function satisfying (∆ + 1)u1 = 0 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE4T4oBgHgl3EQfOgxT/content/2301.04965v1.pdf'} +page_content=' Since v(x) = 0 when |x| = j n−2 2 ,m for any m ≥ 1, where j n−2 2 ,m denotes the mth positive zero of J n−2 2 , we have � |x|=j n−2 2 ,m u1 ∂v ∂r dS = � |x| λ∗ℓ+1 +a +(⇐⇒ λ∗n−ℓ+1 +b +< λ∗n−ℓ +b +) this defines a unique partition which +is the unique λ-Fair assignment. If λ∗ℓ +a += λ∗ℓ+1 +a += ℓ we have q ”borderline” +agents, q ≥ 2, reporting λi = (λ∗ℓ +a , λ∗n−ℓ +b +), while pa agents accept a congestion +on a higher than ℓ and pb agents accept more than n − ℓ on b. The λ-Fair +assignments place any subset of borderline agents in Sa together with the pa +“fans” of post a and the rest in Sb with the fans of b. +Note that if the reports are truthful, the sets of FMEq and of λ-Fair assign- +ments coincide. +Example 2: three posts and a “single” λ-Fair assignment +The n = 12 +agents are split in four types labeled α to δ with three agents in each type. +Agents of a given type are not necessarily identical but they report the same +caps as follows, implying that there is just one λ-Fair congestion and six λ-Fair +assignments by permuting the type α agents: +n = 12 +a +b +c +ααα +4 +4 +4 +βββ +8 +2 +2 +γγγ +2 +8 +2 +δδδ +2 +2 +8 +=⇒ λ-Fair assignment: +a +b +c +αβββ +αγγγ +αδδδ +The claim follows by noticing that we can fit λ-Fairly at most 4 agents on +each post, therefore the only possible congestion profile is 4 agents per post. +Here again free mobility equilibrium and λ-Fairness pick the same assignments. +7 + +More generally write cmx(a; λ) for the maximal number of agents we can +fit λ-Fairly at post a: if � +a∈A cmx(a; λ) = n then all λ-Fair assignments gen- +erate the congestion profile s(a) = cmx(a; λ) for all a. The converse property +holds as well: all λ-Fair assignments have the same congestion profile only if +� +a∈A cmx(a; λ) = n. We omit the easy proofs. +Example 3: three posts and multiple λ-Fair assignments +n = 15 +a +b +c +ααααα +7 +7 +1 +βββββ +7 +1 +7 +γγγγγ +1 +7 +7 +=⇒ +a +b +c +ααααα +γγγγγ +βββββ +and +a +b +c +ααα +ααγγγγγ +βββββ +are λ-Fair +These λ-Fair assignments have very different congestion and welfare implica- +tions. +Both types of λ-Fair assignments may or may not describe a FMEq, +depending on individual preferences. For instance an α-agent in the symmetric +assignment on the left may prefer (b, 6) to (a, 5). So the converse of statement +ii) in Lemma 1 does not hold. +2.2 +competitiveness +Note that λ-Fairness, in addition to the multiplicity just illustrated, can easily +allow inefficient assignments. For instance in Example 2 the six assignments of +the α-agents respect λ-Fairness but if each α-agent prefers a different allocation +(x, 4) only one of these is efficient. Competitive assignments do not have this +problem and their recommendation is unambiguous as well. +2.2.1 +deterministic competitiveness +We use the notation z ∨ y = max{z, y}. +Definition 2 Fix a problem (A, N, ⪰i, i ∈ N). The assignment P is com- +petitive (Comp) iff +for all a ∈ A and i ∈ Sa : (a, sa) ⪰i (x, sx ∨ 1) for all x ∈ A +The difference with the familiar envy freeness property is in the treatment +of empty posts: an agent prefering an empty post x to her own post demands to +move there; in this case our agent cannot assume that the post remains empty +when she gets there ((a, 0) is not an allocation) so Comp requires (a, sa) ⪰i +(x, 1). But any assignment where all agents share a single post are automatically +envy free, so the concept must be adapted to our model. +A FMEq assignment is competitive ”up to one unit of congestion on an +occupied post”; conversely at a competitive assignment P the corresponding +strategies xi = a ⇐⇒ i ∈ Sa form a FMEq. Both claims follow at once from +the definitions. +8 + +Proposition 2: Fix a problem (A, N, (ui)i∈N). +i) All competitive assignments have the same congestion profile (except possibly +at some posts occupied by at most one agent), and the same welfare profile across +agents. +ii) A competitive assignment is weakly efficient, and efficient if preferences are +strict and/or if all posts are occupied. +iii) A competitive assignment is λi-Fair if λi describes a truthful top-n set of +⪰i for all i. +Proof +i) Uniqueness At every congestion profile s ∈ ∆N(A; n) (see(2)) we define agent +i’s demand D(i; s) = {a|(a, sa ∨ 1) ⪰i (x, sx ∨ 1) for all x}: the assignment P is +competitive if and only if a ∈ D(i; s) whenever i ∈ Sa. +Fix P = (Sx)x∈A, P ∗ = (S∗ +x)x∈A both competitive and s. t. s ̸= s∗. Assume +first that A∗ = {a ∈ A|s∗ +a ∨ 1 > sa ∨ 1} is non empty and note that in A∗ we +have s∗ +a > sa, 1. Fixing an agent i we claim that if D(i; s∗) intersects A∗ at a +then D(i; s) must be a subset of A∗. +If the claim fails there is some b ∈ D(i; s) outside A∗ such that, for all a in +A∗: (b, sb ∨ 1) ⪰i (a, sa ∨ 1) ≻i (a, s∗ +a) (the strict preference because s∗ +a > sa, 1). +By the choice of b we also have sb ∨ 1 ≥ s∗ +b ∨ 1; these two facts together give +(b, s∗ +b ∨1) ≻i (a, s∗ +a), and as a was arbitrary in A∗ it follows that D(i; s∗) cannot +intersect A∗, contradiction. +Now for each i ∈ ∪a∈A∗S∗ +a the claim says D(i; s) ⊆ A∗ therefore � +a∈A∗ sa ≥ +� +a∈A∗ s∗ +a, contradicting s∗ +a > sa in A∗. We conclude that A∗ must be empty. +So P, P ∗ must be such that s∗ +a ∨ 1 = sa ∨ 1 for all a: this means that +s∗ +a ̸= sa can only happen when one of s∗ +a,sa is 0 and the other is 1, as claimed +in statement i). +We check now that all agents are indifferent between the two assignments. +Say agent i is in Sa and S∗ +b : from sa, s∗ +b ≥ 1 and Comp we have +(a, sa) ⪰i (b, sb ∨ 1) and (b, s∗ +b) ⪰i (a, s∗ +a ∨ 1) +(3) +If sa = s∗ +a and sb = s∗ +b we are done. If sa ̸= s∗ +a and sb = s∗ +b then sa = 1 > 0 = s∗ +a +and s∗ +b ≥ 1 so that (3) gives (a, 1) ⪰i (b, sb) = (b, s∗ +b) ⪰i (a, 1) as desired; the +last subcase sa ̸= s∗ +a and sb ̸= s∗ +b is just as easy. +A simple example with multiple competitive congestion profiles has all agents +except 1 and 2 refusing the three posts a, b, c, while 1 and 2 refuse all but a, b, c +and they are indifferent between (a, 1), (b, 1) and (c, 1). +ii) Efficiency Assume, to the contrary, that P = (Sx) is competitive and Pareto +inferior to Q = (Tx). Say i assigned to a at P is assigned to b at Q (a, b not +necessarily distinct), and suppose that post b is occupied at P: sb ≥ 1. Then +by Comp and the weak Pareto improvement we have +(b, Sb) ⪯i (a, Sa) ⪯i (b, Tb) =⇒ sb ≥ tb +(4) +and sb > tb if agent i improves strictly at Q. If all posts are occupied at P then +(4) implies s = t and we have a contradiction.. +9 + +If instead for some agent i goes from a at P to c at Q and c is empty at P, +sc = 0, we have +(c, 1) ⪯i (a, Sa) ⪯i (c, Tc) =⇒ tc = 1 and (a, Sa) ≃i (c, 1) +(5) +If preferences are strict we conclude again that P is Pareto optimal. Even +if they are not we see that i is a weak Pareto optimum (not all agent benefit +strictly). +The argument above explains how a competitive assignment can be Pareto +inferior: in the situation (5) moving i from a to c and changing nothing else +is a Pareto improvement to a new assignment �P where agent i’s competitive +demand is now a if sa ≥ 2 (because (a, sa − 1) ≻i (c, 1)). If sa = 1 then �P is +still competitive and welfare-wise indifferent to P. +A simple example of this situation has two posts a, b, all but agent 1 refus- +ing a and agent 1 indifferent between (a, 1) and (b, n); the unique competitive +assignment has everyone at b, and moving agent 1 to a is Pareto improving.But +only happens if ⪯i has some indifference and some post is unoccupied. Moreover +the PO improvement cannot be strict. +iii) λi-Fairness This follows from Lemma 1 and the remark that a compet- +itive assignment is an FMEq (just before Proposition 2). ■ +Dampening the appeal of competitive assignments explained by Proposition +2, it is easy to find examples where none exists. +Example 4: two posts and no competitive assignment +The truthful +reports by the four agents allow only two λ-Fair assignments: +reports λ: +m = 2, n = 4 +a +b +α, α′ +2 +2 +β +3 +1 +γ +1 +3 +=⇒ λ-Fair: +a +b +α, β +α′, γ +and +a +b +α′, β +α, γ +(6) +If both agents α, α′ prefer (a, 2) to (b, 2) no assignment is competitive. +A similar situation happens in example 2 when the three α agents have +identical preferences over the 3 allocations (x, 4). +2.2.2 +fractional competitiveness +We assume now that the agents’ preferences are represented by vNM expected +utility functions ui(a, s) over A × [n]. For a finite set Z the notation ∆R(Z; w) +is the simplex of sum w with non negative real coordinates in Z. +We extend each ui to a function ui(a, x) by linear interpolation between the +two rounded up and down values of x, ⌊x⌋ and ⌈x⌉: +ui(a, x) = +⌈x⌉ − x +⌈x⌉ − ⌊x⌋ui(a, ⌊x⌋) + x − ⌊x⌋ +⌈x⌉ − ⌊x⌋ui(a, ⌈x⌉) +10 + +so ui(a, x) is continuous and strictly decreasing in x ∈ [0, n], the interval in +R.In what follows x is the expected congestion at some post a when a ran- +dom (integer) congestion takes only the values ⌊x⌋ and ⌈x⌉ and ui(a, x) is the +corresponding vNM utility of agent i. +As in the proof of Proposition 2, we define agent i’s competitive demand at +the expected congestion profile σ ∈ ∆R(A; n): +D(ui, σ) = arg max +x∈A ui(x, σx ∨ 1) +This demand ignores the effect of agent i’s own presence at post x but correctly +anticipates that the ex post congestion will round up or down the expected one +σa. +Definition 3:The fractional congestion profile σ∗ ∈ ∆R(A; n) is competitive +iff +σ∗ ∈ +� +i∈N +∆R[D(ui, σ∗); 1] +(7) +The pair ({P k}K +k=1, L) of K deterministic assignments P k together with a lot- +tery L over these, implements the competitive profile σ∗ if, first, the expected +congestion over these K assignments is σ∗: EL(sk +a) = σ∗ +a for all a; and second +for all k ∈ [K], all i and all a we have: +i ∈ Sk +a =⇒ a ∈ D(ui, σ∗) and sk +a = ⌊σ∗ +a⌋ or ⌈σ∗ +a⌉ +(8) +Property (7) means that we can achieve the expected congestion profile σ∗ +by assigning randomly each agent (with a well chosen probability distribution) +over the posts in her competitive demand. Property (8) and the choice of the +lottery explain how it will be done: together they provide an ex post test of +fairness of which the next two results explain the power. After these statements +we contrast our new concept with the ex ante envy freeness property familiar +to the random assignment literature. +Lemma 2 In any problem (A, N, ui, i ∈ N) there is a unique competitive +congestion profile σ∗, except possibly at some posts where σ∗ +a ≤ 1. +We can +implement it by (typically several) pairs ({P k}K +k=1, L) as in Definition 3. +Proof: The existence of a competitive profile σ∗ follows by applying the +Kakutani fixed point argument to the convex compact valued correspondence +Γ(σ) = � +i∈N ∆R(D(ui, σ); 1) mapping ∆R(A; n) into itself. To check that the +graph of Γ is closed (implying that Γ is upper-semi-continuous) we take a se- +quence (σt, τ t)∞ +t=1 converging to (σ, τ) and s. t. +τ t = +� +i∈N +τ t +i and τ t +i ∈ ∆R[D(ui, σt); 1] +We can find a subsequence such that all sequences {τt +i} converge, and all sets +D(ui, σt) are constant in t, which proves the claim. +The proof that σ∗ is unique, except perhaps when σ∗ +a ≤ 1 follows exactly +that of statement i) in Proposition 2, so we omit it. +11 + +In the ”left case” the set B is now where both s, s∗ are ≤ 1; welfare equiv- +alence holds: because we postulate that an agent evaluates being assigned to a +post with expected congestion ≤ 1 as being alone there. +We turn to the existence of an implementing pair ({P k}K +k=1, L). Property +(?? means that there is a semi-stochastic (each row sums to 1) matrix π∗ = +[π∗ +ia] ∈ [0, 1]N×A where each column a to σa, and π∗ +ia > 0 can only happen if +a ∈ D(ui, σ∗). Define the set Π of semi-stochastic matrices π s.t. for all a and i +{π∗ +ia > 0 =⇒ a ∈ D(ui, σ∗)} and ⌊σ∗ +a⌋ ≤ +� +i∈N +πia ≤ ⌈σ∗ +a⌉ +This set is a convex compact polytope, non empty as it contains π∗. We claim +that each extreme point of Π is deterministic, i. e., its entries are all integers +or zero: then π represents a deterministic assignment P meeting (8) and π∗ is +a convex combination of such extreme points, which gives the desired collection +P k and lottery L. +We prove the claim by contradiction: pick an extreme point π of Π and +represent π as the bipartite graph G on N ×A containing the edge ia iff πia > 0. +Extract from G the subgraph G0 of its fractional entries ia : 0 < πia < 1 and +let F be the set of posts a s.t. � +i∈N πia is fractional (not an integer or zero). +If F is non empty it contains some post a: at least one edge ia is in G0; then +at least one other edge ib is in G0 (π is semi-stochastic at i); if b ∈ F then we +can add a small ε to πia and take away ε from πib without leaving Π; or vice +versa: this contradicts the extremality of π, so b is not in F after all. But then +there is another edge jb in G0 because � +i∈N πib is an integer, and again there +is some new edge jc in G0; if c = a or c ∈ F we can again perturb a little the +entries of π in two opposite ways and reach a contradiction; this construction +must stop at a new post in F or cycle back to a. The claim is proved and the +proof is complete. ■ +The key properties of the solution concept we just proved always exists +are: first, that any two deterministic assignments P k, P ℓ that can be selected +ex post (both are in the support of L) yield approximately identical utili- +ties; and second, each P k shares (approximately) the properties of determin- +istic competitive assignments in Proposition 2. Agent i’s approximation pa- +rameter is herworst utility loss from one additional unit of congestion δi = +max(a,s)∈A×[n]{ui(a, s) − ui(a, s + 1)}. +Theorem 1 +i) For any set of deterministic assignments {P k}K +k=1 as in Lemma 2 agent i’s +utility at two such assignments P k, P ℓ differ by at most 2δi: for all k, ℓ ∈ [K] +i ∈ Sk +a ∩ Sℓ +b =⇒ |ui(a, sk +a) − ui(a, sℓ +a)| ≤ 2δi +Each deterministic assignment P k +ii) assigns each agent i to a post in her competitive demand D(ui, σ∗), and +implements at each post the competitive congestion σ∗ rounded up or down. +12 + +iii) is 2δi-competitive: +for all a, i: i ∈ Sk +a =⇒ ui(a, sk +a) ≥ ui(x, sk +x ∨ 1) − 2δi for all x +iv) is weakly 2δi-efficient in utility: for any deterministic assignment Q = +(Ta)a∈A there is at least three agents i such that +i ∈ Tb =⇒ ui(a, sk +a) ≥ ui(b, tb) − 2δi +(9) +2δi-Pareto superior to P k. +Proof Statement i) Fix P k, P ℓ, a, b and i ∈ Sk +a ∩ Sℓ +b. Then by (8) a, b are +both in D(ui, σ∗) hence ui(a, σ∗ +a ∨ 1) = ui(b, σ∗ +b ∨ 1). By (8) again and the +definition of δi +|ui(a, sk +a) − ui(a, σ∗ +a ∨ 1)| ≤ δi and |ui(b, sℓ +b) − ui(b, σ∗ +b ∨ 1)| ≤ δi +and the desired inequality follows. +Statement ii) is in Lemma 2. For statement iii) we fix i ∈ Sk +a and x then use +(8) one more time: a ∈ D(ui, σ∗) gives ui(a, σ∗ +a ∨1) ≥ ui(x, σ∗ +x ∨1); |sk +a −σ∗ +a| ≤ 1 +implies |ui(a, sk +a)− ui(a, σ∗ +a ∨1)| ≤ δi and similarly we get |ui(x, sk +x)− ui(a, σ∗ +x ∨ +1)| ≤ δi; finally we combine the three inequalities. +Statement iv) We fix P k and Q = (Tx)x∈A. Let B be the set of posts b such +that sk +b ≤ tb and tb ≥ 1, B is clearly non empty. for any i in Tb let a be the +post assigned to i by P k (a = b is possible): by statement iii) above we have +ui(a, sk +a) ≥ ui(b, sk +b ∨ 1) − 2δi and ui(b, sk +b ∨ 1) ≥ ui(b, tb) by our choice of b. So +Q does not improve P k by more than 2δi for all agents in ∪b∈BTb. +In the following example with at least two posts: N = Sa, N⧹{1, 2, 3} = Ta, +{1, 2, 3} = Tb, the argument above shows that agents 1, 2, 3 meet (9) but other +agents may not because they compare (a, n) to (a, n − 3). We let the reader +check that if ∪b∈BTb is of size 1 or 2 then (9) holds for every agent. ■ +We illustrate fractional competitiveness in Example 4 ((6)) by choosing sim- +ple vNM utilities where one extra unit of congestion costs one unit of utility to +everyone: +uα = uα′ : + + +s +a +b +1 +4 +3 1 +2 +2 +3 +2 1 +2 +3 +2 +1 1 +2 +4 +1 +1 +2 + + +uβ : + + +s +a +b +1 +4 +2 1 +2 +2 +3 +1 1 +2 +3 +2 +1 +2 +4 +1 +− 1 +2 + + +uγ : + + +s +a +b +1 +2 1 +2 +4 +2 +1 1 +2 +3 +3 +1 +2 +2 +4 +− 1 +2 +1 + + +After checking that the top four allocations of each agent match those given +by (6) and that α, α′ prefer (a, 2) to (b, 2), we note that tossing a fair coin +between the two λ-Fair assignments does not work: it maintain the congestion +profile σ = (2, 2) and the competitive demands a for agents α, α′, β and b for +agent γ, so property (?? fails. To make a and b both competitively attractive to +agents α, α′ (D(uα; σ∗) = {a, b}) we must implement a slightly larger congestion +on a than on b adjusted so that uα(a, σa) = uα(b, σb): this gives σ∗ = (2 1 +4, 1 3 +4). +13 + +In addition to the λ-Fair assignments we must put a positive probability on the +λ-unfair assignment Sa = {α, α′, β}, Sb = {γ}, and we get the following pair +({P k}K +k=1, L) +L = 1 +4 + + +1 +0 +1 +0 +1 +0 +0 +1 + + + 3 +8 + + +1 +0 +0 +1 +1 +0 +0 +1 + + + 3 +8 + + +0 +1 +1 +0 +1 +0 +0 +1 + + = +a +b +α +5/8 +3/8 +α′ +5/8 +3/8 +β +1 +0 +γ +0 +1 +(10) +delivering the expected utilities (uα, uα′, uβ, uγ) = (2 9 +16, 2 9 +16, 2 3 +4, 3 1 +4) different +for agents α, α′ than the competitive evaluations uα(a, σ∗ +a) = uα(b, σ∗ +b) = 2 3 +4: +this “error” comes from the competitive approximation ignoring the impact of +my actual post on the realised assignment. +The familiar ex ante concept of envy freeness does not make this error, +but rests on a significantly more complicated argument. Upon learning the pair +({P k}K +k=1, L) in full detail, each agent i must prefer the probability distributions +it implements over her own allocations to those distributions for other agents +(preferences can be in the stochastic dominance or in the expected utility sense). +In the example the fair draw of a λ-Fair assignment resolves the envy between +α, α′, but they both envy β. Lottery (10) does not work either because α, α′ +envy both β and γ at lottery (10).4 The set of envy free and efficient5 fractional +assignments is in fact the segment +x + + +1 +0 +1 +0 +1 +0 +0 +1 + + + y + + +1 +0 +1 +0 +0 +1 +0 +1 + + + y + + +1 +0 +0 +1 +1 +0 +0 +1 + + + y + + +0 +1 +1 +0 +1 +0 +0 +1 + + where x+ 3y = 1 and 0 ≤ x ≤ 2 +11 +quite different from our competitive recommendation. +Two problems already apparent in our simple example are the multiplicity +of envy free fractional assignments, and the increased complexity to compute +them. +3 +the canonical guarantee with weighted con- +gestion +Each agent i brings now a congestion weight wi, a strictly positive real number, +to her assigned post. The total congestion W = � +i∈N wi plays the role of the +number n of agents before: it is a known parameter to every agent. +An assignment is as before a partition P = (Sa)a∈A of N where Sa is the +set of agents assigned to post a and at most m − 1 of these sets can be empty. +4They prefer 1 +4 (a, 3)+ 3 +4 (a, 2) to their own lottery 1 +4(a, 3)+ 3 +8(a, 2)+ 3 +8(b, 2) in the stochastic +dominance sense, but 1 +4(b, 1) + 3 +4(b, 2) only in the expected utility sense. +5Recall that we must exclude trivial assignments with everyone in the same post. +14 + +Agent i’s preferences only depends upon her assigned post a and the congestion +� +i∈Sa wi denoted wSa at that post. They are represented by a utility function +ui(a, x) over the set the set A × [wi, W] of agent i’s feasible allocations; the +function ui is continuous and strictly decreasing in x. +In the Ex Ante state Agent i only knows her own weight wi, the total +congestion W and the set of posts A; in particular she does not know from how +many other agents the congestion W − wi will come. Thus the weighted model +is not a direct generalisation of the anonymous congestion one: here an agent +must consider that the weights of other participants could be (much) bigger or +smaller than her own, a more complicated challenge. +As before each agent selects a profile of caps λi = (λia)a∈A on the congestion +she accepts on every post. The report λia = 0 still allows her to refuse to be +assigned at a. If she potentially accepts post a the cap λia is a real number +in [wi, W]: λia = wi means she only accepts a is she is alone there, while by +reporting λia = W she accepts any congestion at post a. +Definition 4 Fix A, N and (wi)i∈N. Given the report λi = (λia) ∈ [wi, W]A +by each agent i, an assignment P is λ-Fair iff +wSa ≤ λia for all a ∈ A and all i ∈ Sa +To achieve as in Proposition 1 feasibility of decentralised individual reports +the constraints we must impose on λi are more complicated. With the notation +[[z]] for the number of strictly positive coordinates of z ∈ RA ++, we limit agent i’s +choice to the following subset G(A; wi; W) of ({0} ∪ [wi, W])A: +for each a : λia = 0 or wi ≤ λia ≤ W; and +� +a∈A +λia = W + ([[λi]] − 1)wi +(11) +Agent i chooses first the (strict and possibly empty) subset B of the posts +she refuses, then the sum of the positive caps on A⧹B decreases in |B| as +W +(m−1−|B|)wi. The smallest sum is W when she insists on being assigned to +a certain post but cannot limit congestion there; the largest sum is W +(m−1)wi +when she does not rule out any post; becoming more flexible in the sense of +accepting one more post adds wi to the total sum of caps. +For instance our agent improves her options by reporting λia = W, λib = +wi, λix = 0 for x ̸= a, b, rather than λia = W, λix = 0 for x ̸= a, if she prefers +post b alone to post a with full congestion W. +The two definitions of permissible congestion caps in the unweighted and +weighted models give equal veto rights to each participant. Given λi ∈ G(A; wi; W) +the set of allocations acceptable to agent i is the union of intervals [wi, λia] for +each post she does not reject: in view of (11) its size is +� +a:λia>0 +(λia − wi) = W − wi +while the size of the set of agent i’s feasible allocations A×[wi, W] is m(W −wi): +the report λi accepts 1 +m-th of all feasible allocations, just like in our anonymous +congestion model where λi accepts n allocations out of n × m feasible ones. +15 + +Theorem 2 Given A, N and (wi)i∈N let each agent i choose a profile of +caps λi ∈ G(A; wi; W). Then there exists at least one λ-Fair assignment P. +Maximality property: Fix A, W, an agent i∗ with weight wi∗ and an arbitrary +vector of caps λi∗ ∈ G(A; wi∗; W). Then there is a set of agents N⧹{i∗} with +weights wi s. t. � +i∈N⧹{i∗ wi = W − wi∗, and caps λi ∈ G(A; wi; W), such that +in any λ-Fair assignment agent i∗ is at a where the congestion arbitrarily close +to λi∗a. +Proof +Existence. As in Proposition 1 we combine a greedy algorithm and an induc- +tion on the number of agents. +Step A. We can clearly get rid of the “inactive” posts s. t. λia = 0 for all i. +As this causes no confusion we still write A for the set of active posts. We pick +an arbitrary active post a and construct recursively a set S ⊆ A s. t. +∀i ∈ S : λia ≥ wS and ∀j ∈ N⧹S : λja < wS + wj +(12) +Label the agents from 1 to n so that λi ≥ λi+1 for all i ∈ [n − 1]. For any +two disjoints subsets S, T in > 0?? we say that S rejects T if λia < wS + wT for +some i ∈ S; otherwise we say that S accepts T . Note that if all labels in T are +(weakly) smaller than all in S, and S rejects T , then T rejects S as well; and S +accepts T if T accepts S. We construct the desired set S recursively. In each +step we either find S or add one agent to the provisional set of the previous +step. +step 1. If all single agents j ≥ 2 reject {1} then S = {1} meets (12) and we +are done. Otherwise we pick the smallest label ℓ1 accepting {1}, which implies +that {1} accepts {ℓ1} as well, and we form the provisional set S1 = {1, ℓ1}. If +ℓ1 = n we are done by choosing S1 so going into step 2 we have ℓ1 < n. +step k + 1. Because a subset S has not yet been found, in the provisional +set Sk with largest label ℓk < n all agents i accept Sk⧹{i}: λia ≥ wSk for all +i ∈ Sk. Moreover all agents j < ℓk outside Sk have rejected some earlier Sk′, +so they also reject the larger set Sk. +If all agents j > ℓk reject Sk as well we are done by choosing Sk. Otherwise +we pick the smallest label ℓk+1 after ℓk s.t. ℓk+1 accepts Sk: this implies that +Sk accepts ℓk+1 as well, so we set Sk+1 = Sk ∪{ℓk+1} and we have λia ≥ wSk+1 +for all i ∈ Sk+1. We are done if ℓk+1 = n otherwise we go to the next step. +Step B. We assign S found in step A to post a, and consider the residual +problem in �A = A⧹{a} with the agents in N⧹S and total congestion � +W = +W − wS. For each j ∈ �A such that λja > 0 inequality (12) and equation (11) +give +λj � +A > W − wS + ([[λj]] − 2)wj = � +W + ([[�λj]] − 1)wj +where in �λj we drop λja. +For each j ∈ �A such that λja = 0 inequality (12) has no bite and equation +(11) give +λj � +A = W + ([[λi]] − 1)wi > � +W + ([[λj]] − 1)wj = � +W + ([[�λj]] − 1)wj +16 + +So in the reduced problem the induction argument shows that we can assign +λ-Fairly N⧹S to �A. +Maximality We fix W an agent i∗ with weight w∗ and a report λi∗. Call B +the set of posts a s.t. λi∗a > 0 and q is its size so equation (11) is λi∗B = +W + (q − 1)wi. +We add q agents, one written ib for each b ∈ B and their weight is wib = +λib − w∗; the total congestion is then wi∗ + � +b∈B(λi∗b − wi∗) = W as desired. +Next assume each agent ib reports W on b and zero elsewhere. In each λ-Fair +assignment of the problem and the reports we just constructed, each ib is at b +and i∗ can be at any post in B. If we pick an arbitrary post b∗ in B and a small +enough positive ε, we can substract (q − 1)ε from from wib∗ and add ε to each +other wib without changing any report: then the only λ-Fair assignment has i∗ +at b∗ experiencing a congestion very close to λib∗ . ■ +As explained above, agent i’s truthful report λi identifies the subset of her +top W −wi allocations (a well defined unique subset of A×[ww, W]). For exactly +the same reason as when congestion is anonymous, in any equilibrium of the +free mobility game each agent i consumes an allocation in her top-(W − wi) set +(staement ii) in Lemma 1). But we know from [15] that FMEq do not always +exist when congestion is weighted, which makes the FMGame less relevant. +When congestion is weighted, the analysis of λ-Fairness is more difficult +already with two posts: while in the unweighted model there is only one λ-Fair +congestion profile (Example 1 section 2), we have three in the following three +agent example: +a +b +w +λ1 +9 +7 +6 +λ2 +6 +6 +2 +λ3 +8 +4 +2 +=⇒ λ-Fair +a +b +1 +2, 3 , +a +b +1, 3 +2 , +a +b +2, 3 +1 +D(i; σ) = arg max +a {ui(a, σa ∨ wi)} +4 +Concluding comments +take home points +Whether congestion is anonymous or weighted, the canon- +ical guarantee allows each participant to independently veto all but +1 +m of the +set of feasible allocations, where m is the number of posts (Proposition 1 and +Theorem 2). +Competitiveness, when it exists in the deterministic model with anonymous +congestion, identifies a single assignment in terms of congestion and welfare: +combining efficiency with natural ex ante and ex post fairness properties, it is an +appealing normative solution to the congested assignment problem (Proposition +2). +Randomised competitiveness achieves similar results up to an approximation +factor capturing the influence of one unit of congestion on welfare (Theorem 1). +17 + +open questions +When congestion is weighted competitiveness is easy to de- +fine in the deterministic model and its impact is similar to what Proposition 2 +describes. But a plausible definition in the randomised model is elusive. +In the extent literature a non congested random assignment is envy free if, +ex ante, the distribution over my ex post allocations stochastically dominates +that of any other participant: is this property feasible in the randomised model +with congestion (together with efficiency)? The probabilistic serial algorithm +([6]) gives a constructive proof of existence in the absence of congestion but has +no obvious analog when congestion must be taken into account. +In synthetic or empirical data for the deterministic model with anonymous +congestion, how likely is the existence of a competitive assignment? +extensions of the model +The addition of post-specific lower and upper ca- +pacity constraints is natural in most of our motivating examples. Under anony- +mous congestion we generalise the canonical guarantee simply by asking each +profile of caps to respect these constraints. 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Equity, Envy and Efficiency, Journal of Economic Theory, +29(2), 217-244. +20 + diff --git a/_tFLT4oBgHgl3EQfwS-J/content/tmp_files/load_file.txt b/_tFLT4oBgHgl3EQfwS-J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2a29cfa205349fd415c60aea2e1929065ff3588 --- /dev/null +++ b/_tFLT4oBgHgl3EQfwS-J/content/tmp_files/load_file.txt @@ -0,0 +1,629 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf,len=628 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='12163v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='TH] 28 Jan 2023 The congested assignment problem Anna Bogomolnaia∗and Herv´e Moulin† January 18, 2023 Abstract We propose a fair and efficient solution for assigning agents to m posts subject to congestion, when agents care about both their post and its congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Examples include assigning jobs to busy servers, students to crowded schools or crowded classes, commuters to congested routes, workers to crowded office spaces or to team projects etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Congestion is anonymous (it only depends on the number n of agents in a given post).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' A canonical interpretation of ex ante fairness allows each agent to choose m post-specific caps on the congestion they tolerate: these requests are mutually feasible if and only if the sum of the caps is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' For ex post fairness we impose a competitive requirement close to envy freeness: taking the congestion profile as given each agent is assigned to one of her best posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If a competitive assignment exists, it delivers unique congestion and welfare profiles and is also efficient and ex ante fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' In a fractional (randomised or time sharing) version of our model, a unique competitive congestion profile always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' It is approximately implemented by a mixture of ex post deterministic assignments: with an approxination factor equal to the largest utility loss from one more unit of congestion, the latter deliver identical welfare profiles and are weakly efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Our approach to ex ante fairness generalises to the model where each agent’s congestion is weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Now the caps on posts depend only upon own weight and total congestion, not on the number of other agents con- tributing to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Remarkably in both models these caps are feasible if and only if they give to each agent the right to veto all but 1 m of their feasible allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Acknowledgments: critical comments and bibliographic references by Felix Fischer and Ioannis Caragiannis were very helpful;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' we also thank participants in seminars and conferences at the Hebrew University of Jerusalem, Singapore Management University, the Indian Statistical In- stitute Dehli, and the University of Auckland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Keywords: assignment, congestion, guarantee, competitive, free mobil- ity, envy free ∗Adam Smith Business School, University of Glasgow and Centre d’Economie de la Sor- bonne, CNRS, Paris †Adam Smith Business School, University of Glasgow 1 1 Introduction We take a normative viewpoint on the assignment of agents to excludable public items subject to congestion that we call ”posts”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Examples include: how to assign fairly and efficiently students to crowded schools, jobs to busy servers, workers to shared office spaces or team projects, messages or commuters in a congested communication or road network, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='. The vast microeconomic literature on the bilateral assignment of agents to positions ([21], [18], [6],etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='.) includes the special case of the “many agents to one position” problem, particularly rich in applications to school choice ([1]), assignment of students to classes ([7]), workers to employers etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='. These models routinely incorporate upper or lower bounds on the filling of each post: max- imal capacity of a classroom or a school, minimal quotas for some subsets of agents in some posts ([12]) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='. But when these hard constraints do not bite, congestion remains an important component of agents’ preferences: some par- ents will accept more crowded classes if the school’s academic context is better;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' an idle low quality server may be more appealing than a top quality but busy one, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='. We propose what we believe to be the first mechanism design paper incorporating the potentially complicated impact of congestion on welfare into the definition of a fair and efficient assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Starting with the Wardrop equilibrium of transportation models ([10]) a long and distinguished microeconomic literature discusses congestion games in the “free mobility” regime: each player chooses freely her post, and the congestion results of the players’ non cooperative equilibrium behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' A decentralised view of congestion is surely an appropriate model for rush hour traffic, but in many other assignment problems (students to classes, workers to office spaces, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='.) the free mobility game is impractical and/or normatively repugnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' More on free mobility games and their connection to our approach in the subsection below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Our methodology is simple: we apply the two tests of fairness standing out from more than seven decades of formal research on the fair division of private commodities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The first one captures the worst case welfare level (the guarantee) that a participant can secure when, ex ante, she is only aware of her own preferences and the physical description of the assignment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='1 Our second test, competitiveness, is very close to – yet different from – the familiar envy-freeness property ([11], [23]): taking the congestion at each post as a given (independent of my own moves), I am assigned to one of my preferred (post × congestion) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We discuss the guarantee approach in two models: when congestion is anony- mous (aka unweighted), and when each agent’s contribution to congestion is weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We identify in each case a canonical guarantee based on different ex ante information (different interpretations of the worst case) but iplementing effectively the same veto rights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 1The seminal example is Steinhaus’ cake-cutting mechanism, in which each agent can guarantee that her share will be worth at least 1 n-th of her valuation for the entire cake, irrespective of the actions of the other n − 1 participants ([22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 2 We develop the competitive approach when congestion is anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Com- petitiveness may not be feasible in the deterministic model, but if it is it selects an essentially unique efficient assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Competitiveness is always feasible in the randomised (or time sharing) version of the model, where it still delivers an approximately unique recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' overview of our results In section 2 congestion is anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We have m posts and n agents, each one bringing one unit of congestion where they are assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' An agent cares about her allocation (a, sa) (to post a shared with sa − 1 other agents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The canonical guarantee allows her to report for each post a the maximal congestion λia she accepts if assigned to a (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=', λia = 0 if she refuses a altogether).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Provided the sum � a λia is n, any profile λ = (λi) of one such request per agent i is simultaneously satisfied by at least one assignment :Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We call such assignment λ-Fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' To identify the λ- Fair assignments requires much less than a full report of individual preferences: the vector λi is cognitively simple and reveals little of agent i’s full preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' But the set of λ-Fair assignments can still be large and may include assign- ments with sharply different welfare consequences, and/or seriously inefficient ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Both issues are solved by the ex post test of competitiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' An assignment is competitive if I weakly prefer my allocation (a, sa) to any other agent’s allocation (b, sb) as well as to (c, 1) if post c is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Proposition 2 explains why a competitive assignment is a compelling nor- mative solution: if preferences over allocations are strict there is only one such assignment and it is efficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' with general preferences the corresponding con- gestion and welfare profiles are still unique and the latter is efficient (at least weakly and often strongly Pareto optimal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' λ-Fairness is preserved as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' To remedy the possible absence of any competitive assignment in the deter- ministic model, we allow randomised assignments and assume vNM expected utilities (subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Our concept of competitiveness is “ex post”: it se- lects first the unique competitive expected congestion profile σ∗ (Definition 3 and Lemma 2), then implements it by a distribution over finitely many deterministic assignments in which an agent is assigned to a post in her competitive demand at σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' These assignments are approximately (up to twice the worst utility loss of one unit of congestion) identical congestion-wise and welfare-wise: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' In an example where the two concepts make disjoint recommendations we contrast our ex post definition of competitiveness with the familiar ex ante no- tion of envy freeness in the random assignment literature: we believe that our concept is easier to explain and accept in particular because its recommendation is essentially unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' In the weighted congestion model (section 3) each agent i brings wi units of congestion: this could measure the impact of trucks versus cars on traffic, of jobs with different processing times, of students with different claims on the resources of the school, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='. As usual severely complicates the congestion prob- lem: for instance the existence of a free mobility equilibrium is guaranteed under anonymous but not under weighted congestion ([15]), and the computationally 3 challenging knapsack problems arise ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The canonical guarantee still allows each agent to cap the maximal conges- tion that she tolerates on each post, but this time it only depends upon this agent’s own weight wi and the total weight of the other agents, not on the num- ber of agents in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The system of conditions detailed in Theorem 2 ensuring that the individual reports (profiles of caps) are mutually feasible is similar to – though less transparent than – the previous one in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The key fact is that in both models, the canonical guarantee allows each agent to veto exactly all but the top 1 m-th share of her feasible allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The concluding section 4 briefly discusses several possible extensions or vari- ants of our problem that can inspire further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Some relevant literature The vast literature on non cooperative/ free mo- bility congestion games includes the seminal instance of potential games ([19], [16]) and offers powerful existence results of Nash and strong equilibria ([15], [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Free mobility games also play a key role in the hedonic model of coalition formation ([2], [3]) in particular local public goods ([4], [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' They (the conges- tion games) host the seminal discussion of the price of anarchy ([14], [20]) and of algorithmic mechanism design ([17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The two fairness properties organising our discussion are closely related to the Nash equilibria of the corresponding free mobility game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If congestion is anonymous this game always has at least one Nash equilib- rium and each equilibrium implements a λ-Fair assignment, provided each λi vetoes all but agent i’s top n allocations (Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Conversely a free mobility equilibrium is competitive ”up to one unit of congestion” but not necessarily efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The connection is more tenuous if congestion is weighted because the free mobility game may not have any Nash equilibrium ([15]) but the properties just mentioned in Lemma 1 still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The classic combinatorial optimisation knapsack (aka bin packing) problems ([8], see also makespan minimisation as in [17]) discuss like us how to fill bins (posts) with indivisible balls (agents);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' but the balls are just objects in those problems and the concern is about the ”welfare” of the bins (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=', to respect a capacity constraint) while we focus on the welfare of the balls and treat the bins as objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' One exception is [9] where each ball has its own maximal acceptable conges- tion in each bin: this is exactly like the reported profiles of caps λi above, with the difference that the caps are exogenous in [9] so that we may not be able to assign all balls to some bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The paper shows the complexity of computing the maximal number of balls we can assign and evaluates the price of anarchy of the free mobility equilibrium for this metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 2 anonymous congestion We have m posts denoted a, b, · · · , and their set is A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 4 Each agent i in the finite set N of cardinality n must be assigned to some post a in A, which creates one unit of congestion at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' An assignment of agents to posts is a partition P = (Sa)a∈A of N where Sa is the set of agents assigned to post a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' the sets Sa, Sb are mutually disjoint and at most m − 1 of them can be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The corresponding congestion at post a is sa = |Sa| (the cardinality of Sa);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' we call s = (sa)a∈A the congestion profile of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We use the notation [q] for the interval {1, q} in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' An agent’s ordinal preferences depends upon his assigned post a and the congestion sa at that post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Agent i’s (transitive and complete) preferences ⪰i bear on the set A × [n] of feasible allocations (a, sa) and are strictly decreasing in sa (indifferences between allocations at different posts are allowed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' When the assignment is randomised in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2 we assume that the agents’ preferences are represented by vNM expected utility functions ui(a, s) over A × [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='1 the canonical guarantee In the ex ante state agent i only knows the set A and n, the number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' He selects for each post a the largest congestion λia he accepts if assigned to a: λia ∈ [n] ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' So λia = 0 means that i refuses to be assigned to a, λia = 1 that he wants to be alone if assigned there, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='. Definition 1 Given A, N, and a profile of caps λi = (λia)a∈A, the assign- ment P is λ-Fair iff sa ≤ λia for all a ∈ A and i ∈ Sa (1) In what set can we allow each agent to choose his profile of caps so that, when the n agents choose independently in that set, the constraints (1) are jointly compatible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Proposition 1 Given A, N let each agent i ∈ N choose a profile of caps λi = (λia)a∈A in the following simplex ∆N(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' n) with integer coordinates: � a∈A λia = n and λia ≥ 0 for all a ∈ A (2) Then there exists at least one λ-Fair assignment P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The constraints (2) cannot be relaxed: if agents are allowed to report at least one profile λi such that � a∈A λia ≤ n−1, there may exist no λ-Fair assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Proof Step 1 existence Notation: if ⪰ is a preference over the set Z, a top-q set of ⪰ in Z is a subset X of size q in Z such that y ⪰ z for all y ∈ X and z ∈ Z⧹X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We use a simple greedy algorithm and an induction argument on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If λia = 0 for each i we set Sa = ∅ and it remains to prove the claim on the residual problem (A⧹{a}, N, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We clean up in this way all posts that nobody accepts so we can assume from now on that maxi λia ≥ 1 for all a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 5 Pick any post a and order the caps λia, i ∈ N as λ∗1 ≥ λ∗2 ≥ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Write �k for the largest k s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' λ∗k ≥ k (well defined because λ∗1 ≥ 1) and pick a top-�k subset Sa of the ordering {i ⪰a j ⇔ λia ≥ λja} in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Then {λia}i∈Sa = {λ∗k}1≤k≤�k and λja ≤ �k for each j ∈ N⧹Sa by definition of �k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Assigning Sa to a meets inequalities (1) for a, and in the residual problem (A⧹{a}, N⧹Sa, λ) we have � b∈A⧹{a} λib ≥ n − �k = |N⧹Sa| so the induction argument applies: Proposition 1 implies at once that any set of caps s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' � a∈A λia ≥ n for each i ensures the existence of a λ-Fair assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Step 2 maximality If agent i reports caps s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' � a∈A λia ≤ n − 1 and everyone else reports λi as well, a λ-Fair assignment cannot have more than λia agents posted at a, for each a, so that not everyone can be assigned to a post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' ■ Agent i will be assigned to a given post a by reporting λia = n, λib = 0 for b ̸= a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' At the other extreme reporting2 λia = ⌊ n m⌋ or ⌈ n m⌉ for all a guarantees a congestion level uniform (up to one unit) accross all posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If agent i with preferences ⪰i, clueless about other agents’ preferences, max- imises his welfare in the worst case he will select the caps λia so that the union of the intervals {a} × [λia] is a top-n set of ⪰i in A × [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='3 Indeed i cannot rule out the case where everyone else has identical preferences ⪰iand reports identical caps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Then an assignment rule treating equals equally can give agent i anyone of the allocations (a, λia), in particular a worst one in this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Thus i’s cautious report can also be viewed as a truthful message about his preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We now relate Proposition 1 to a classic result in the congestion games literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Given a problem (A, N, ⪰i, i ∈ N) we define the non cooperative free mobility game: each agent chooses a post a in A, then consumes the allocation (a, qa) where qa − 1 is the number of other agents who chose a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Lemma 1 Fix a problem (A, N, ⪰i, i ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' i) The free mobility game has at least one Nash equilibrium (thereafter FMEq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' ii) At a free mobility equilibrium each agent i’s allocation is in a top-n set of ⪰i in A × [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Therefore a FMEq assignment is a λ-Fair assignment for each agent i for at least one cautious/truthful report of this agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Statement i) is the main result in Milchtaich (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Proof of statement ii) Fix a FMEq x = (xi)i∈N ∈ AN and use the nota- tion s(a|x) for the resulting congestion of post a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The equilibrium property is (xi, s(xi|x)) ⪰i (a, s(a|x) + 1) for all a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' combining this with (xi, s(xi|x)) ≻i (xi, s(xi|x) + 1) we see that the only allocations that i could strictly prefer to (xi, s(xi|x)) are in {xi} × [s(xi|x) − 1] and {a} × [s(a|x)] for each a ̸= xi: their number is n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' ■ We can use statement i) to prove Proposition 1: attach to each profile λi in its premises a strict preference ≻i on A × [n] of which the top n allocations cover the union of the intervals {a} × [λia].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Then a free mobility equilibrium at 2Notation: ⌊x⌋ and ⌈x⌉ are the largest (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' smallest) integer bounded above (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' below) by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 3This set may not be unique if ⪰ihas some indifferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 6 those preferences implements a λ-Fair assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' This is of course an overkill as the proof of Milchtaich’s result is much more difficult than the easy induction argument for Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Note that the existence of a FMEq is not guaranteed in the weighted congestion model (section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We illustrate the power of the λ-Fairness constraint (1) and its connection to the FMEq assignments in some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If all agents report the same profile of caps λ0 this is clearly the unique λ- Fair congestion profile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' we start with two more examples where this uniqueness still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Example 1: two posts If m = 2 and there are several λ-Fair assignments they have the same congestion and welfare profiles and are easy to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Fix A = {a, b}, the vectors of caps λi, i ∈ N, and arrange the caps on a and on b decreasingly as λ∗k a , λ∗k b , 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Because λia + λib = n for all i, there is for any k ∈ [n] (at least) one agent s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' λi = (λ∗k a , λ∗n−k+1 b ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' There is clearly a single ℓ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' λ∗ℓ a ≥ ℓ ≥ λ∗ℓ+1 a ⇐⇒ λ∗n−ℓ b ≥ n − ℓ ≥ λ∗n−ℓ+1 b The assignment P is λ-Fair if Sa is a top-ℓ set of the ordering ≥a (by weakly decreasing λia) in N – equivalently Sb is a top-(n − ℓ) set of the ordering ≥b in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If λ∗ℓ a > λ∗ℓ+1 a (⇐⇒ λ∗n−ℓ+1 b < λ∗n−ℓ b ) this defines a unique partition which is the unique λ-Fair assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If λ∗ℓ a = λ∗ℓ+1 a = ℓ we have q ”borderline” agents, q ≥ 2, reporting λi = (λ∗ℓ a , λ∗n−ℓ b ), while pa agents accept a congestion on a higher than ℓ and pb agents accept more than n − ℓ on b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The λ-Fair assignments place any subset of borderline agents in Sa together with the pa “fans” of post a and the rest in Sb with the fans of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Note that if the reports are truthful, the sets of FMEq and of λ-Fair assign- ments coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Example 2: three posts and a “single” λ-Fair assignment The n = 12 agents are split in four types labeled α to δ with three agents in each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Agents of a given type are not necessarily identical but they report the same caps as follows, implying that there is just one λ-Fair congestion and six λ-Fair assignments by permuting the type α agents: n = 12 a b c ααα 4 4 4 βββ 8 2 2 γγγ 2 8 2 δδδ 2 2 8 =⇒ λ-Fair assignment: a b c αβββ αγγγ αδδδ The claim follows by noticing that we can fit λ-Fairly at most 4 agents on each post, therefore the only possible congestion profile is 4 agents per post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Here again free mobility equilibrium and λ-Fairness pick the same assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 7 More generally write cmx(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' λ) for the maximal number of agents we can fit λ-Fairly at post a: if � a∈A cmx(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' λ) = n then all λ-Fair assignments gen- erate the congestion profile s(a) = cmx(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' λ) for all a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The converse property holds as well: all λ-Fair assignments have the same congestion profile only if � a∈A cmx(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' λ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We omit the easy proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Example 3: three posts and multiple λ-Fair assignments n = 15 a b c ααααα 7 7 1 βββββ 7 1 7 γγγγγ 1 7 7 =⇒ a b c ααααα γγγγγ βββββ and a b c ααα ααγγγγγ βββββ are λ-Fair These λ-Fair assignments have very different congestion and welfare implica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Both types of λ-Fair assignments may or may not describe a FMEq, depending on individual preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' For instance an α-agent in the symmetric assignment on the left may prefer (b, 6) to (a, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' So the converse of statement ii) in Lemma 1 does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2 competitiveness Note that λ-Fairness, in addition to the multiplicity just illustrated, can easily allow inefficient assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' For instance in Example 2 the six assignments of the α-agents respect λ-Fairness but if each α-agent prefers a different allocation (x, 4) only one of these is efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Competitive assignments do not have this problem and their recommendation is unambiguous as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='1 deterministic competitiveness We use the notation z ∨ y = max{z, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Definition 2 Fix a problem (A, N, ⪰i, i ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The assignment P is com- petitive (Comp) iff for all a ∈ A and i ∈ Sa : (a, sa) ⪰i (x, sx ∨ 1) for all x ∈ A The difference with the familiar envy freeness property is in the treatment of empty posts: an agent prefering an empty post x to her own post demands to move there;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' in this case our agent cannot assume that the post remains empty when she gets there ((a, 0) is not an allocation) so Comp requires (a, sa) ⪰i (x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' But any assignment where all agents share a single post are automatically envy free, so the concept must be adapted to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' A FMEq assignment is competitive ”up to one unit of congestion on an occupied post”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' conversely at a competitive assignment P the corresponding strategies xi = a ⇐⇒ i ∈ Sa form a FMEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Both claims follow at once from the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 8 Proposition 2: Fix a problem (A, N, (ui)i∈N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' i) All competitive assignments have the same congestion profile (except possibly at some posts occupied by at most one agent), and the same welfare profile across agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' ii) A competitive assignment is weakly efficient, and efficient if preferences are strict and/or if all posts are occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' iii) A competitive assignment is λi-Fair if λi describes a truthful top-n set of ⪰i for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Proof i) Uniqueness At every congestion profile s ∈ ∆N(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' n) (see(2)) we define agent i’s demand D(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' s) = {a|(a, sa ∨ 1) ⪰i (x, sx ∨ 1) for all x}: the assignment P is competitive if and only if a ∈ D(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' s) whenever i ∈ Sa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Fix P = (Sx)x∈A, P ∗ = (S∗ x)x∈A both competitive and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' s ̸= s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Assume first that A∗ = {a ∈ A|s∗ a ∨ 1 > sa ∨ 1} is non empty and note that in A∗ we have s∗ a > sa, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Fixing an agent i we claim that if D(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' s∗) intersects A∗ at a then D(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' s) must be a subset of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If the claim fails there is some b ∈ D(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' s) outside A∗ such that, for all a in A∗: (b, sb ∨ 1) ⪰i (a, sa ∨ 1) ≻i (a, s∗ a) (the strict preference because s∗ a > sa, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' By the choice of b we also have sb ∨ 1 ≥ s∗ b ∨ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' these two facts together give (b, s∗ b ∨1) ≻i (a, s∗ a), and as a was arbitrary in A∗ it follows that D(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' s∗) cannot intersect A∗, contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Now for each i ∈ ∪a∈A∗S∗ a the claim says D(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' s) ⊆ A∗ therefore � a∈A∗ sa ≥ � a∈A∗ s∗ a, contradicting s∗ a > sa in A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We conclude that A∗ must be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' So P, P ∗ must be such that s∗ a ∨ 1 = sa ∨ 1 for all a: this means that s∗ a ̸= sa can only happen when one of s∗ a,sa is 0 and the other is 1, as claimed in statement i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We check now that all agents are indifferent between the two assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Say agent i is in Sa and S∗ b : from sa, s∗ b ≥ 1 and Comp we have (a, sa) ⪰i (b, sb ∨ 1) and (b, s∗ b) ⪰i (a, s∗ a ∨ 1) (3) If sa = s∗ a and sb = s∗ b we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If sa ̸= s∗ a and sb = s∗ b then sa = 1 > 0 = s∗ a and s∗ b ≥ 1 so that (3) gives (a, 1) ⪰i (b, sb) = (b, s∗ b) ⪰i (a, 1) as desired;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' the last subcase sa ̸= s∗ a and sb ̸= s∗ b is just as easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' A simple example with multiple competitive congestion profiles has all agents except 1 and 2 refusing the three posts a, b, c, while 1 and 2 refuse all but a, b, c and they are indifferent between (a, 1), (b, 1) and (c, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' ii) Efficiency Assume, to the contrary, that P = (Sx) is competitive and Pareto inferior to Q = (Tx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Say i assigned to a at P is assigned to b at Q (a, b not necessarily distinct), and suppose that post b is occupied at P: sb ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Then by Comp and the weak Pareto improvement we have (b, Sb) ⪯i (a, Sa) ⪯i (b, Tb) =⇒ sb ≥ tb (4) and sb > tb if agent i improves strictly at Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If all posts are occupied at P then (4) implies s = t and we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='. 9 If instead for some agent i goes from a at P to c at Q and c is empty at P, sc = 0, we have (c, 1) ⪯i (a, Sa) ⪯i (c, Tc) =⇒ tc = 1 and (a, Sa) ≃i (c, 1) (5) If preferences are strict we conclude again that P is Pareto optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Even if they are not we see that i is a weak Pareto optimum (not all agent benefit strictly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The argument above explains how a competitive assignment can be Pareto inferior: in the situation (5) moving i from a to c and changing nothing else is a Pareto improvement to a new assignment �P where agent i’s competitive demand is now a if sa ≥ 2 (because (a, sa − 1) ≻i (c, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If sa = 1 then �P is still competitive and welfare-wise indifferent to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' A simple example of this situation has two posts a, b, all but agent 1 refus- ing a and agent 1 indifferent between (a, 1) and (b, n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' the unique competitive assignment has everyone at b, and moving agent 1 to a is Pareto improving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='But only happens if ⪯i has some indifference and some post is unoccupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Moreover the PO improvement cannot be strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' iii) λi-Fairness This follows from Lemma 1 and the remark that a compet- itive assignment is an FMEq (just before Proposition 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' ■ Dampening the appeal of competitive assignments explained by Proposition 2, it is easy to find examples where none exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Example 4: two posts and no competitive assignment The truthful reports by the four agents allow only two λ-Fair assignments: reports λ: m = 2, n = 4 a b α, α′ 2 2 β 3 1 γ 1 3 =⇒ λ-Fair: a b α, β α′, γ and a b α′, β α, γ (6) If both agents α, α′ prefer (a, 2) to (b, 2) no assignment is competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' A similar situation happens in example 2 when the three α agents have identical preferences over the 3 allocations (x, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2 fractional competitiveness We assume now that the agents’ preferences are represented by vNM expected utility functions ui(a, s) over A × [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' For a finite set Z the notation ∆R(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' w) is the simplex of sum w with non negative real coordinates in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We extend each ui to a function ui(a, x) by linear interpolation between the two rounded up and down values of x, ⌊x⌋ and ⌈x⌉: ui(a, x) = ⌈x⌉ − x ⌈x⌉ − ⌊x⌋ui(a, ⌊x⌋) + x − ⌊x⌋ ⌈x⌉ − ⌊x⌋ui(a, ⌈x⌉) 10 so ui(a, x) is continuous and strictly decreasing in x ∈ [0, n], the interval in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='In what follows x is the expected congestion at some post a when a ran- dom (integer) congestion takes only the values ⌊x⌋ and ⌈x⌉ and ui(a, x) is the corresponding vNM utility of agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' As in the proof of Proposition 2, we define agent i’s competitive demand at the expected congestion profile σ ∈ ∆R(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' n): D(ui, σ) = arg max x∈A ui(x, σx ∨ 1) This demand ignores the effect of agent i’s own presence at post x but correctly anticipates that the ex post congestion will round up or down the expected one σa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Definition 3:The fractional congestion profile σ∗ ∈ ∆R(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' n) is competitive iff σ∗ ∈ � i∈N ∆R[D(ui, σ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 1] (7) The pair ({P k}K k=1, L) of K deterministic assignments P k together with a lot- tery L over these, implements the competitive profile σ∗ if, first, the expected congestion over these K assignments is σ∗: EL(sk a) = σ∗ a for all a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' and second for all k ∈ [K], all i and all a we have: i ∈ Sk a =⇒ a ∈ D(ui, σ∗) and sk a = ⌊σ∗ a⌋ or ⌈σ∗ a⌉ (8) Property (7) means that we can achieve the expected congestion profile σ∗ by assigning randomly each agent (with a well chosen probability distribution) over the posts in her competitive demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Property (8) and the choice of the lottery explain how it will be done: together they provide an ex post test of fairness of which the next two results explain the power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' After these statements we contrast our new concept with the ex ante envy freeness property familiar to the random assignment literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Lemma 2 In any problem (A, N, ui, i ∈ N) there is a unique competitive congestion profile σ∗, except possibly at some posts where σ∗ a ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We can implement it by (typically several) pairs ({P k}K k=1, L) as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Proof: The existence of a competitive profile σ∗ follows by applying the Kakutani fixed point argument to the convex compact valued correspondence Γ(σ) = � i∈N ∆R(D(ui, σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 1) mapping ∆R(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' n) into itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' To check that the graph of Γ is closed (implying that Γ is upper-semi-continuous) we take a se- quence (σt, τ t)∞ t=1 converging to (σ, τ) and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' τ t = � i∈N τ t i and τ t i ∈ ∆R[D(ui, σt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 1] We can find a subsequence such that all sequences {τt i} converge, and all sets D(ui, σt) are constant in t, which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The proof that σ∗ is unique, except perhaps when σ∗ a ≤ 1 follows exactly that of statement i) in Proposition 2, so we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 11 In the ”left case” the set B is now where both s, s∗ are ≤ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' welfare equiv- alence holds: because we postulate that an agent evaluates being assigned to a post with expected congestion ≤ 1 as being alone there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We turn to the existence of an implementing pair ({P k}K k=1, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Property (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' means that there is a semi-stochastic (each row sums to 1) matrix π∗ = [π∗ ia] ∈ [0, 1]N×A where each column a to σa, and π∗ ia > 0 can only happen if a ∈ D(ui, σ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Define the set Π of semi-stochastic matrices π s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' for all a and i {π∗ ia > 0 =⇒ a ∈ D(ui, σ∗)} and ⌊σ∗ a⌋ ≤ � i∈N πia ≤ ⌈σ∗ a⌉ This set is a convex compact polytope, non empty as it contains π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We claim that each extreme point of Π is deterministic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=', its entries are all integers or zero: then π represents a deterministic assignment P meeting (8) and π∗ is a convex combination of such extreme points, which gives the desired collection P k and lottery L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We prove the claim by contradiction: pick an extreme point π of Π and represent π as the bipartite graph G on N ×A containing the edge ia iff πia > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Extract from G the subgraph G0 of its fractional entries ia : 0 < πia < 1 and let F be the set of posts a s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' � i∈N πia is fractional (not an integer or zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If F is non empty it contains some post a: at least one edge ia is in G0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' then at least one other edge ib is in G0 (π is semi-stochastic at i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' if b ∈ F then we can add a small ε to πia and take away ε from πib without leaving Π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' or vice versa: this contradicts the extremality of π, so b is not in F after all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' But then there is another edge jb in G0 because � i∈N πib is an integer, and again there is some new edge jc in G0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' if c = a or c ∈ F we can again perturb a little the entries of π in two opposite ways and reach a contradiction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' this construction must stop at a new post in F or cycle back to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The claim is proved and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' ■ The key properties of the solution concept we just proved always exists are: first, that any two deterministic assignments P k, P ℓ that can be selected ex post (both are in the support of L) yield approximately identical utili- ties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' and second, each P k shares (approximately) the properties of determin- istic competitive assignments in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Agent i’s approximation pa- rameter is herworst utility loss from one additional unit of congestion δi = max(a,s)∈A×[n]{ui(a, s) − ui(a, s + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Theorem 1 i) For any set of deterministic assignments {P k}K k=1 as in Lemma 2 agent i’s utility at two such assignments P k, P ℓ differ by at most 2δi: for all k, ℓ ∈ [K] i ∈ Sk a ∩ Sℓ b =⇒ |ui(a, sk a) − ui(a, sℓ a)| ≤ 2δi Each deterministic assignment P k ii) assigns each agent i to a post in her competitive demand D(ui, σ∗), and implements at each post the competitive congestion σ∗ rounded up or down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 12 iii) is 2δi-competitive: for all a, i: i ∈ Sk a =⇒ ui(a, sk a) ≥ ui(x, sk x ∨ 1) − 2δi for all x iv) is weakly 2δi-efficient in utility: for any deterministic assignment Q = (Ta)a∈A there is at least three agents i such that i ∈ Tb =⇒ ui(a, sk a) ≥ ui(b, tb) − 2δi (9) 2δi-Pareto superior to P k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Proof Statement i) Fix P k, P ℓ, a, b and i ∈ Sk a ∩ Sℓ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Then by (8) a, b are both in D(ui, σ∗) hence ui(a, σ∗ a ∨ 1) = ui(b, σ∗ b ∨ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' By (8) again and the definition of δi |ui(a, sk a) − ui(a, σ∗ a ∨ 1)| ≤ δi and |ui(b, sℓ b) − ui(b, σ∗ b ∨ 1)| ≤ δi and the desired inequality follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Statement ii) is in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' For statement iii) we fix i ∈ Sk a and x then use (8) one more time: a ∈ D(ui, σ∗) gives ui(a, σ∗ a ∨1) ≥ ui(x, σ∗ x ∨1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' |sk a −σ∗ a| ≤ 1 implies |ui(a, sk a)− ui(a, σ∗ a ∨1)| ≤ δi and similarly we get |ui(x, sk x)− ui(a, σ∗ x ∨ 1)| ≤ δi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' finally we combine the three inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Statement iv) We fix P k and Q = (Tx)x∈A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Let B be the set of posts b such that sk b ≤ tb and tb ≥ 1, B is clearly non empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' for any i in Tb let a be the post assigned to i by P k (a = b is possible): by statement iii) above we have ui(a, sk a) ≥ ui(b, sk b ∨ 1) − 2δi and ui(b, sk b ∨ 1) ≥ ui(b, tb) by our choice of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' So Q does not improve P k by more than 2δi for all agents in ∪b∈BTb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' In the following example with at least two posts: N = Sa, N⧹{1, 2, 3} = Ta, {1, 2, 3} = Tb, the argument above shows that agents 1, 2, 3 meet (9) but other agents may not because they compare (a, n) to (a, n − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We let the reader check that if ∪b∈BTb is of size 1 or 2 then (9) holds for every agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' ■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='We illustrate fractional competitiveness in Example 4 ((6)) by choosing sim- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='ple vNM utilities where one extra unit of congestion costs one unit of utility to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='everyone: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='uα = uα′ : ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='1 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='\uf8f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='After checking that the top four allocations of each agent match those given ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='by (6) and that α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' α′ prefer (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 2) to (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' we note that tossing a fair coin between the two λ-Fair assignments does not work: it maintain the congestion profile σ = (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 2) and the competitive demands a for agents α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' β and b for agent γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' so property (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' To make a and b both competitively attractive to agents α, α′ (D(uα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' σ∗) = {a, b}) we must implement a slightly larger congestion on a than on b adjusted so that uα(a, σa) = uα(b, σb): this gives σ∗ = (2 1 4, 1 3 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 13 In addition to the λ-Fair assignments we must put a positive probability on the λ-unfair assignment Sa = {α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' β},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Sb = {γ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' and we get the following pair ({P k}K k=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' L) L = 1 4 \uf8ee \uf8ef\uf8ef\uf8f0 1 0 1 0 1 0 0 1 \uf8f9 \uf8fa\uf8fa\uf8fb + 3 8 \uf8ee \uf8ef\uf8ef\uf8f0 1 0 0 1 1 0 0 1 \uf8f9 \uf8fa\uf8fa\uf8fb + 3 8 \uf8ee \uf8ef\uf8ef\uf8f0 0 1 1 0 1 0 0 1 \uf8f9 \uf8fa\uf8fa\uf8fb = a b α 5/8 3/8 α′ 5/8 3/8 β 1 0 γ 0 1 (10) delivering the expected utilities (uα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' uα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' uβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' uγ) = (2 9 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 2 9 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 2 3 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 3 1 4) different for agents α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' α′ than the competitive evaluations uα(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' σ∗ a) = uα(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' σ∗ b) = 2 3 4: this “error” comes from the competitive approximation ignoring the impact of my actual post on the realised assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The familiar ex ante concept of envy freeness does not make this error, but rests on a significantly more complicated argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Upon learning the pair ({P k}K k=1, L) in full detail, each agent i must prefer the probability distributions it implements over her own allocations to those distributions for other agents (preferences can be in the stochastic dominance or in the expected utility sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' In the example the fair draw of a λ-Fair assignment resolves the envy between α, α′, but they both envy β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Lottery (10) does not work either because α, α′ envy both β and γ at lottery (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='4 The set of envy free and efficient5 fractional assignments is in fact the segment x \uf8ee \uf8ef\uf8ef\uf8f0 1 0 1 0 1 0 0 1 \uf8f9 \uf8fa\uf8fa\uf8fb + y \uf8ee \uf8ef\uf8ef\uf8f0 1 0 1 0 0 1 0 1 \uf8f9 \uf8fa\uf8fa\uf8fb + y \uf8ee \uf8ef\uf8ef\uf8f0 1 0 0 1 1 0 0 1 \uf8f9 \uf8fa\uf8fa\uf8fb + y \uf8ee \uf8ef\uf8ef\uf8f0 0 1 1 0 1 0 0 1 \uf8f9 \uf8fa\uf8fa\uf8fb where x+ 3y = 1 and 0 ≤ x ≤ 2 11 quite different from our competitive recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Two problems already apparent in our simple example are the multiplicity of envy free fractional assignments, and the increased complexity to compute them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 3 the canonical guarantee with weighted con- gestion Each agent i brings now a congestion weight wi, a strictly positive real number, to her assigned post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The total congestion W = � i∈N wi plays the role of the number n of agents before: it is a known parameter to every agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' An assignment is as before a partition P = (Sa)a∈A of N where Sa is the set of agents assigned to post a and at most m − 1 of these sets can be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 4They prefer 1 4 (a, 3)+ 3 4 (a, 2) to their own lottery 1 4(a, 3)+ 3 8(a, 2)+ 3 8(b, 2) in the stochastic dominance sense, but 1 4(b, 1) + 3 4(b, 2) only in the expected utility sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 5Recall that we must exclude trivial assignments with everyone in the same post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 14 Agent i’s preferences only depends upon her assigned post a and the congestion � i∈Sa wi denoted wSa at that post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' They are represented by a utility function ui(a, x) over the set the set A × [wi, W] of agent i’s feasible allocations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' the function ui is continuous and strictly decreasing in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' In the Ex Ante state Agent i only knows her own weight wi, the total congestion W and the set of posts A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' in particular she does not know from how many other agents the congestion W − wi will come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Thus the weighted model is not a direct generalisation of the anonymous congestion one: here an agent must consider that the weights of other participants could be (much) bigger or smaller than her own, a more complicated challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' As before each agent selects a profile of caps λi = (λia)a∈A on the congestion she accepts on every post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The report λia = 0 still allows her to refuse to be assigned at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If she potentially accepts post a the cap λia is a real number in [wi, W]: λia = wi means she only accepts a is she is alone there, while by reporting λia = W she accepts any congestion at post a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Definition 4 Fix A, N and (wi)i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Given the report λi = (λia) ∈ [wi, W]A by each agent i, an assignment P is λ-Fair iff wSa ≤ λia for all a ∈ A and all i ∈ Sa To achieve as in Proposition 1 feasibility of decentralised individual reports the constraints we must impose on λi are more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' With the notation [[z]] for the number of strictly positive coordinates of z ∈ RA +, we limit agent i’s choice to the following subset G(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' wi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' W) of ({0} ∪ [wi, W])A: for each a : λia = 0 or wi ≤ λia ≤ W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' and � a∈A λia = W + ([[λi]] − 1)wi (11) Agent i chooses first the (strict and possibly empty) subset B of the posts she refuses, then the sum of the positive caps on A⧹B decreases in |B| as W +(m−1−|B|)wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The smallest sum is W when she insists on being assigned to a certain post but cannot limit congestion there;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' the largest sum is W +(m−1)wi when she does not rule out any post;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' becoming more flexible in the sense of accepting one more post adds wi to the total sum of caps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' For instance our agent improves her options by reporting λia = W, λib = wi, λix = 0 for x ̸= a, b, rather than λia = W, λix = 0 for x ̸= a, if she prefers post b alone to post a with full congestion W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The two definitions of permissible congestion caps in the unweighted and weighted models give equal veto rights to each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Given λi ∈ G(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' wi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' W) the set of allocations acceptable to agent i is the union of intervals [wi, λia] for each post she does not reject: in view of (11) its size is � a:λia>0 (λia − wi) = W − wi while the size of the set of agent i’s feasible allocations A×[wi, W] is m(W −wi): the report λi accepts 1 m-th of all feasible allocations, just like in our anonymous congestion model where λi accepts n allocations out of n × m feasible ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 15 Theorem 2 Given A, N and (wi)i∈N let each agent i choose a profile of caps λi ∈ G(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' wi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Then there exists at least one λ-Fair assignment P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Maximality property: Fix A, W, an agent i∗ with weight wi∗ and an arbitrary vector of caps λi∗ ∈ G(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' wi∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Then there is a set of agents N⧹{i∗} with weights wi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' � i∈N⧹{i∗ wi = W − wi∗, and caps λi ∈ G(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' wi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' W), such that in any λ-Fair assignment agent i∗ is at a where the congestion arbitrarily close to λi∗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Proof Existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' As in Proposition 1 we combine a greedy algorithm and an induc- tion on the number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Step A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We can clearly get rid of the “inactive” posts s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' λia = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' As this causes no confusion we still write A for the set of active posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We pick an arbitrary active post a and construct recursively a set S ⊆ A s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' ∀i ∈ S : λia ≥ wS and ∀j ∈ N⧹S : λja < wS + wj (12) Label the agents from 1 to n so that λi ≥ λi+1 for all i ∈ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' For any two disjoints subsets S, T in > 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' we say that S rejects T if λia < wS + wT for some i ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' otherwise we say that S accepts T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Note that if all labels in T are (weakly) smaller than all in S, and S rejects T , then T rejects S as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' and S accepts T if T accepts S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We construct the desired set S recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' In each step we either find S or add one agent to the provisional set of the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If all single agents j ≥ 2 reject {1} then S = {1} meets (12) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Otherwise we pick the smallest label ℓ1 accepting {1}, which implies that {1} accepts {ℓ1} as well, and we form the provisional set S1 = {1, ℓ1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If ℓ1 = n we are done by choosing S1 so going into step 2 we have ℓ1 < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' step k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Because a subset S has not yet been found, in the provisional set Sk with largest label ℓk < n all agents i accept Sk⧹{i}: λia ≥ wSk for all i ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Moreover all agents j < ℓk outside Sk have rejected some earlier Sk′, so they also reject the larger set Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If all agents j > ℓk reject Sk as well we are done by choosing Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Otherwise we pick the smallest label ℓk+1 after ℓk s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' ℓk+1 accepts Sk: this implies that Sk accepts ℓk+1 as well, so we set Sk+1 = Sk ∪{ℓk+1} and we have λia ≥ wSk+1 for all i ∈ Sk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We are done if ℓk+1 = n otherwise we go to the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Step B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We assign S found in step A to post a, and consider the residual problem in �A = A⧹{a} with the agents in N⧹S and total congestion � W = W − wS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' For each j ∈ �A such that λja > 0 inequality (12) and equation (11) give λj � A > W − wS + ([[λj]] − 2)wj = � W + ([[�λj]] − 1)wj where in �λj we drop λja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' For each j ∈ �A such that λja = 0 inequality (12) has no bite and equation (11) give λj � A = W + ([[λi]] − 1)wi > � W + ([[λj]] − 1)wj = � W + ([[�λj]] − 1)wj 16 So in the reduced problem the induction argument shows that we can assign λ-Fairly N⧹S to �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Maximality We fix W an agent i∗ with weight w∗ and a report λi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Call B the set of posts a s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' λi∗a > 0 and q is its size so equation (11) is λi∗B = W + (q − 1)wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' We add q agents, one written ib for each b ∈ B and their weight is wib = λib − w∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' the total congestion is then wi∗ + � b∈B(λi∗b − wi∗) = W as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Next assume each agent ib reports W on b and zero elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' In each λ-Fair assignment of the problem and the reports we just constructed, each ib is at b and i∗ can be at any post in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' If we pick an arbitrary post b∗ in B and a small enough positive ε, we can substract (q − 1)ε from from wib∗ and add ε to each other wib without changing any report: then the only λ-Fair assignment has i∗ at b∗ experiencing a congestion very close to λib∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' ■ As explained above, agent i’s truthful report λi identifies the subset of her top W −wi allocations (a well defined unique subset of A×[ww, W]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' For exactly the same reason as when congestion is anonymous, in any equilibrium of the free mobility game each agent i consumes an allocation in her top-(W − wi) set (staement ii) in Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' But we know from [15] that FMEq do not always exist when congestion is weighted, which makes the FMGame less relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' When congestion is weighted, the analysis of λ-Fairness is more difficult already with two posts: while in the unweighted model there is only one λ-Fair congestion profile (Example 1 section 2), we have three in the following three agent example: a b w λ1 9 7 6 λ2 6 6 2 λ3 8 4 2 =⇒ λ-Fair a b 1 2, 3 , a b 1, 3 2 , a b 2, 3 1 D(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' σ) = arg max a {ui(a, σa ∨ wi)} 4 Concluding comments take home points Whether congestion is anonymous or weighted, the canon- ical guarantee allows each participant to independently veto all but 1 m of the set of feasible allocations, where m is the number of posts (Proposition 1 and Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Competitiveness, when it exists in the deterministic model with anonymous congestion, identifies a single assignment in terms of congestion and welfare: combining efficiency with natural ex ante and ex post fairness properties, it is an appealing normative solution to the congested assignment problem (Proposition 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Randomised competitiveness achieves similar results up to an approximation factor capturing the influence of one unit of congestion on welfare (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' 17 open questions When congestion is weighted competitiveness is easy to de- fine in the deterministic model and its impact is similar to what Proposition 2 describes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' But a plausible definition in the randomised model is elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' In the extent literature a non congested random assignment is envy free if, ex ante, the distribution over my ex post allocations stochastically dominates that of any other participant: is this property feasible in the randomised model with congestion (together with efficiency)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' The probabilistic serial algorithm ([6]) gives a constructive proof of existence in the absence of congestion but has no obvious analog when congestion must be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' In synthetic or empirical data for the deterministic model with anonymous congestion, how likely is the existence of a competitive assignment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' extensions of the model The addition of post-specific lower and upper ca- pacity constraints is natural in most of our motivating examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Under anony- mous congestion we generalise the canonical guarantee simply by asking each profile of caps to respect these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' But it is not clear how to adapt competitiveness even in the deterministic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' More general preferences about the congestion level, in particular single- peaked, are common in the literature on the formation of coalitions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} +page_content=' Initiating a normative discussion of congested assignment in this entirely real- istic context is a very tall challenge that will hopefully inspire further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfwS-J/content/2301.12163v1.pdf'} 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Unsupervised Volumetric Animation (UVA). Selected animation results for faces and bodies. Given a driving image sequence +and a source image (not shown), UVA renders realistic animations and simultaneously generates novel views of the animated object. With +our reconstruction loss, our method also generates high-fidelity depth and normals, and identifies semantically meaningful object parts. +Abstract +We propose a novel approach for unsupervised 3D +animation of non-rigid deformable objects. +Our method +learns the 3D structure and dynamics of objects solely +from single-view RGB videos, and can decompose them +into semantically meaningful parts that can be tracked and +animated. Using a 3D autodecoder framework, paired with +a keypoint estimator via a differentiable PnP algorithm, +our model learns the underlying object geometry and parts +decomposition in an entirely unsupervised manner. This al- +lows it to perform 3D segmentation, 3D keypoint estimation, +novel view synthesis, and animation. We primarily evaluate +the framework on two video datasets: VoxCeleb 2562 and +TEDXPeople 2562. In addition, on the Cats 2562 image +dataset, we show it even learns compelling 3D geometry +from still images. Finally, we show our model can obtain +animatable 3D objects from a single or few images 1. +1Code and visual results available on our project website. +∗ Work done while interning at Snap. +1. Introduction +The ability to realistically animate a dynamic object +seen in a single image enables compelling creative tasks. +Such applications range from tractable and cost-effective +approaches to visual effects for cinema and television, to +more lightweight consumer applications (e.g., enabling ar- +bitrary users to create “performances” by famous modern or +historical figures). However, this requires understanding the +object’s structure and motion patterns from a single static +depiction. Efforts in this field are primarily divided into +two approaches: those that outsource this understanding to +existing, off-the-shelf models specific to an object category +that capture its particular factors of variation; and those that +learn the object structure from the raw training data itself. +The former group employs supervision, and thus requires +knowledge about the animated object (e.g., the plausible +range of shapes and motions of human faces or bodies). +The latter group is unsupervised, providing the flexibility +needed for a wider range of arbitrary object categories. +Significant progress has been made recently in the do- +1 +arXiv:2301.11326v1 [cs.CV] 26 Jan 2023 + +main of unsupervised image animation. Methods in this +category typically learn a motion model based on object +parts and the corresponding transformations applied to +them. Initially, such transformations were modeled using +a simple set of sparse keypoints. Further works improved +the motion representation [52, 55], learned latent motion +dictionaries [64], kinematic chains [59] or used thin-plate +spline transformations [81]. However, broadly speaking, +all such works propose 2D motion representations, warping +the pixels or features of the input image such that they +correspond to the pose of a given driving image. +As +such, prior unsupervised animation methods offer means +to perform 2D animation only, and are inherently limited +in modeling complex, 3D effects, such as occlusions, +viewpoint changes, and extreme rotations, which can only +be explained and addressed appropriately when considering +the 3D nature of the observed objects. +Our work fundamentally differs from prior 2D works in +that it is the first to explore unsupervised image animation in +3D. This setting is substantially more challenging compared +to classical 2D animation for several reasons. First, as the +predicted regions or parts now exist in a 3D space, it is quite +challenging to identify and plausibly control them from +only 2D videos without extra supervision. +Second, this +challenge is further compounded by the need to properly +model the distribution of the camera in 3D, which is a +problem in its own right [40], with multiple 3D generators +resorting to existing pose predictors to facilitate the learning +of the underlying 3D geometry [5,58]. Finally, in 3D space, +there exists no obvious and tractable counterpart for the bias +of 2D CNNs, which are essential for unsupervised keypoint +detection frameworks for 2D images [53]. +We offer a solution to these challenges. Our framework +maps an embedding of each object to a canonical volumetric +representation, parameterized with a voxel grid, containing +volumetric density and appearance. To allow for non-rigid +deformations of the canonical object representation, we +assume the object consists of a certain number of rigid parts +which are softly assigned to each of the points in the canon- +ical volume. A procedure based on linear blend skinning is +employed to produce the deformed volume according to the +pose of each part. Rather than directly estimating the poses, +we introduce a set of learnable 3D canonical keypoints for +each part, and leverage the 2D inductive bias of 2D CNNs +to predict a set of corresponding 2D keypoints in the current +frame. We propose the use of a differentiable Perspective- +n-Point (PnP) algorithm to estimate the corresponding pose, +explicitly linking 2D observations to our 3D representation. +This framework allows us to propagate the knowledge from +2D images to our 3D representation, thereby learning rich +and detailed geometry for diverse object categories using +a photometric reconstruction loss as our driving objective. +The parts are learned in an unsupervised manner, yet they +converge to meaningful volumetric object constituents. For +example, for faces, they correspond to the jaw, hair, neck, +and the left and right eyes and cheeks. For bodies, the same +approach learns parts to represent the torso, head, and each +hand. Examples of these parts are given in Fig. 1. +To simplify the optimization, we introduce a two-stage +strategy, in which we start by learning a single part such that +the overall geometry is learned, and proceed by allowing the +model to discover the remaining parts so that animation is +possible. When the object is represented with a single part, +the model can perform 3D reconstruction and novel view +synthesis. When more parts are used, our method allows us +to not only identify meaningful object parts, but to perform +non-rigid animation and novel view synthesis at the same +time. Examples of images animated using our Unsupervised +Volumetric Animation (UVA) are given in Fig. 1. +We train our framework on three diverse datasets con- +taining images or videos of various objects. We first show +that our method learns meaningful 3D geometry when +trained on still images of cat faces [79]. +We then train +our method on the VoxCeleb [38] and TEDXPeople [17] +video datasets to evaluate 3D animation. Since our method +is the first to consider unsupervised 3D animation, we +further introduce evaluation metrics assessing novel view +synthesis and animation quality when only single-view data +is available. +2. Related work +3D-aware image and video synthesis experienced sub- +stantial progress over the last two years. Early works [40, +41, 51] used Neural Radiance Fields (NeRFs) [37] as a 3D +representation to synthesize simple objects and often con- +sidered synthetic datasets [51, 70]. They spurred a line of +works that scaled the generator and increased its efficiency +to attain high-resolution 3D synthesis [5, 18, 45, 58, 72]. +These works rely on different types of volumetric repre- +sentations such as a coordinate-MLP [6], voxel-grids [39], +tri-planes [5, 58], generative manifolds [10], multi-plane +representations [82], and signed distance functions [42]. +Further works combined implicit video synthesis [57, 76] +techniques with that of volumetric rendering [18] to gener- +ate 3D-aware videos [1]. A common requirement of these +methods is access to the ground truth camera distribution +(e.g., [6, 18, 45, 51, 80]) or even the known camera poses +for each training image [5, 10, 58, 82]. This gives a strong +inductive bias towards recovering the proper 3D geome- +try [58,82]. Our work shows that it is possible to learn rich +geometry and object parts decomposition in a completely +unsupervised manner in a non-adversarial framework. +Unsupervised 3D reconstruction. +Unsupervised recon- +struction of 3D objects from image or video collections +is a long standing problem [23, 32, 67, 68, 73–75]. Initial +attempts [23] utilize image collections and try to predict +2 + +camera, mesh displacement parameters and texture, render +the mesh and use reconstruction as the main guiding signal. +Later work [68] proposes to further improve this pipeline by +incorporating additional knowledge about object symmetry. +However, those works did not model deformations, which +was addressed later in works [32, 67, 73–75] that propose +to train on video datasets. +Most of them [32, 73–75] +optimize the object parameters for each frame, and thus +can not be trained on a large dataset of videos. +On the +contrary, Dove [67] infers the articulation parameters from +individual frames, which allows training on large video +dataset. However, Dove [67] is a mesh based method, thus +rendering quality is limited. Moreover, all of these methods +utilize additional annotations such as template shapes [32], +camera poses [75], 2D keypoints [75], optical flow [73–75] +or ground truth object masks [32,67,73–75]. Instead, in our +method everything, including object masks, was obtained in +an purely unsupervised way from video data only. +Supervised image animation requires an off-the-shelf +keypoint predictor [62, 77, 78] or a 3D morphable model +(3DMM) estimator [14,15] run through the training dataset +prior to training. To train such an estimator, one needs to +have large amounts of labeled data for the task at a hand. +Supervised animation works are typically designed for only +one object category, such as bodies [33, 62] or faces [78]. +Among them, some support only a single object identity [4], +others single- or few-shot cases [46,62]. +Thanks to significant advances in neural rendering and +3D-aware synthesis, several works extended supervised +animation to the 3D domain. +Initially, a dataset with +multiview videos was required to train animatable radiance +fields [43]. +Later, HumanNeRF [65] and NeuMan [21] +showed the feasibility of leveraging only a monocular video +of the same subject. However, these models require fitting +of a 3D model of human bodies to every frame of a +video. With some exceptions [46], such methods do not +support multiple identities with the same framework. In +contrast, our method features a space in which all objects +are represented in their canonical, animation-ready form. +Unsupervised image animation is the most related group +of works to ours. These works do not require supervision +beyond photometric reconstruction loss and, hence, support +a variety of object categories with one framework [36, 52, +53,55]. A key focus area of such works is to design appro- +priate motion representations for animation [35,52,55,66]. +A number of improved representations have been proposed, +such as those setting additional constraints on a kinematic +tree [59], and thin-plate spline motion modelling [81]. A +further work, titled Latent Image Animator [64], learned +a latent space for possible motions. +Interestingly, a di- +rection in the latent space is found to be responsible for +generating novel views of the same subject. As we confirm +experimentally, similarly to 2D image generators [24], the +direction cannot be reliably used to synthesize the novel +views. +Several recent works [11, 63], propose to use +mixed schemes where pose of the object is supervised and +expression is learned, such approaches works well for faces +however did not generalize to other categories. +3. Method +This section presents our method for unsupervised 3D +animation of non-rigid deformable objects. +Our model +trains on a set of images {Fi, αi}Nf +i=1, where Fi ∈ RH×W ×3 +is an image frame, αi ∈ N is an object identifier2, and Nf +is the number of frames in a video. The primary training +objective of our framework is the reconstruction task. Given +a frame Fi with identity αi, we reconstruct it using four core +components (see Fig. 2). First, Canonical Voxel Generator +G maps a learnable identity-specific embedding e ∈ RNe +to an object’s volumetric representation in the canonical +pose, parametrized as a voxel grid. Following [52, 53, 55], +we assume that each non-rigid object can be represented +as a set of moving rigid parts. +In this way, our voxel +generator segments the volume and assigns each 3D point +to its corresponding object’s part (Sec. 3.1). +Next, we +define 2D keypoint predictor C with and the differentiable +PnP [28] algorithm to estimate each part’s pose (position +and orientation) in a given RGB frame Fi (Sec. 3.2). +Subsequently, we employ a method based on linear blend +skinning [29] to map the canonical object volume into a +deformed one which represents the object in the current +frame (Sec. 3.3). Finally, we use volumetric rendering [37] +to render the colors to the image space (Sec. 3.4). +3.1. Canonical Voxel Generator +We use a voxel grid V to parametrize the volume since +we found it to provide the best trade-off between generation +efficiency, expressivity and rendering speed. +Given an +object’s embedding e ∈ RNe, we use Canonical Voxel +Generator G to produce a volume cube of size S: +G(e) = V = +� +V Density∥V RGB∥V LBS� +, +(1) +where V Density ∈ RS3 is the object’s (discretized) density +field in the canonical pose and V RGB ∈ RS3×3 is its +(discretized) RGB radiance field. To animate an object, we +assume that it can be modeled as a set of rigid moving parts +p ∈ {1, 2, ..., Np} [52,53,55], so we use V LBS ∈ RS3×Np +to model a soft assignment of each point of the volume to +one of the Np parts. Notably, we do not use any encoder to +produce identity embeddings e and instead optimize them +directly during training [3]. Examples of canonical density, +parts, and rendered canonical radiance are shown in Fig. 2. +2We assume that we know which object instance appears in a video. In +practice, this assumption is easily satisfied by assigning the same identity +to all the frames of a given video. +3 + +Canonical Parts, +Density & 3D Keypoints +Deformed Density & Parts +ACIXicbVDLSsNAFJ3UV42vqEs3g0VxISEJonVXcONKqtgHNCFMptN26OTBzEQob/ixl9x40KR7sSfcZJmoa0XBs7jXu7cEySMCmlZX1plZXVtfaO6qW9t7+zuGfsHbRGnHJMWjlnMuwEShNGItCSVjHQTlAYMNIJxje53kiXNA4epSThHghGkZ0QDGSvKN+il8O1zKH3bdfWcODlxCgKhaZqlnN35yTS3CuC6vlGzTKsouAzsEtRAWU3fmLn9GKchiSRmSIiebSXSyxCXFDMy1d1UkAThMRqSnoIRConwsuLCKTxRSh8OYq5eJGh/p7IUCjEJAxUZ4jkSCx6ufif10vloO5lNEpSI8XzRIGZQxzOCfcoJlmyiAMKcqr9CPEIcYalC1VUI9uLJy6DtmPal6dxf1BrXZRxVcASOwRmwRVogFvQBC2AwTN4Be/gQ3vR3rRPbTZvrWjlzCH4U9r3Dz2wnhk=R1, t1 +R2, t2 +... +RNp, tNp +Volumetric +Rendering +2D keypoint +predictor +Reconstruction supervision +Driving +Rendered +Canonical Voxel Generator +Volumetric +Skinning +2D Keypoints +Differentiable +PnP +Poses +AB+XicbVDLSsNAFL2pr1pfUZduBovgqiRFfOyKbly4qGgf0MYymU7boZNJmJkUSsifuHGhiFv/xJ1/46TNQlsPDBzOuZd75vgRZ0o7zrdVWFldW98obpa2tnd29+z9g6YKY0log4Q8lG0fK8qZoA3NKftSFIc+Jy2/PFN5rcmVCoWikc9jagX4KFgA0awNlLPtptPSTfAeiSD5O76IU17dtmpODOgZeLmpAw56j37q9sPSRxQoQnHSnVcJ9JegqVmhNO01I0VjTAZ4yHtGCpwQJWXzJKn6MQofTQIpXlCo5n6eyPBgVLTwDeTWUi16GXif14n1oNL2EijUVZH5oEHOkQ5TVgPpMUqL51BMJDNZERlhiYk2ZVMCe7il5dJs1pxzyvV+7Ny7SqvowhHcAyn4MIF1OAW6tAhN4hld4sxLrxXq3PuajBSvfOYQ/sD5/ALsCk7M= +V LBS +ACB3icbVDLSsNAFJ34rPUVdSnIYBFcSEmq+NgV7EJwU8E+oE3DZDpth04mYWYihJCdG3/FjQtF3PoL7vwbJ20W2nrgwuGce7n3Hi9kVCrL+jYWFpeWV1YLa8X1jc2tbXNntymDSGDSwAELRNtDkjDKSUNRxUg7FAT5HiMtb3yd+a0HIiQN+L2KQ+L4aMjpgGKktOSaB81e0vWRGgk/qREuqYrT9ATe9pLTWuqGrlmytYEcJ7YOSmBHX/Or2Axz5hCvMkJQd2wqVkyChKGYkLXYjSUKEx2hIOpy5BPpJM/UniklT4cBEIXV3Ci/p5IkC9l7Hu6MztZznqZ+J/XidTg0koDyNFOJ4uGkQMqgBmocA+FQrFmuCsKD6VohHSCsdHRFHYI9+/I8aVbK9nm5cndWql7lcRTAPjgEx8AGF6AKbkAdNAGj+AZvI348l4Md6Nj2nrgpHP7IE/MD5/ACF6mXE= +V Density, K3D +p +AB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9ktUvVW0IPgpYL9gHYt2TbxmaTJckKZel/8OJBEa/+H2/+G9N2D9r6YODx3gwz84KYM21c9vJrayurW/kNwtb2zu7e8X9g6aWiSK0QSXqh1gTkTtGY4bQdK4qjgNWMLqa+q0nqjST4t6MY+pHeCBYyAg2VmrePqSV60mvWHL7gxomXgZKUGeq/41e1LkRUGMKx1h3PjY2fYmUY4XRS6CaxpiM8IB2LBU4otpPZ9dO0IlV+iUypYwaKb+nkhxpPU4CmxnhM1QL3pT8T+vk5jwk+ZiBNDBZkvChOjET1GfKUoMH1uCiWL2VkSGWGFibEAFG4K3+PIyaVbKXrVcuTsr1S6zOPJwBMdwCh6cQw1uoA4NIPAIz/AKb450Xpx352PemnOymUP4A+fzBxiWjs= +K2D +AB8XicbVDLSgMxFL3js9ZX1aWbYBFclZki6rLoQpcV7APbUjLpnTY0kxmSjFCG/oUbF4q49W/c+Tem01lo64HA4Zx7ybnHjwXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1FGiGDZYJCLV9qlGwSU2DcC27FCGvoCW/74Zua3nlBpHskHM4mxF9Kh5AFn1FjpsRtSM9JBejvtl8puxc1AlomXkzLkqPdLX91BxJIQpWGCat3x3Nj0UqoMZwKnxW6iMaZsTIfYsVTSEHUvzRJPyalVBiSIlH3SkEz9vZHSUOtJ6NvJLOGiNxP/8zqJCa56KZdxYlCy+UdBIoiJyOx8MuAKmRETSyhT3GYlbEQVZcaWVLQleIsnL5NmteJdVKr35+XadV5HAY7hBM7Ag0uowR3UoQEMJDzDK7w52nlx3p2P+eiKk+8cwR84nz/PFpEDG +Embedding space +Figure 2. Unsupervised Volumetric Animation consists of a canonical voxel generator G mapping a point in the latent space to the +canonical density, radiance and canonical parts. In the embedding space we show canonical shapes rendered under identity camera (faces +have the same pose with mouth open). For each part, a set of canonical 3D keypoints K3D is learnt during training. The 2D keypoint +predictor uses a driving image to predict a set of 2D keypoints K2D, corresponding to K3D. The differentiable PnP algorithm is used to +predict the pose of each part. Canonical density, radiance, poses and parts are then used to compute the deformed density and radiance via +volumetric skinning. We then volumetrically render the deformed radiance to produce the rendered image. Note, that our approach does +not use any knowledge about the object being animated, and is supervised using the reconstruction loss. Zoom-in for greater detail. +3.2. Unsupervised Pose Estimation +As described in Sec. 3.1, we assume that an object +movement can be factorized into a set of rigid movements +of each individual object’s part p. +However, detecting +3D part poses, especially in an unsupervised way, is a +difficult task. MRAA [55] shows that estimating 2D parts +and their poses in an unsupervised fashion is an under- +constrained problem, which requires specialized inductive +biases to guide the pose estimation towards the proper +solution. We incorporate such an inductive bias by framing +pose prediction as a 2D landmark detection problem which +CNNs can solve proficiently due to their natural ability to +detect local patterns [20]. +To lift this 2D bias into 3D, we estimate the poses of +3D parts by learning a set of 3D keypoints in the canonical +space and detecting their 2D projections in the current frame +using a 2D CNN. We then use a differentiable Perspective- +n-Point (PnP) formulation to recover the pose of each part +since we know its corresponding 2D and 3D keypoints. +More formally, PnP is a problem where, given a set of +the 3D keypoints K3D ∈ RNk×3, a set of corresponding +2D projections K2D ∈ RNk×2 and the camera intrinsics +parameters, one need to find a camera pose T = [R, t] ∈ +R3×4, such that K3D project to K2D when viewed from +this pose. Note that, while T represents the pose of the +camera with respect to the part, in our framework we +consider the camera extrinsics to be constant and equal to +the identity matrix, i.e. +a part moves while the camera +remains fixed. Recovering a part’s pose with respect to the +camera is performed by inverting the estimated pose matrix +Tp = [Rp, tp] = [R−1, −R−1t]. +To implement this idea, we introduce Nk learnable +canonical 3D keypoints K3D +p +for each part, totaling Nk × +Np. These 3D keypoints are shared among all the objects +in a dataset, which are directly optimized with the rest of +the model’s parameters. Then, we define a 2D keypoints +prediction network C, which takes frame Fi as input and +outputs Nk 2D keypoints K2D +p +for each part p, where each +2D keypoint corresponds to its respective 3D keypoint. The +pose of part p is thus recovered as: +T −1 +p += PnP +� +K2D +p , K3D +p +� += PnP +� +C(Fi), K3D +p +� +. +(2) +Crucially, in this formulation K3D +p +are shared for all the +objects in the dataset, thus all objects will share the same +canonical space for poses. +This property is essential +for performing cross-subject animations, where poses are +estimated on frames depicting a different identity. +We used the EPnP [28] implementation from Py- +torch3D [44], since we found it to be significantly faster +and more stable than the methods based on declarative +layers [8,16]. +3.3. Volumetric Skinning +In this section, we describe the procedure to deform +the canonical volumetric object representation into its rep- +resentation in the driving pose. +The deformation can +4 + +be completely described by establishing correspondences +between each point xd in the deformed space and points xc +in the canonical space. We establish such correspondence +through Linear Blend Skinning (LBS) [29]: +xd = +Np +� +p=1 +wc +p(xc) (Rpxc + tp) , +(3) +where wc +p(x) is a weight assigned to each part p. Intuitively, +LBS weights segment the object into different parts. As an +example, a point with LBS weight equal to 1.0 for the left +hand, will always move according to the transformation for +the left hand. Unfortunately, during volumetric rendering +we typically need to query canonical points using points in +the deformed space, requiring solving Eq. (3) for xc. This +procedure is prohibitively expensive [30], so we rely on +the approximate solution introduced in HumanNeRF [65], +which defines inverse LBS weights wd +p such that: +xc = +Np +� +p=1 +wp(xd) +� +R−1 +p xd − R−1 +p tp +� +, +(4) +where weights wd +p are defined as follows: +wp(xd) = +wc +p(R−1 +p xd − R−1 +p tp) +�Np +p=1 wcp(R−1 +p xd − R−1 +p tp) +. +(5) +This approximation has an intuitive explanation, i.e. given +the deformed point, we project it using the inverse Tp to the +canonical pose and check if it corresponds to the part p in +canonical pose. It is easy to see that if each point has a strict +assignment to a single part and there is no self-penetration +in the deformed space, the approximation is exact. In our +work, we parametrize wc +p as the channel-wise softmax of +V LBS. Examples of the parts are given in Figs. 1 & 2. +3.4. Volumetric Rendering +We render the deformed object using differentiable +volumetric rendering [37]. +Given camera intrinsics and +extrinsics, we cast a ray r through each pixel in the image +plane and compute the color c associated to each ray by +integration as: +c(r) = +� tf +tn +e− +� t +tn σ(r(s))dsσ(r(t))c(r(t))dt, +(6) +where σ and c are functions mapping each 3D point along +each ray r(t) to the respective volume density and radiance. +In our framework, we parametrize σ as V Density and c +as V RGB which can be efficiently queried using trilinear +interpolation. +We train the model using a camera with +fixed extrinsics initialized to the identity matrix, and fixed +intrinsics. Note that, to reduce computational resources, we +render images directly from voxels without any additional +MLP, nor did we employ any upsampling technique. +We assume that the background is flat and it is not +moving. We model it as a plate of fixed, high density. This +density is modeled with a single dedicated volume, while +the color is obtained from V RGB. +3.5. Training +Learning a 3D representation of an articulated object +from 2D observations without additional supervision is a +highly ambiguous task, prone to spurious solutions with +poor underlying geometry that leads to corrupted renderings +if the camera is moved away from the origin. We devise a +two-stage training strategy that promotes learning of correct +3D representations. First, we train the model with only a +single part, i.e. Np = 1. This allows the model to obtain +meaningful estimation of the object geometry. +Thus we +name this pretraining a Geometry phase or G-phase. During +the second phase, we introduce Np = 10 parts, allowing +the model to learn the pose of each part. We copy all the +weights from the G-phase. Moreover, for C the weight of +the final layer is extended such that all the part predictions +are the same as in the first stage, while for G, we just add +additional weights for V LBS initialized to zero. +The model is trained using a range of losses. +Reconstruction loss. +We use perceptual reconstruction +loss [22] as the main driving loss. Similarly to FOMM [52] +we use a pyramid of resolutions: +Lr = +� +l +� +i +���VGGi(Dl ⊙ ˆF) − VGGi(Dl ⊙ F) +��� , +(7) +where VGGi is the ith-layer of a pretrained VGG-19 [56] +network, and Dl is a downsampling operator corresponding +to the current resolution in the pyramid. The same loss is +enforced for F low. +Unsupervised background loss. Contrary to 2D frame- +works for unsupervised animation that use motion cues to +separate background from foreground objects, our generator +G mostly relies on appearance features, thus it is harder for +it to disentangle the background from the foreground. In the +first stage, we encourage the model to correctly disentangle +the background from the foreground leveraging a coarse +background mask B that we obtain in an unsupervised +manner from MRAA [55]. Given the occupancy map O +for the foreground part obtainable by evaluating Eq. (6) +excluding the background, we enforce a cross entropy loss: +Lbkg = +� +i +O log(1 − B) + (1 − O) log(B), +(8) +the background mask B is very coarse and we observe is +necessary only in the earliest iterations to avoid degenerate +solutions, thus we reduce the contribution of this loss each +epoch, we specify the exact schedule in Appx A. +5 + +Pose losses. +Finally, to regularize the PnP-based pose +prediction we add two regularization terms: equivariance +and projection. +First one is a standard technique for +regularizing unsupervised keypoints [52,55]: +Leq = |A ◦ C(F) − C(W(F, A))| , +(9) +where A is a random affine transformation, and W is a +warping operation. The intuition behind this loss is that +when an image is deformed, its keypoints should undergo +a similar deformation. Second, we explicitly minimize the +K3D reprojection error with K2D: +Lproj = +� +p +��K2D +p +− Π(K3D +p , Tp) +�� , +(10) +where Πp projects the points according to the estimated +camera pose Tp. This loss enforces keypoints to comply +with the predicted camera transformation Tp, improving the +stability of the PnP algorithm. +The final loss is the sum of all terms with equal weights. +Note that for the second stage Lbkg is not used. +3.6. Inference +Despite our model learns the embedding space for the +identities in the training set only, it can be used to model +previously unseen identities. Given an image of an unseen +identity Ftest and a randomly initialized embedding etest, +we optimize the reconstruction loss Lr (Eq. 7) with respect +to the embedding etest. This procedure produces a volu- +metric representation with detailed geometry, but imperfect +textures. We address this issue by finetuning the generator +G, following the pivotal tuning procedure [49]. In order +to avoid significant distortions to the geometry, during this +finetuning stage we regularize V Density and V LBS to stay +close to their values prior to finetuning. +Note that we +only optimize with respect to the appearance and do not +modify the 2D keypoint predictor C, ensuring that motion +can be transferred from different objects. Additional details +concerning this embedding are provided in Appx A. +4. Experiments +Evaluating animation, whether 2D or 3D, is a challeng- +ing task as there is no ground truth for the animated images. +We are not aware of prior works in unsupervised volumetric +animation, hence, in this section we establish an evaluation +protocol for this task. Our protocol makes use of established +metrics in unsupervised 2D animation, when applicable, +and introduces procedures to evaluate the quality of the +synthesized 3D geometry and animation under novel views. +Datasets. To evaluate our method we use three publicly +available datasets: 1) Cats [79], consisting of 9,993 images +of cat faces. +We used 9,793 for training and 200 for +testing. 2) For VoxCeleb [38], we employed the same pre- +processing as FOMM [52], using 19522 face videos for +training and 100 for testing. 3) TEDXPeople [17] is a video +dataset of TEDx speakers. +Using timestamps provided +by [17], we extract continuous video chunks. More details +can be found in Appx B. In total, we employ 40896 videos +for training, and retain 100 videos for testing. +4.1. Geometry from Image Data +Our method learns high-fidelity geometry from images +or videos without camera or geometry supervision. This +is a challenging setting, even for recent 3D-GANs, as they +require camera supervision. In this setting, we compare +the quality of inferred geometry to a state-of-the-art 3D- +GAN, EpiGRAF [58], trained with ground truth camera +poses. As both UVA and EpiGRAF render non-absolute +depth, to evaluate its quality, we use the Pearson correlation +coefficient. +Given a test image, we reconstruct it by +inversion, and obtain depth using volumetric rendering. +We then measure the correlation between the predicted +depth and the depth estimated with an off-the-shell depth +estimator [12]. +For a fair comparison, during inversion, +we do not provide camera poses to EpiGRAF and instead +find them during the optimization, in combination with the +rest of the parameters. UVA provides higher-quality depth, +while not requiring camera supervision during training, +reaching a correlation value of 0.63. EpiGRAF reaches only +0.53, often failing to render accurate depth for non-frontal +cameras (see Fig. 3a). +4.2. Animation Evaluation +Unsupervised animation in 3D is a new task introduced +in this work. A key feature of 3D animation is the ability +to change the viewpoint from which the object is rendered +during animation. +Commonly used animation datasets, +however, do not typically offer multi-view data. To evaluate +viewpoint consistency without access to multi-view data, +we introduce three new metrics: Average Yaw Deviation +(AYD), Average Shape Consistency (ASC), and Average +Pose Consistency (APC). In more detail, given an object, +we rotate it along the y-axis using a set of predefined +angles. We then fit an SMPL [9] model for humans and +a 3DMM [13] for faces to the frontal and rotated views of +the objects. These models estimate the root angle, defining +how the object is oriented with respect to the camera; a +shape parameter, defining the identity of the object; and +a parameter defining its pose (in terms of joint rotations +for SMPL and facial expression parameters for 3DMM). +To evaluate the ability of the model to rotate the object by +the required angle to produce novel views, we use AYD. +In particular, we compute the y-axis component of the +root angle between the rotated and non-rotated object, and +compare it with the known yaw of the camera, used to +render that view. We use ASC to compare the consistency +6 + +Input +UVA +EpiGRAF +Our Np=10 +Our Np=1 +No G-phase Np=10 +Our full model +Our with only 1 part +No geometry-phase +No BG Np=1 +Direct Np=1 +No unsupervised background +Directly predict poses +(a) Depth comparisons +(b) Qualitative ablation results of methods in Tab. 2 +Figure 3. (a) Typical depth examples of embedded images using our method (UVA) and EpiGRAF [58]. Note, UVA’s depth contains +sharper details regardless of the pose. (b) We show a block for each method, with novel views (top), and depth, normals, parts (bottom). +Source +Source +Driving +UVA +MRAA +Driving +UVA +MRAA +Driving +UVA +MRAA +Driving +UVA +MRAA +Source +Source +Source +Source +Driving +UVA +MRAA +Driving +UVA +MRAA +Figure 4. 2D animation. Example 2D animations on bodies and faces from our method (UVA) and a state-of-the-art work, MRAA [55]. +As UVA models objects in canonical 3D space, it better preserves an object’s shapes when animated. Zoom-in for greater detail. +of the shape parameters between the frontal and the rotated +views. A lower ASC indicates that the identity is better +preserved during rotation. APC is used to measure how +much the pose is altered during rotation, with a lower APC +indicating better preservation of the object pose. +These +metrics enable evaluating the capabilities of competing +models in generating view-consistent results. +Appx C +contains full details on these metrics. +No prior unsupervised animation method [52, 55, 64] +offers a built-in ability to generate the data under novel +views. Thus, for [52,55] we introduce a simple, depth-based +method to generate novel views. First, we predict the depth +from a monocular depth predictor [12] and normalize it to +make it compatible with our camera intrinsics. Then, for +each method, we estimate parts and their affine transforma- +tions. We choose a central 2D keypoint for each part, and +augment it with 4 additional keypoints in its neighborhood. +Using the depth, we lift the keypoints in 3D and re-project +them into the novel viewpoint. From these new keypoints, +a new affine transformation is estimated and used to drive +the view synthesis. +We then evaluate against LIA [64], +which expresses animation as linear navigation in a latent +space. +Interestingly, for the VoxCeleb [38] dataset, we +found one of the components of its latent space to correlate +with the rotation of the head along the y-axis. Exploiting +this finding, we fit a linear model mapping the magnitude of +the movement along this latent component to the produced +head rotation, and use it to generate the head under novel +viewpoints. +We also use the standard 2D reconstruction metrics: L1, +AKD/MKR [53], AED [53]. However, we emphasize that +such metrics favor 2D methods, which can solve the 2D +animation problem by copying pixels from the source to +the target view, at the cost of limited 3D understanding +and consistency. In contrast, UVA renders view-consistent +pixel values from a 3D representation, making this shortcut +unavailable. A significant gap may also be introduced by +the single-image embedding procedure we adopt. +Note, +however, that as our embedding procedure seamlessly sup- +ports the use of multiple source frames at inference time, +a shared representation can be optimized, pooling infor- +mation from all available frames to improve performance. +7 + +AB8nicbVDLSgNBEJyNrxhfUY9eBoPgxbAbfB0DXvQWwTxgs4bZyWwyZHZmekVwpLP8OJBEa9+jTf/xkmyB0saCiqunuChPBDbjut1NYWV1b3yhulra2d3b3yvsHLaNSTVmTKqF0JySGCS5ZEzgI1k0I3EoWDsc3Uz9hPThiv5AOEBTEZSB5xSsBK/pl38Zh1Kd0itX3Ko7A14mXk4qKEejV/7q9hVNYyaBCmKM7kJBnRwKlgk1I3NSwhdEQGzLdUkpiZIJudPMEnVunjSGlbEvBM/T2RkdiYcRzazpjA0Cx6U/E/z08hug4yLpMUmKTzRVEqMCg8/R/3uWYUxNgSQjW3t2I6JpQsCmVbAje4svLpFWrepfV2v15pX6Xx1FER+gYnSIPXaE6ukUN1EQUKfSMXtGbA86L8+58zFsLTj5ziP7A+fwBn3KQ3A=−15� 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Novel view synthesis. We report typical examples of novel views synthesized using a single input image. For bodies, we show a +narrower range, as the TEDXPeople dataset is biased towards frontal poses. +VoxCeleb +TEDXPeople +Method +AYD↓ +ASC↓ +APC↓ +L1↓ +AKD↓ +AED↓ +AYD↓ +ASC↓ +APC↓ +L1↓ +(AKD↓, MKR↓) +AED↓ +FOMM [52] +0.655 +0.129 +0.177 +0.0413 +1.289 +0.134 +0.507 +0.028 +1.07 +0.0318 +(3.248, 0.009) +0.120 +MRAA [55] +0.173 +0.123 +0.174 +0.0424 +1.250 +0.131 +0.181 +0.023 +0.702 +0.0262 +(2.282, 0.007) +0.101 +LIA [64] +0.207 +0.130 +0.190 +0.0529 +1.437 +0.138 +- +- +- +- +- +- +Our 1 frame +0.051 +0.078 +0.144 +0.0655 +1.737 +0.226 +0.128 +0.019 +0.635 +0.0474 +(3.571, 0.017) +0.163 +Our 5 frame +0.045 +0.091 +0.112 +0.0418 +1.378 +0.111 +0.107 +0.021 +0.571 +0.029 +(2.373, 0.014) +0.086 +Table 1. Comparison with 2D animation methods. Novel view synthesis for AYD, ASC & APC from yaw in range −45◦ to +45◦. +We demonstrate the results of our model with one and +five frames. We also note that, despite a wide range of a +viewpoints in different videos, subjects in each individual +video in VoxCeleb and TEDXPeople have very limited 3D +rotations, as they primarily face towards the camera. Thus, +standard reconstruction metrics do not reflect the model’s +capacity to perform complex 3D transformations. +We provide the quantitative results in Tab. 1. As the +affine transformations of FOMM [52] are mostly based on +edge detection that is not very robust, any minor modi- +fication of this transformations for novel view synthesis +leads to significant movement. +Thus, FOMM has the +worst AYD among all methods. +Affine estimation in +MRAA, in contrast, is significantly more robust, and thus +it has a significantly lower AYD. However, we observe +that MRAA does not have enough knowledge about the 3D +structure of the objects, treating them as planes—while they +roughly preserve the shape and pose for small angles, for +larger angles objects become thinner, until they eventually +disappear. LIA has a rotation direction that is entangled +with the other movements, and thus it has the lowest ASC +and APC. Finally, our model is the best at preserving shape +and expressions, as judged by the ASC and APC. Moreover, +our model also provides the most meaningful rotations as +judged by the AYD. When standard reconstruction metrics +are considered, our 5 frame model performs on par with +the baselines. +However, as previously mentioned, these +metrics do not reflect the ability of the model to preform +complex 3D movements. This point is further highlighted in +Fig. 4, when the pose of the source and driving images differ +significantly, MRAA fails to properly reflect that, while our +model produces more consistent results. +Interesting, we +also note that, as our model is based on learning a 3D prior +and not copying pixels, it can filter out some occlusions, as +seen in the third column in Fig. 4, while MRAA produces +artifacts in the occluded region. +Method +AYD↓ +ASC↓ +APC↓ +L1↓ +AKD↓ +AED↓ +Direct Np = 1 +0.707 +0.160 +0.239 +0.0723 +3.582 +0.326 +No BG Np = 1 +0.301 +0.117 +0.216 +0.0702 +2.410 +0.263 +Our Np = 1 +0.141 +0.113 +0.210 +0.0637 +2.170 +0.242 +No G-phase Np = 10 +1.08 +0.145 +0.226 +0.0620 +1.993 +0.243 +Our Np = 10 +0.051 +0.078 +0.144 +0.0655 +1.737 +0.226 +Table 2. Ablation results on the VoxCeleb dataset. +4.3. Ablation Studies +We evaluate the key design choices made in our frame- +work. First, we compare our PnP-based part pose predictor +with direct part pose prediction (Direct). +As directly +predicting an R3×3 rotation matrix could produce solutions +not corresponding to a rigid rotation, we adopt the 6D +rotation parameterization from [83]. The geometry learned +by this approach is essentially flat. +We compare our +method and Direct only in the geometry phase of training +(e.g., when Np = 1), as it does not produce sufficiently +accurate geometry to proceed with the next phase. +We +also demonstrate the effect of the unsupervised background +loss Lbkg by training the model without this loss (No +BG). Finally, we investigate the importance of two-phase +training, learning a model with multiple parts without the +geometry phase No G-phase. We show numerical results in +Tab. 2 and qualitative examples in Fig. 3b. Our full model +achieves the best scores, and generates higher fidelity novel +views and geometric details. The utility of the geometry +phase is clearly demonstrated by the scores and qualitative +results, which, without this phase, produce corrupted results +and do not learn representative parts. While it produces +meaningful depth, the model trained without Lbkg fails to +separate the background and foreground. +4.4. Geometry Evaluation for Synthetic Objects +To further evaluate the quality of the learned geometry, +we ran experiments on images from two synthetically +rendered datasets providing ground truth depth: 1.) that of +Khan et al. [25], which provides high-quality, portrait-style +8 + +facial images; and 2.) +SURREAL [60], which provides +full-body renderings of animated subjects. +We use 112 +image for faces and 60 images for bodies, cropped such +that they roughly correspond to the cropping used in the +respective real datasets used for training. These datasets +contain subjects with widely varying identities, poses, +hairstyles, and attire, rendered in different environments +and lighting conditions. However, for these experiments +we did not rely on synthetic data for training, instead using +models pretrained on 2D images from VoxCeleb [38] or +TEDXPeople [17] for faces or bodies, respectively. Despite +the domain gap between our training and evaluation data, +we are able to obtain high-quality depth estimates for these +synthetic renderings using models trained only on real, +in-the-wild images. +Given a synthetic input image, we +invert it, then compute the Pearson correlation coefficient +between our method’s inferred depth and the ground truth. +For these experiments, as we are only concerned with the +geometry of the target object, we masked out the depth +for background regions, computing the correlation only +between the depths of foreground pixels. We compare our +predicted depth with the general purpose state-of-the-art +depth predictor Omnidata [12]. The depth correlation for +Omnidata is 0.602 for faces and 0.470 for bodies, while +for our method they are 0.793 and 0.568, respectively. In +Fig. 6, we show the image along with the reconstructed +depth. +These results demonstrate that our unsupervised +method learns meaningful geometric representations, even +for significantly out-of-distribution inference data. +5. Limitations +Our model addresses, for the first time, the task of +unsupervised 3D animation. +While our model obtains +compelling results on this challenging task, here we note +some limitations: +• Our method assumes the object can be represented +with a voxel cube of size 643. +We notice that +when generating novel views involving large camera +displacements from the original pose, some seam-like +artifacts may appear. We believe they are due to the +small size of the voxel cube and errors in predicting +precisely the distance of the part, which could lead to +a slight displacement between different parts. +• For each test identity, our model makes use of an +optimization-based procedure to compute the respec- +tive identity embedding and fine-tune the generator. +This procedure increases the inference cost of our +model, but needs to be performed only once for each +test identity, thus the cost of the procedure is amortized +when producing a large number of frames. +• Our model renders frames at a resolution of 256×256, +which is lower than the ones typically supported by +Depth +Depth +Image +Image +(a) Khan et al. [25] +(b) SURREAL [60] +Figure 6. Visualization of predicted depth for synthetic datasets. +We show the input image and the depth predicted by our method. +state of the art 2D animation methods. +This is a +common limitation of 3D methods based on volumet- +ric rendering, and we expect continuous progress in +efficient volumetric representations and rendering to +enable the generation of higher resolution images. +• Our method can learn geometry only from the views +that were observed in the training dataset, thus, for +the back side of the face in VoxCeleb [38] and the +back side of the body in TEDXPeople [17], no precise +geometry is learned. +6. Conclusion +Our approach for unsupervised volumetric animation +demonstrates a significant step towards 3D animation of +dynamic objects. While trained exclusively on real-world +monocular 2D videos, our method obtains high quality +geometry, object parts, 3D segmentation and normals. Due +to the unsupervised nature of our work, the same approach +applies to a variety of object categories without using +explicit labels or other cumbersome supervision. +This +understanding of the underlying geometry and structure of +the object, allows our method to perform animation and +novel view synthesis at the same time. These properties +open exciting possibilities for employing this information +for future exploration, e.g. controlling an object’s fine- +grained shape, or relighting it for composition into novel +environments. +9 + +References +[1] Sherwin Bahmani, Jeong Joon Park, Despoina Paschali- +dou, Hao Tang, Gordon Wetzstein, Leonidas Guibas, Luc +Van Gool, and Radu Timofte. 3d-aware video generation. +arXiv:2206.14797, 2022. 2 +[2] Jonathan T Barron, Ben Mildenhall, Matthew Tancik, Peter +Hedman, Ricardo Martin-Brualla, and Pratul P Srinivasan. +Mip-nerf: A multiscale representation for anti-aliasing neu- +ral radiance fields. 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Given the embedding e, we +transform it to a 43 voxel cube with 512 features using +a fully connected layer. +This voxel cube is then passed +through four 3D-Residual Blocks with upsampling. The +3D-Residual block consists of two 3 × 3 × 3 convolutions +and two batch normalization in the main branch, and one 1× +1 × 1 convolution in the residual branch. We choose ReLU +for our non-linearity. Each 3D-Residual block reduces the +number of features two times, while also increasing the +resolution two times using nearest neighbor upsampling. +After the final upsampling layer, the voxel cube has size +643 and 32 features. We make use of a final projection layer +implemented as a 1 × 1 × 1 convolution preceded by batch +normalization to obtain 1+3+10 features for density, RGB, +and LBS volumes, respectively. +Neural rendering. For neural rendering, we use fixed +camera extrinsics equal to the identity matrix, and a fixed +intrinsics matrix assuming a field of view of 0.175 [41]. +Based on this configuration, we define the region in which +we sample points for rendering to be the cube span- +ning the region [−1.0088, 1.0088] × [−1.0088, 1.0088] × +[9.5000, 11.5000]. The camera in our settings looks in the +positive z direction. We call this region the rendering cube. +We use the rendering cube intersections with casted rays to +define the near and far camera planes, and uniformly sample +128 points between them. We note that during the G − +phase, there is no need to capture small parts, as a single +part is employed. Thus, to speed up training, we reduce +the number of sampling points to 48 in this phase. When +mapping the rendering cube to the respective volumes, we +increase the rendering cube by a factor of 1.075. For the +Cats [79] dataset, we scale the rendering cube by a factor +of 1.2. In this manner, we increase the amount of actual +space that can be covered by the modeled volume, allowing +for a larger set of transformations. Following NeRF [37], +we apply perturbations to the sampled point in two ways +during training. First, we perturb the position of each point +along the ray. Second, we add noise to the sampled densities +before rendering. We use a standard deviation of 0.5 for the +noise. As the geometry is mostly discovered at the begining +of training, we linearly decrease it during each phase of +training down to zero at 100k training steps. +Perspective-n-Point. We obtain a solution to the PnP +problem leveraging the differentiable EPnP [28] implemen- +tation from PyTorch3D [44]. For each part p, we predict +Nk = 53 = 125 keypoints, for a total of Np × Nk = 125 × +10 = 1250 keypoints. Each keypoint is represented with +an unconstrained three-dimensional parameter, followed by +sigmoid and normalization to ensure that each individual +keypoint always lies inside the rendering cube. We initialize +3D keypoints for each part such that they form a regular +cubical grid with 5 equally spaced keypoints on each side +of the cube. +Due to possible incoherent arrangements +between 2D and 3D keypoints, the estimated part poses +may correspond to object positions outside the rendering +box. This behavior would prevent the affected parts from +learning density, since they would bring no contribution +to the rendered image. To address this issue, we add an +additional loss that pushes the estimated pose of parts with +no associated density to the the pose of the largest part (i.e. +with the most density). We formulate this loss as follows: +Linit = +� +p +max(0, t − σp)|Tp − Tpmax|, +where t = 0.01 is a density threshold, σp is the density +associated with part p (which is the mean of all densities +for all sampled points, multiplied by the LBS weight) and +Tpmax is the pose of the part with the maximal density. For +the first phase, when a single part is learned, we use the +identity matrix in place of Tpmax. +Training. +In the first phase, we train the model for +200 epochs with a batch size of 128 on 8 A100 GPUs. +To equally balance the contribution of each identity, each +epoch consists of a single frame randomly sampled from +each video in the dataset. +In this phase, we decrease +the contribution of the Lbkg by a factor of 0.8 every 10 +epochs. For the Cats [79] dataset, the method is trained +for 1000 epochs on a single A100 GPU with a batch size +of 16. In the second phase, we train the model for 1000 +epochs using a batch size of 16 on 8 A100 GPUs. The +learning rate is decayed 10 times at 600 epochs and 900 +epochs. Finally, to encourage the network to discover small +parts such as hands in TEDXPeople [17] dataset, during +this phase we also employ co-part segmentation obtained +without supervision from MRAA. The loss consists in the +cross-entropy loss between MRAA [55] co-parts and our +rendered LBS weights. As with Lbkg, this loss is decreased +by a factor of 0.8 every 10 epochs. For both phases, we use +Adam [27] with lr = 5e − 4 and β = (0.5, 0.999). +Inference. +To embed test images, we rely on a two +stage procedure, similar to PTI [49]. First, we optimize the +embedding e for a particular image using the reconstruction +loss Lr used during training, and employing Adam [27] +with lr = 1e − 2. The optimization is run for 3000 steps, +and the learning rate is decayed by a factor of 10 every +14 + +750 steps. Similarly to StyleGAN2 [24], we add random +noise to the embedding to promote exploration. We select +an initial standard deviation of 0.5 for this noise and decay +it until reaching zero at step 1500. For the second stage, we +fine-tune all the generator parameters. To avoid forgetting +the useful geometry prior learned during training, we add +an additional geometry regularization loss: +Lgeo = | ˆV Density − V Density| + | ˆV LBS − V LBS|, +where ˆV Density, ˆV LBS are the density and LBS weights +from the first stage, respectively. We also employ additional +data augmentation at alternate steps by applying random +Euclidean transformations for the source image, for which +we sample rotation angles and translations in the [−0.1, 0.1] +range. Note that, as our model assumes the background is +static, we only enforce this loss for the foreground using +the rough background mask obtained from the first stage. +We train for 500 steps in this stage. For optimization with +5 input frames, we increase the number of steps in the +second stage to 3000. +Since 5 images may not fit into +a single GPU, we use a batch size equal to 2. Inverting +one image takes roughly 10 minutes on an A100 GPU, +while inverting five images takes roughly 20 minutes. Our +PnP-based part pose estimation algorithm introduces some +instability in the estimation of the distance of the part from +the camera. While this instability does not produce artifacts +when rendering the object from limited rotation angles, +it becomes more noticeable when rendering from extreme +camera angles. +To mitigate such effects, we devise an +inference-time filtering strategy to smooth abrupt changes +in the estimated depth of each part. +For each part, we +estimate the distance from the camera origin to the center of +the rendering cube. We then compute the mean of all these +distances for each part lp. Finally, we rescale the vector +from the camera origin to the center of the rendering cube, +such that they have the same length lp in all frames. +B. Dataset details +We employ three training datasets: +VoxCeleb [38], +TEDXPeople [17] and Cats [79]. +We adopt the Vox- +Celeb [38] preprocessing of FOMM [52], and preprocess +Cats [79] in the same way as [10, 58]. +For TEDXPeo- +ple [17], we first download the videos listed in [17], then, +using the provided timestamps, select continuous chunks +of videos starting at the provided timestamp and lasting +at most 512 frames. +In each chunk, we detect human +keypoints and bounding boxes for each frame using [69]. +We clamp the predicted bounding box at the hip joints at +the bottom of the frame, then increase its size by a factor +of 1.2 so as to capture the subject’s full upper body, then +make it square. +We process the video chunk frame-by- +frame, adding each processed frame to the current video +sample. If, in some frames, the human is not detected or the +bounding box moves significantly from the initial position, +we stop the current video sample and start collecting a new +sample at the next detection of a human. To further clean +the dataset, we discard video samples that are too short +(less than 64 frames), to small (less than 256 pixels on +any side), have significant background movement (detected +using simple L1 error on pixel values), have no movement +in foreground (which most likely indicate that the detected +human is a static image visualized during the presentation) +or have a width similar to the height (which indicates a +failure of the hip predictor). We select only views marked +as ”front” in the original annotations [17], and from each +YouTube video, we take at most three different samples. +Our final dataset consists of 40896 different samples from +17451 different YouTube videos. +C. Metric details +A critical aspect of 3D animation is the ability to syn- +thesize novel views of the observed target object. However, +evaluating this ability is challenging, as animation datasets +typically lack multi-view observations. We thus introduce +metrics that, given a triplet composed of a source frame, a +driving frame, and a result rendered under a target camera, +can quantitatively evaluate the quality of the rendered novel +view: +• Average Yaw Deviation (AYD): this evaluates whether +the object is rendered from the target camera perspec- +tive. Given the yaw angle between the camera and +the object in the driving frame Θd and in the rendered +frame Θr, and the yaw angle of the novel view camera +with respect to the original camera Θc, we define +AYD = |Θd − (Θc + Θr)| +• Average Shape Consistency (ASC): this evaluates +whether the identity of the rendered object is the one in +the source frame. Given an identity code for the source +frame cs +s ∈ RNcs and an identity code for the rendered +frame cs +r, we define ASC = |cs +s−cs +r| +Ncs +• Average Pose Consistency (APC): this evaluates +whether the object is rendered in the pose given by the +driving frame. Given a pose code for the driving frame +cp +d and a pose code for the rendered frame cp +r ∈ RNcp , +we define APC = |cp +d−cp +r| +Ncp +We obtain Θ, cs and cp in a way that is specific to the given +object category. For faces, we compute the head yaw angle +Θ using the 6DOF head pose estimator 6DRepNet [19], +which we find robust to extreme head poses and possible +corrupted regions in the rendered frames. Given an image, +the model directly provides the estimated yaw angle, which +we convert to radians prior to the computation. To compute +cs and cp we use the DECA [13] 3DMM. We select this +model due to its robustness to large head rotations, its +fast, encoder-based inference, and its ability to disentangle +15 + +the head shape from the current expression. In particular, +we define cs as the inferred Ncs = 100 FLAME [31] +face shape parameter, which encodes the identity of the +subject. We define cp as the concatenation of the inferred +50 expression parameters with the estimated jaw rotation +in axis-angle representation for Ncp = 53, which together +capture the particular facial pose. We choose not to make +use of the estimated head yaw angle, since we find it less +robust than the one inferred from our adopted 6DOF head +pose estimator. For human bodies, we fit the SMPL [34] +body model to each frame using 3DCrowdNet [9]. +We +choose 3DCrowdNet due to its fast inference time and its +robustness to partially-occluded subjects which are frequent +in the TEDXPeople [17] dataset, where only the upper half +of the body is typically present in the frame. 3DCrowdNet +requires a set of 2D human body keypoints to be detected +for each frame. We first detect person bounding boxes using +Faster R-CNN [47] and use VitPose [71] to detect the 2D +human body keypoints, which we find to work robustly +even in the presence of artifacts in the images. Given the +fitted SMPL model, we define cs as the inferred Ncs = 10 +body shape parameters, and cp as the concatenation of the +inferred angles for a selected set of 13 joints in axis-angle +representation corresponding to the joints situated above +the ‘belly button’ joint for a total of cp = 39 elements. +Selection of the joints ensures that joints that are typically +not present in the TEDXPeople dataset, and thus cannot be +reliably estimated, will not negatively affect the precision of +the evaluation. We extract the yaw angle Θ by transforming +the root joint axis angle rotation inferred by the model into +the corresponding rotation matrix M = MyMxMz, and +extract the yaw angle Θ of the My component representing +the y-axis rotation matrix as follows: +pitch = arcsin M1,2 +cos(Θ) = +M2,2 +cos(pitch) +sin(Θ) = +M0,2 +cos(pitch) +Θ = arctan2(sin(Θ), cos(Θ)), +where the case of cos(pitch) = 0 is disregarded, since in +practice we never render objects from high-pitch angles. +We now define the evaluation protocols followed for the +animation and novel view synthesis tasks. For the animation +task, we consider each test set video and select the first +frame of each video as the source frame. We consider as +driving frames five video frames, equally spaced along the +duration of the test video. We then generate the object in the +source frame in the pose of each driving frame under novel +views, produced by rotating the object with the following +Θ angles: 0, ± π +12, ± π +6 , ± π +4 . +The triplets built from all +combinations of source, driving and rendered frames are +used for the computation of AYD, ASC and APC. For the +novel view synthesis, we consider as source and driving +frame the same, first frame of each video. This allows pure +evaluation of the novel view synthesis capabilities of the +method. We render each frame under the set of 256 linearly +sampled Θ angles in the range [− π +2 , + π +2 ] and compute +AYD, ASC and APC using all the available frame triplets. +D. Baseline details +MRAA and FOMM. MRAA and FOMM rely on affine +transformations to transfer motion. +Thus, in order to +perform novel view synthesis a natural idea would be to +modify these affine transformations such that they represent +the object in the novel view. +We achieve this with the +following procedure. +First, near each region center, we +sample 4 additional keypoints in a small distance = 0.05 +forming a cross centered around the central keypoint for the +region. The region center and these additional keypoints are +then lifted to the 3D space using a depth map obtained from +the driving image with off-the-shelf depth estimator [12]. +As we need to recover depth for the object in the pose of +the frame for which to perform novel view synthesis, we +use absolute depth for animation to ensure the rendered +frame pose is the same as the driving frame. These points +are then projected to the target view using the desired +camera parameters. +Finally, we estimate a new affine +transformation from these projected points. Note the off- +the-shelf depth estimator only provides relative depth, and +it is thus not possible to utilize it directly. To overcome +this issue, we leverage depth obtained from our method and +find a linear mapping between our depth and off-the-shelf +depth. This linear mapping is consists of dscale and dshift +parameters and can be find in closed form: +dscale = Cov(d, ˆd) +V ar( ˆd) +, +dshift = E [d] − dscaleE[ ˆd], +where d is the depth map from our method, ˆd is the off-the- +shelf depth map, E is sample mean, V ar is sample variance, +and Cov is sample covariance. +LIA. LIA expresses animation as navigation inside a +learned latent space. Given an embedding zs in this latent +space for the source image, animation is expressed as zd = +zs + w = zs + � aidi, with zd expressing the latent +code corresponding to the animated result and vector w +expressed as the summation of a set of learned motion +directions d multiplied by corresponding magnitudes a, +which form an orthogonal basis of the latent space. The +set of learned motion directions represents the main types +of motions performed by the objects. +Interestingly, we +find that for the VoxCeleb dataset, d2, the second of such +directions, is correlated with y-axis head rotation. While +this movement is undesirably entangled with other motion +components such as x-axis head rotation, we exploit this +16 + +VoxCeleb +TEDXPeople +Method +AYD↓ +ASC↓ +APC↓ +AYD↓ +ASC↓ +APC↓ +FOMM [52] +0.801 +0.145 +0.194 +0.639 +0.029 +1.14 +MRAA [55] +0.760 +0.133 +0.177 +0.686 +0.022 +0.861 +LIA [64] +0.188 +0.132 +0.198 +- +- +- +Our 1 frame +0.155 +0.119 +0.171 +0.248 +0.023 +0.941 +Our 5 frame +0.153 +0.126 +0.184 +0.244 +0.024 +0.959 +Table 3. The results of generating novel views of the first frame of +each video sequence. Camera angles range from −90◦ to +90◦. +finding to produce novel views. Since no immediate corre- +spondence between magnitude a2 added to such direction +and Θ exists a priori, we build a linear model mapping +changes in Θ between the source and driving frame with a2. +To build such a linear model, we consider the first frame of +each video and produce novel views using values of a2 in +the range of [−17, +17] degrees. For each generated novel +view, we evaluate the corresponding changes in Θ between +the source and driving frame using 6DRepNet [19] and use +such data to fit our linear model a2 = 7.453Θ. Given a +desired Θ angle, we leverage the linear model to devise the +magnitude a2 and produce znovel = zd + a2d2, which is +decoded to the frame under the novel view. Note that since +the linear model directly optimizes the Θ error on the test +set, we expect the AYD metric produced for such baseline +to be biased toward lower values. +E. Novel view synthesis +In this section, we evaluate the capabilities of the anima- +tion methods to perform novel view synthesis without ani- +mation. To this end, we simply rotate the image along the y +axis on the set of angles from −90◦ to +90◦. The results are +provided in Tab. 3, and confirm the findings from Sec. 4.2. +While MRAA has favourable ASC and APC errors, it has +very high AYD. This behavior is expected, because the +method is simply performing a translation of the subject, +rather than rotating it according to the provided yaw angles. +This ensures the pose and identity remain preserved, at the +cost of performing poor novel view synthesis. LIA, on the +other hand, has a low AYD, while ASC and APC are high, +which confirms that LIA has entangled latent directions that +prevent novel view synthesis without significantly altering +the pose and identity. Our model achieves the best AYD, +which suggests that it performs the most accurate camera +manipulations. +F. Canonical visualization +In order to better demonstrate the representation learned +by our model, we visualize some of the training identities +in the canonical pose, i.e. where Tp is the identity matrix +for all parts. Note that, since there are no prior assumptions +on how the object should be placed in the rendering cube, +parts seen from the camera with identity matrix extrinsics +may be arbitrary. Thus, we select the camera from which +objects will look reasonable. Note that Tp is still the same +(a) VoxCeleb [38] +(b) TEDXPeople [17] +Figure 7. Visualization of canonical spaces. +for all parts and all objects. The visual results are presented +in Fig 7, which clearly shows that all objects have the same +pose, which is a crucial property for animation. +G. Failed Experiments +Canonical space representation. +During our initial +experiments we tried many different representations for +canonical space: Triplanar [5], MLP [37], CP and VM +decomposed cubes [7]. However, we found that decom- +posed solutions such as Triplanar [5] and VM [7] are biased +towards flat geometry, while an MLP [37] is extremely +slow. Note that the Triplanar [5] representation also utilizes +a small MLP, thus in our experiments it was slower than +directly sampling our Voxel cube. +Pose prediction. Before reaching the PnP formulation, +we tried many different approaches for pose prediction. +First, we started with Direct approaches, and we tested sev- +eral architectures and rotation representations [83]. How- +ever, all of them failed to produce meaningful geometry. +We also tried an optimization-based approach for motion, +i.e. having a pose parameter for each frame in the dataset. +While this produced decent results for a single video, when +the number of frames scales to millions, this approach +quickly becomes infeasible. +Different PnP. We tested several different PnP imple- +mentations. +We found that implementations based on +declarative layers [8,16] are extremely slow, and using them +in our setting would have been unfeasible. We also tested an +implementation from the Kornia Library [48] that is based +on DLT. However, it did not produce any meaningful results +and produced divergence of the model. Our final choice +was the EPnP [28] implementation from Pytorch3D [44]. +However, we would like to note that it was only working +in PyTorch 10.1 and not PyTorch 11, where it was not +converging. We discovered the problem was the initially +unstable gradients of the pinverse function in the newer +version. We think that this instability can be solved with +better initialization for the 2d points, however we left this +investigation for future work. +Depth and normal supervision. To help discovering +the geometry we also tried to utilize depth and normal +supervision from an off-the-shelf predictor [12]. Note that, +17 + +because the normals supervision require computation of +second order derivatives and we rely on voxel sampling with +grid sample, we need a second derivative of grid sample, +which is not implemented in PyTorch3. Thus, we develop +a custom cuda kernel for the second derivative. While the +depth and normal supervision helps to improve results for +one-phase training, we found it to be unnecessary with two- +phase training. +Upsampler. We render images in full resolution, how- +ever in prior experiments we utilize an upsampler. While +this method works faster and consumes less memory, it pro- +duces less detailed geometry and worse view consistency. +Different multipart representations. +We also tried +two different representations for describing objects with +multiple parts. +The first had a shared radiance V RGB +volume, but a separate density for each part, while the +second used different radiance and density volume for each +part. Both of these strategies produce reasonable results. +However, for them it is much harder to discover a large +number of parts, and they usually degrade to solutions with +only one or two parts being used. +Few shot NeRF regularization. We also tested several +few-shot NeRF regularization techniques: entropy loss on +the NeRF weights [26], loss on the weights from MiP +NeRF [2], surface normals regularization [61] and warping +loss from MVCGAN [80]. We found that all of them are +unnecessary with our two phase training strategy. +Discriminator. To regularize the novel views, we also +try to employ a Discriminator, similarly to 3D-GANs [5, +58]. In more detail, we first predict the pose of the object +and then try to rotate this pose to generate the object in +the novel view. This image is subsequently passed to the +discriminator. However, we found it hard to find proper +rotation ranges, thus the discriminator reduced the quality +of the geometry in our experiments. +H. Ethical considerations +Dataset usage. +The primary datasets used in our +experiments, VoxCeleb [38] and TEDXPeople [17], con- +tain publicly available videos of notable figures in public +venues, e.g. celebrities giving interviews and speakers +giving presentations to large audiences. These datasets have +been released by and employed for prior academic research, +e.g. the works we use for our comparisons and evaluations. +Other datasets, such as Khan et al. [25] and SUR- +REAL [60] which are used for our ground-truth depth +inference evaluations, contain realistic but synthetic images +rendered from 3D models of human faces and bodies, +respectively. SURREAL [60] uses body scans and motion +capture sequences generated from 3D capture of the appear- +ances and performances of subjects who consented to have +this captured and released for academic purposes. Khan et +3https://github.com/pytorch/pytorch/issues/34704 +al. [25] contains facial images generated by perturbing +characteristics such as facial identity, hair and clothing +for models in a standard 3D modeling and rendering +framework, and thus do not correspond to any particular +person whose identity may be at risk of being revealed. +Each of these datasets are both publicly available and +have been used for prior academic works. As such, there +are no particular concerns about violating the privacy or +anonymity of our test subjects. +Potential for bias in synthesis results. As with other +data-driven methods for performance-driven animation, the +amount of variation in characteristics such as gender, age, +body type, and ethnicity that can be handled by our methods +with the source and driving subjects while producing plau- +sible synthesis results is dependent on the amount of such +variations contained in the dataset. +While the variations +in the real and synthetic images used in our experiments +are limited by those in the aforementioned datasets used in +our evaluations, e.g. in typical celebrity videos and TEDx +presentations, our method has no particular limitations +towards such subjects, and thus could be deployed on +other datasets containing different identity characteristics. +Deploying this approach in a manner which is fair and +robust with respect to such variations for non-academic +purposes, such as commercial applications, would require +employing a dataset that is appropriately representative of +the possible target identities, and evaluating the results +to ensure consistent behavior across these demographics. +However, for the academic evaluations presented here, our +evaluations suggest that our approach works as expected +given the datasets we use, and thus could generalize to +other training datasets fairly easily. Finally, as our approach +only relies on unconstrained video sequences for training, +acquiring the data needed to adapt to new subjects is +fairly straightforward, provided that the appropriate video +sequences can be collected for training. As such, there are +no particular concerns related to unfair bias in our approach. +Possibly misuse. As with other works in the domain of +realistic, performance-driven animation, our work carries +with it the possibility of use for deceptive activities, e.g., +creating plausible videos of public figures as misinforma- +tion to advance a political agenda. However, we maintain +that, while this is clearly a valid concern for the near future, +developing and studying such technology in public forms +such as this work raises awareness of this potential, and with +it the skepticism of viewers towards potentially misleading +videos. +Furthermore, publicly describing our work and +results allows for the advancement of forensic methods to +identify when such manipulations have occurred. We thus +believe that our work helps to prevent the secretive devel- +opment and deployment of these techniques for malicious +ends which are not known or detectable either to average +media consumers or professional forensic analysts. +18 + diff --git a/jdFIT4oBgHgl3EQfqSsD/content/tmp_files/load_file.txt b/jdFIT4oBgHgl3EQfqSsD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..993fe1b8a62cb3a1af61fe43d74db2fad1ac1e13 --- /dev/null +++ b/jdFIT4oBgHgl3EQfqSsD/content/tmp_files/load_file.txt @@ -0,0 +1,1134 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFIT4oBgHgl3EQfqSsD/content/2301.11326v1.pdf,len=1133 +page_content='Unsupervised Volumetric Animation Aliaksandr Siarohin Snap Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFIT4oBgHgl3EQfqSsD/content/2301.11326v1.pdf'} +page_content=' Willi Menapace∗ University of Trento Ivan Skorokhodov∗ KAUST Kyle Olszewski Snap Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFIT4oBgHgl3EQfqSsD/content/2301.11326v1.pdf'} +page_content=' Jian Ren Snap Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFIT4oBgHgl3EQfqSsD/content/2301.11326v1.pdf'} +page_content=' Hsin-Ying Lee Snap Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFIT4oBgHgl3EQfqSsD/content/2301.11326v1.pdf'} +page_content=' Menglei Chai Snap Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFIT4oBgHgl3EQfqSsD/content/2301.11326v1.pdf'} +page_content=' Sergey Tulyakov Snap Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFIT4oBgHgl3EQfqSsD/content/2301.11326v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFIT4oBgHgl3EQfqSsD/content/2301.11326v1.pdf'} +page_content='Driving ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFIT4oBgHgl3EQfqSsD/content/2301.11326v1.pdf'} +page_content='Novel view synthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFIT4oBgHgl3EQfqSsD/content/2301.11326v1.pdf'} +page_content='. +Copyright 2023 the authors. +& Smola, 2002) and the use of statistical mechanics tech- +niques to solve probabilistic problems (Hastings, 1970; +Gelfand, 2000). Here we suggest another connection be- +tween physics and ML, which relates to representation of +observables: When features and labels are represented in a +mathematical form that involves investigator choices, the +ML method (or any relevant function) ought to be written in +a form that is exactly equivariant to changes in those investi- +gator choices. These ideas appear in the physics literature as +early as Einstein (1915). They are given in the introduction +of The Classical Groups by Weyl (1946) as a motivation to +study group theory. The first sentences of Modern Classical +Physics by Thorne & Blandford (2017) say +[...] a central theme will be a Geometric Principle: +The laws of physics must all be expressible as +geometric (coordinate-independent and reference- +frame-independent) relationships between geo- +metric objects (scalars, vectors, tensors, ...) that +represent physical entities. +This principle leads to the important physical symmetries +of coordinate freedom and gauge symmetry; a small gen- +eralization would include units covariance. Each of these +symmetries has led to fundamental results in physics. Some +of these ideas are also exploited in ML, in particular in the +geometric deep learning literature (Bronstein et al., 2021; +Weiler et al., 2021). We argue—in this purely conceptual +contribution—that analogs of these symmetries could have +big impacts in ML, and enormously increase the scope of +group-equivariant methods in ML. +In natural science there are two types of symmetries (see, for +example, Section 4.1 of Rovelli & Gaul 2000). The first kind +is passive, arising from the arbitrariness of the mathemati- +cal representation described above. An example familiar to +machine learners is the equivariance of functions on graphs +to the relabeling of the graph nodes. This is an exact, pas- +sive symmetry; graph neural network architectures (GNNs) +build this passive symmetry in by design (Bruna et al., 2013; +Duvenaud et al., 2015; Gilmer et al., 2017). An example fa- +miliar to physicists is what we call units covariance, which +is the requirement that any correct description of the world +have inputs and outputs with the correct units. This passive +symmetry leads to remarkable results; we discuss these in +Section 3. +arXiv:2301.13724v1 [stat.ML] 31 Jan 2023 + +The passive symmetries of machine learning +2 +The second kind of symmetries is active. These are the ones +that must be established by observations and experiments. +The fundamental laws of physics do not seem (at current +precision) to depend on position, orientation, or time, which +in turn imply conservation of momentum, angular momen- +tum and energy (the celebrated theorem of Noether 1918). +Of course the motion of a particle depends on time and +position! But the fundamental laws governing the motion +don’t themselves appear to depend on the absolute value of +the time, nor the position of the experimental apparatus. Ac- +tive symmetries like these are empirical and can be falsified +by experimental tests. Both active and passive symmetries +can be expressed in terms of group or groupoid actions and +equivariances, but their epistemological content and range +of applicability are very different. +In this contribution, we argue that passive symmetries apply +to essentially all data analysis problems. They have implica- +tions for how we structure ML methods. While we provide +some examples, most of our contributions are conceptual: +Our contributions: +• We introduce the concept of passive symmetries to ML. +While this is an old concept in classical physics, it has +not been applied widely in ML. +• We give a formal definition of passive symmetries in +terms of group actions and explain how passive sym- +metries are always in play in data problems. +• We illustrate with toy examples how enforcing passive +symmetries can improve regressions. +• We demonstrate that imposing passive symmetries can +lead to the discovery of important hidden objects in a +data problem. +• We draw connections with causal inference. One is +that all causal graphs and mechanistic models are con- +strained to be consistent with the passive symmetries. +Another is that the determination that a data problem +has all the inputs necessary to express the symmetry +exactly looks like a causal inference. +• We provide guidance on how to structure ML models +so that they respect the passive symmetries. We call out +some current standard practices that prevent models +from obeying symmetries. +Some of the ideas seem deceptively simple, yet they have +profound implications. Since ML has not yet absorbed these +notions as naturally as physics has, we need to discuss analo- +gies to physics in some detail. We have tried to make the +discussion accessible to an ML audience; the Appendices in- +cludes a glossary that can be used to translate ideas between +physics and ML. +Figure 1. Figure depicting the difference between active and pas- +sive transformations. (Left panel): The passive or alias transforma- +tion corresponding to a rotation of the coordinates through an angle +θ in the xy-plane. The equivariance of the dynamics with respect to +this tranformation is a passive symmetry. (Right panel): The active +or alibi transformation corresponding to a rotation of the double +pendulum state through an angle −θ in the xy-plane. Equivariance +with respect to this transformation is an active symmetry. +2. Passive symmetries +Passive symmetries arise from redundancies or free param- +eters or investigator choices in the representation of data. +They are to be contrasted with the active symmetries, which +arise from observed or empirical invariances of the laws of +physics with respect to parameters, like position, velocity, +particle labeling, or angle. Passive symmetries can be es- +tablished with no need of observations, as they arise solely +from the principle that the physical world is independent +of the mathematical choices we make to describe it. The +groups involved in coordinate freedom can be large and +complicated (for example, groups of reparameterizations). +A big part of the literature on equivariant ML is implicitly +or explicitly looking at active symmetries. This is possibly +because in most problems the coordinate system is fixed +before the problem is posed, and both training and test data +are expressed in those fixed coordinates. If a data set is +made with a fixed coordinate system, but still exhibits an +observable invariance or equivariance with respect to (say) +rotations, then that represents an active symmetry. However, +cases of exact active symmetries are rare; they only really +appear in natural-science contexts like protein folding or +cosmology. For example, in a protein folding problem, the +way the protein folds may not depend on its orientation +in space (rendering the problem actively O(3) equivariant). +This finding relies on the (empirical) observation that the lo- +cal gravitational field (on Earth) does not affect the folding. +This may be approximately true or assumed or experimen- +tally established; it is an active symmetry. In contrast, the +fact that the protein folds in a way that doesn’t depend on +the coordinate system chosen to describe it is absolutely +and always true; it is not experimentally established; it is a +passive symmetry. +The relationship between active and passive symmetries is +reflected in the relationship between what are sometimes +called active and passive transformations, or alibi and alias + +Z +x +x +yThe passive symmetries of machine learning +3 +transformations, depicted in Figure 1. An active or alibi +transformation is one in which the objects of study are +moved (rotated, translated, interchanged, etc.). A passive or +alias transformation is one in which the coordinate system in +which the objects of study are described is changed (rotated, +translated, relabeled, etc.). Mathematically, the two kinds +of transformations seem very similar: For example, how do +we know whether we rotated all the vectors in our problem +by 30 deg, or else rotated the coordinate system used by +−30 deg? The answer is that if you rotated absolutely all +the vectors (and tensors) in your problem, including possi- +bly many latent physical vectors, then there would be no +mathematical difference. But in real problems, where some +vectors can’t be actively rotated (think, for example of the +local gravitational-field vector, or the vector pointing to- +wards the Sun), or some may not be known or measurable, +the two kinds of transformations are different. +The protein-folding example suggests that it is hard to im- +plement or enforce a true passive symmetry in a real data- +analysis problem. It requires us to incorporate all relevant +contextual information. How do we know if all relevant +features are part of our data set? We could perform the +protein-folding experiment in a closed, isolated environ- +ment to make sure no external forces are in play; this is +impossible for many practical applications, and furthermore +there could still exist fundamental constants that are not +part of our model or knowledge (see Section 5). Another +approach is to perform the experiment multiple times after +actively putting the molecules into different orientations. +If the protein folds differently, we learn that the problem +is not symmetric with respect to the 3d coordinates of the +molecule, and therefore when a rotation is performed there +must be at least one more vector that needs to be rotated +as well (for instance, the gravity vector or the eigenvectors +of a stress tensor, say). This identification of all necessary +inputs to establish the passive symmetry is similar to the +problem of performing interventions to learn the existence +of confounding factors in causal inference. We will come +back to the connections to causality in Section 6. +Once a passive symmetry—and all relevant contextual +information—is identified, we want to write the data analy- +sis problem or learned function such that it is exactly equiv- +ariant with respect to the relevant group: If the representa- +tion or coordinate system of the inputs is changed, the repre- +sentation or coordinate system of the output should change +correspondingly. ML methods that are not constrained to +respect passive symmetries are doomed to make certain +kinds of mistakes. We will provide some examples thereof +in Section 8. +The most restrictive form of the Geometric Principle quoted +in Section 1 states that physical law must be written in +terms of vectors, tensors, and coordinate-invariant scalars. +These objects can only be combined by rules set out in +the Ricci calculus (Ricci & Levi-Civita 1900; sometimes +Einstein summation notation, Einstein 1916). This calculus +was introduced to make objects equivariant to coordinate +diffeomorphisms on curved manifolds. In the Ricci calculus, +objects are written in index notation (a scalar has no indices, +a vector has one index, and a k-tensor has k indices), outer +products are formed, and only certain kinds of sums over +pairs of indices are permitted. When the inputs to a function +are scalars, vectors, and tensors, and the function conforms +to the rules of the Ricci calculus, the function will produce +a geometric output (a scalar, vector, or tensor,1 depending +on the number of unsummed indices), and the function +will be precisely equivariant to rotations and reflections of +the coordinate system. This is how a large class of passive +symmetries are enforced in physics contexts. +There are many other passive symmetries, including co- +ordinate diffeomorphisms, reparameterizations (including +canonical transformations), units covariance (see Section 3), +and gauge. Some of these are easy to implement and some +are difficult; not all passive symmetries have practical im- +plementations available at present. +The passive coordinate diffeomorphism symmetry on curved +manifolds has been tremendously important in the devel- +opment of contemporary physics. Einstein, for example, +found the equations of general relativity by looking at every +expression to some polynomial degree consistent with the +passive symmetry enforced by the Ricci calculus (he called +this family of polynomials covariant2 ) until he found the +one that reduced to Newtonian gravity in the weak-field +limit (Einstein, 1915). Even before that, while not expressed +in terms of passive symmetry, the discovery of special rela- +tivity (Einstein, 1905) can be thought of as the decision to +move an active symmetry (the observation that the speed of +light is the same for all observers) into a passive symmetry +(Lorentz invariance). We stress that these insights were truly +profound (Earman & Glymour, 1978); they revolutionized +physics; not too long before Einstein, such considerations +would have been highly unusual. Similarly, passive symme- +tries do not feature in today’s ML practice—if the develop- +ment of physics is any indication, their potential could be +very significant. +1It should be noted here that with the word “vector” and “tensor” +here we are making specific technical reference to true vectors +and tensors in 3-space, subject to passive O(3) symmetries, like +(physical) velocities, accelerations, and stress tensors. We are not +including arbitrary lists or tables of data or coefficients, which are +sometimes called “vectors” and “tensors” in ML contexts. +2Following the lead of physics, we could in principle call ML +methods equivariant to passive symmetries “covariant”. + +The passive symmetries of machine learning +4 +3. Example: Units covariance +Perhaps the most universal passive symmetry is units +covariance—the behavior of a system doesn’t depend on the +units system in which we write the measured quantities. It +is a passive symmetry with extremely useful consequences. +Consider a mass m near the surface of the Earth, close +enough to the surface such that the gravitational field can +be considered to be determined by a constant (not spatially +varying) vector with magnitude g and direction downwards. +Question 1: If this mass m is dropped (released at rest) from +a height h from above the ground, how much time T does +it take to fall to the ground? Question 2: If this mass m is +launched from the surface at a velocity of magnitude v at an +angle θ to the horizontal, how much horizontal distance L +will it fly before it hits the surface again? Assume that only +m, g, h come in to the solution;3 assume that the height h +and the velocity v are both small enough that air resistance, +say, can be ignored. +The answers to these questions are almost completely de- +termined by dimensional (or units-covariance) arguments. +The mass m has units of kg, the gravitational acceleration +magnitude g has units of m s−2, the velocity magnitude +v has units of m s−1, the time T has units of s, and the +lengths h and L have units of m. The angle θ is dimension- +less. The only possible combination of m, g, h that has units +of time is α +� +h/g, where α is a dimensionless constant, +which doesn’t depend on any of the inputs. The only pos- +sible combination of m, g, v, θ that has units of length is +β(θ) v2/g, where β(θ) is a dimensionless function of only +one dimensionless input. That is, both Questions 1 and 2 +can be answered up to a dimensionless prefactor without +any considerations beyond those of the units of the inputs +and outputs, and without any training data. And both of +those answers don’t depend in any way on the input mass +m, which is a fundamental observation (Einstein, 1915). +This shows that a function can sometimes be inferred from +units covariance only, i.e., from a purely passive symmetry. +Units covariance has been introduced to ML methods pre- +viously (Villar et al., 2022; Bakarji et al., 2022; Xie et al., +2022). It can help with training, predictive accuracy, and +out-of-sample generalization. In particular, out-of-sample +generalization improves because the enforcement of the +symmetry enforces scaling properties of the learned func- +tion. These, in turn, help make predictions when the test +data are outside the range of the training data. +4. Formal definition +Consider X to be the state space of a specific physical sys- +tem (for instance x ∈ X could be the positions, velocities, +3We will return to this seemingly innocuous point below. +masses, spins, and charges of a set of particles at a given +time). We consider a family of maps {Φi : X → Hi}i∈I +where the Hi are spaces of observations or encodings in- +dexed by i ∈ I (for instance Φi(x) could be the total me- +chanical energy of the system x at a set of times measured +in joules, and Φj(x) could be in kcal, etc). The family of +maps {Φi}i∈I not only describes the possible observables +but also a way to express those observables in a specific +coordinate system and with specific units. +We say that two encodings Φi and Φj are compatible if there +exists an invertible morphism βi +j : Φi(X) → Φj(X) that +makes the diagram (1) commute. +X +X +Hi +Hj +id +Φi +Φj +βi +j +(1) +Note that not all observables are compatible. We expect +that observables of different types (for instance, masses +and positions) are not compatible. Which observables are +compatible depend on the definition of the spaces Hi and +the invertible morphisms β between them. +The passive symmetries are the groupoid of invertible mor- +phisms β between compatible encodings that make the di- +agram commute P = {β : Φi(X) → Φj(X) s.t. β ◦ Φi = +Φj, i, j ∈ I}, where the groupoid operator is the com- +position. We note that this is a groupoid and not a group +because not every pair of transformations are composable. +The groupoid of passive symmetries P consists of all the +possible changes of coordinates or automorphisms between +encodings or observables. +Critical to the definition is that we establish beforehand +the spaces of possible observables or encodings Hi, the +observables or encodings Φi; and the class of invertible mor- +phisms β. For instance, Hi can be in the category of smooth +manifolds with smooth diffeomorphisms, or in the category +of vector spaces with invertible linear transformations, or +orthogonal transformations. +For example, take X to be a protein. Each map Φi could +encode a list of positions of each of its atoms in some coor- +dinate system. The passive symmetries include reordering +of the atoms, and changes of the coordinate system by any +invertible morphism. We may restrict Hi to the category of +smooth manifolds where the β’s are diffeomorphisms, or +we may only allow the β’s to be orthogonal transformations +that fix the origin. Restricting the space of valid coordi- +nates naturally restricts the valid changes of coordinates and +therefore the space of passive symmetries. +Active symmetries, on the other hand, can be thought as +transformations of the world that preserve an observable +property. They involve interventions in the physical system, +and therefore they are typically empirical and approximate. + +The passive symmetries of machine learning +5 +Not every passive symmetry corresponds to an active sym- +metry, nor vice versa. +Imposing a passive symmetry on the structure of an ML +model can permit the discovery of scalings, structures, or +missing elements in the physical description of, or predic- +tions about, the problem. We illustrate these ideas with some +toy examples below. The passive symmetries are seemingly +trivial statements about the world, but they lead to strong +constraints on the laws of physics, and deliver scaling ar- +guments that solve real problems in physics. They can also +constrain ML in valuable ways. We conjecture that enforc- +ing passive symmetries in ML and data-analysis tasks will +lead to generalization improvements in a wide range of +circumstances. In particular, we make this conjecture even +for problems in which no (or few) active symmetries are +present. After all, most problems (like reading handwriting +or predicting gravitational trajectories near the surface of +the Earth) are not equivariant to rotations, reflections, and +translations, but they are all, in their data-generating pro- +cesses, exactly coordinate free, and exactly agnostic to the +choices made in data representation. +5. Experiments and examples +Black-body radiation: An important moment in the his- +tory of physics was the discovery that the electromagnetic +radiation intensity Bλ (energy per time per area per solid +angle per wavelength) of thermal black-body radiation can +be described with a simple equation (Planck, 1901) +Bλ(λ) = 2 h c2 +λ5 +1 +exp +h c +λ k T − 1 , +(2) +where h is Planck’s constant, c is the speed of light, λ is +the wavelength of the electromagnetic radiation, k is Boltz- +mann’s constant, and T is the temperature. In finding this +formula, Planck had to posit the existence (and units) of +the constant h = 6.62607015 × 10−34 kg m2 s−1 (Planck’s +original value was presented in erg s, which are different +units but the same dimensions). Prior to the introduction +of h, the only dimensionally acceptable expression for the +black-body radiation intensity was Bλ(λ) = 2 c k T/λ4, +which is the long-wavelength (infrared) or high-temperature +limit. Planck’s discovery solved the “ultraviolet catastrophe” +of classical physics. This is the problem that, classically, +the black-body spectrum, or any thermal object, ought to +contain infinite numbers of excited modes at short wave- +lengths, or high frequencies, and thus infinite energy density. +Planck’s solution seeded the development of quantum me- +chanics, which governs the behavior of all matter at small +scales, and which cuts off the ultraviolet modes through +quantization of energy. +Planck’s problem can be solved almost directly with the +passive symmetry of units covariance. That is, the exponen- +tial cut-off of the intensity appears at a wavelength set by +the temperature and a new constant, that must have units +of action (or action times c, or action divided by k, or one +equivalent in terms of the lattice of dimensional features, +see Villar et al. 2022). +In Figure 2 (left) we perform the following toy experiment: +We generate noisy samples of intensities as a function of +wavelength and temperature according to (2), and the learn- +ing task is to predict the intensity for different values of +wavelengths and temperatures. We perform a units covariant +regression (employing the approach of Villar et al. 2022) us- +ing only λ, T, c, k; a units covariant regression with an extra +dimensional constant found by cross-validation; and a stan- +dard MLP with no units constraints. Our results show that +no units-covariant regression for the intensity as a function +of λ, T, c, k can reproduce accurately the intensity Bλ. How- +ever when the regression is permitted to introduce a new +dimensional constant (and remains units covariant given +the new constant), it finds a constant with units (and, less +precisely, magnitude) that is consistent with h (or h times a +combination of c and k). The units covariant model outper- +forms the baseline MLP. Again, this shows that the passive +symmetry leads to powerful capability. +Springy double pendulum: The double pendulum con- +nected by springs is a toy example often used in equivariant +ML demonstrations (Finzi et al., 2021; Yao et al., 2021; +Villar et al., 2022). The final conditions (position and veloc- +ities of both masses after elapsed time T) are related to the +initial conditions (position and velocities of the masses at +the initial time), and the dynamics is classically chaotic. +The system is subject to a passive O(3) symmetry (equivari- +ance with respect to orthogonal coordinate transformations), +and an active O(2) symmetry (equivariance with respect +to rotations and reflections in the 2-d plane normal to the +gravity). The O(3) symmetry is passive, because it is guar- +anteed by the fact that all vectors must be described in a +coordinate system; nothing physical can change as the vec- +tors undergo passive transformations because of coordinate- +system changes. The O(2) symmetry is active, because it is +an experimental fact that if the initial conditions are changed +by an active or alibi rotation in the plane perpendicular to +gravity, the dynamics and final state rotate accordingly. Here +we can see that the O(2) active symmetry corresponds to +the set of transformations in O(3) that fix the gravity vector. +The O(3) passive symmetry requires that the coordinates +of all relevant vectors are transformed identically, the posi- +tions and momenta of both masses and the gravity vector. +If the model doesn’t contain all relevant vectors as inputs +then the predictions will not be O(3) equivariant. We per- +form an experiment in which we predict the dynamics of the +double pendulum using O(3)-equivariant models. The sym- +metries are implemented by converting the network inputs + +The passive symmetries of machine learning +6 +10 +8 +10 +7 +10 +6 +10 +5 +10 +4 +wavelength +106 +109 +1012 +1015 +1018 +1021 +intensity +test labels +units covariant without extra dimensional constant, RMSE=6.35 +units covariant with extra dimensional constant, RMSE=0.39 +standard MLP (no covariance), RMSE=0.64 +0.8 +0 +0.8 +ex q1 +0.8 +0 +0.8 +ex p1 +0.8 +0 +0.8 +ey q1 +0.6 +0 +0.6 +ey p1 +4 +3.2 +2.4 +ez q1 +1 +0 +1 +ez p1 +1.6 +0.8 +0 +0.8 +ex q2 +0.6 +0 +0.6 +ex p2 +0.8 +0 +0.8 +ey q2 +0.8 +0 +0.8 +ey p2 +5.6 +4.8 +4 +3.2 +ez q2 +0.8 +0 +0.8 +ez p2 +0 +30 +60 +90 +120 +time +0 +30 +60 +90 +120 +time +0 +30 +60 +90 +120 +time +0 +30 +60 +90 +120 +time +Ground truth +Known-g +Learned-g +Ground truth +Known-g +Learned-g +Figure 2. (Left panel) We predict the intensity of black body radiation as a function of wavelength and temperature. For all experiments +we use an MLP consisting of 3 layers with 20 hidden units each. The standard MLP uses wavelength and temperature as features and it +doesn’t require the output to be dimensionally correct. The units covariant without extra constant learns a scaling of the only dimensionally +correct object one can construct with inputs λ, T, c, k (see description main in text). The units covariant with extra dimensional constant +incorporates a constant with units [kg, m, s, K] ∈ [−1, 0, 1]4 as an input feature, it performs a units covariant regression with the original +features λ, T, c, k and the extra constant. It then selects a constant with low validation error and reports the results on the test set. The +constant learned for the depicted plot is 1.61e52 kg−1m−1s−1K−1, which is the same units and similar magnitude to the valid physical +combination c k h−2. (Right panel) Performance of learning the dynamics of the springy double pendulum. We consider the three models +(described in main text): (Known-g) an O(3)-equivariant model where the gravity is an input to the model, (No-g) an O(3)-equivariant +model where the gravity is not given, and (learned-g) an O(3)-equivariant model that uses the position and momenta as well as an +unknown vector that the model learns. The results show that O(3)-equivariance permits the learning of the gravity vector from data with +only minimal impact in performance. See Appendix A for a more detailed description of the experiment. +(scalars and components of vectors) into invariant scalar +quantities according to the Ricci calculus (which explic- +itly encodes O(3)), building the model in the space of the +invariant scalars (as per Villar et al. 2021). The models im- +plemented are (Known-g)—an O(3)-equivariant model that +has the positions, momenta and gravity vector all as features +(similar to the models in Villar et al. 2021; Yao et al. 2021); +(No-g)—an O(3)-equivariant model that that is missing +the gravity vector as an input feature; and (Learned-g)–an +O(3)-equivariant model that has the position, momenta and +an extra unknown vector as features. The latter model op- +timizes model weights along with the unknown vector. In +the right panel of Figure 2 we show the performance of the +three models. We remark that in (Learned-g), the learned +vector in the performed experiments was nearly parallel to +the true (but unknown) gravity vector g; the angle between +the learned and true gravity vector was 0.00016 radians. +6. Connections with causality +There is nothing statistical about the notion of passive sym- +metries, and thus everything we have said above also applies +to causal models (Peters et al., 2017). There are, however, a +few comments specific to causality. +The passive symmetry discussed in Section 3—and indeed +all passive symmetries—can also deliver information per- +taining to the (hard problem of) inference of causal structure: +treating g as a constant, we can construct a structural causal +model with the following vertices: (a) an initial value of v, +(b) a value of m, chosen independently, and (c) a final value +of L, affected by a noise term θ. Time ordering implies that +possible causal arrows are from v, m, θ to L. As argued +above, dimensional analysis rules out the arrow m → L, +leaving us with the non-trivial result that in the causal graph, +only v, θ cause L. As in Section 3, this conclusion can be +reached without any training data or interventions. +That said, dimensional analysis makes a strong assumption, +which is that all relevant quantities for predicting L have +been specified in the list m, g, v, θ. For example, if the pro- +jectile is large enough or the speed v is high enough, air +resistance will come into play, and the size of the object and +the density of air will enter, bringing new variables and new +combinations of variables that matter to the answer. This +difficulty is related to the problem in causal inference of +knowing or specifying all possible confounding variables. +This can also be linked to the notion of experimental inter- +ventions. Suppose we assume that only certain quantities +come into the solution (say, m, g, h). How would we con- +firm this in practice? In essence, this is not a probabilistic +statement, but one about the behavior of a system under +interventions. A set of experiments can indicate that a cer- + +Scenario +Known-g +Learned-g +No-g +State.RelErr +.0052 ± 0.0004 +.0054 ± 0.0011 +.3160 ± .0014The passive symmetries of machine learning +7 +tain outcome (or effect variable) depends on a certain set of +input (cause) variables but is independent of certain other +potential cause variables. In this case, the physical law is +not inferred from dimensional arguments alone, but from a +combination of dimensional and causal arguments. +Even if interventions are not available (e.g., for g), physicists +trying to infer a law will not do so based (purely) on input- +output data: they will have prior knowledge from related +problems informing them as to which variables are relevant. +E.g., we may know from having previously solved a related +problem that we expect a problem to depend on g. This +is a form of qualitative transfer that we expect will also +become relevant for model transfer in ML (Rojas-Carulla +et al., 2018). +7. Connections to current ML practice +Most ML implementations don’t impose exact symmetries. +Sometimes they approximate equivariances by means of +data augmentation (Chen et al., 2020; Huang et al., 2022). +In the present work we focus on exact symmetries: Given +data spaces X and Y and a group G acting on X and Y , +equivariant ML restricts the function space to those satis- +fying f(g · x) = g · f(x) for all f ∈ F, g ∈ G, x ∈ X. +There are two main approaches to perform optimization in +the space of equivariant functions: +• Explicitly parameterizing the space of equivariant func- +tions via equivariant layers or weight sharing (Cohen & +Welling, 2016; Kondor & Trivedi, 2018; Thomas et al., +2018; Geiger & Smidt, 2022; Finzi et al., 2020; 2021). +• Finding a set of invariant features and expressing the +equivariant functions in terms of those features (Villar +et al., 2021; Blum-Smith & Villar, 2022). +The two approaches are theoretically equivalent, but their +practical implementation may be dramatically different. For +example, efficiently computing a complete generating set +of invariant/equivariant features may be prohibitive. On the +other hand, in some cases it may hard to construct a perfectly +invariant/equivariant layer (or family of approximating func- +tions); more often, it may be possible to construct such a +family, but we may lack proof that they are universal approx- +imators of all invariant/equivariant functions within some +well-defined context, even in a limiting sense. +Aside from the previously mentioned results in Convolu- +tional/Graph Neural Networks, another example of success- +ful exact universal parametrization of a family of functions +is the implementation of symplectic networks, which ex- +actly preserve a (not-necessarily-known) differential 2-form +on a manifold by acting on ambient space (Jin et al., 2020; +Burby et al., 2020). Even the non-trivial diffeomorphism +symmetries of general relativity have been considered for +ML (Weiler et al., 2021). +Equivariant ML models can predict the properties and be- +haviour of physical systems (see Cheng et al. 2019), and +have plenty of scientific applications (Batzner et al., 2022; +Musaelian et al., 2022; St¨ark et al., 2022; Yu et al., 2021; +Wang et al., 2022). The implicit bias, generalization error, +and sample complexity of equivariant ML models have been +recently studied (Lawrence et al., 2021; Bietti et al., 2021; +Elesedy & Zaidi, 2021; Elesedy, 2021; Mei et al., 2021). +8. Dos and Don’ts +MacKay famously wrote (see Muldoon 2021) +Principal Component Analysis is a dimensionally +invalid method that gives people a delusion that +they are doing something useful with their data. +If you change the units that one of the variables +is measured in, it will change all the “principal +components” +This comment is aligned with our mission, but also mis- +leading: If a rectangular data set contains only data with +identical units (that is, all features of all records have the +same units), then PCA does excactly the right thing. That +said, if a rectangular data set has features with different units +(for example, if every record contains a position, a tempera- +ture, a voltage, and a few intensities), then indeed the output +of PCA will be extremely sensitive to the units system in +which the features are recorded. If PCA is run on such a +data set, the subsequent data model or data manipulations +will be, by construction, asymmetric or not consistent with +the passive symmetry of units covariance. The choice to use +PCA on such a data set is the choice to be wrong. +Consider a kernel function with inputs that are lists of fea- +tures with different units. If the kernel function involves, +say, an exponential of a sum of squares of differences of the +input features, the output of the kernel function cannot obey +the passive symmetry of units covariance. Quantities with +different units cannot be summed, and dimensional quanti- +ties cannot be exponentiated. On the other hand, if a kernel +function can be chosen that is units covariant (e.g., if all +features have the same units, or if the kernel is constructed +from tensor products of kernels, each of which uses only one +type of input), then the result of a kernel algorithm can be +covariant. These considerations are relevant for maximum +margin hyperplane (in SVMs), eigenvectors in kernel PCA +(Sch¨olkopf & Smola, 2002), or Gaussian processes. +Learning involves optimization. Optimization is of a scalar +cost function (a number, which is a function of many param- +eters). If passive geometric groups are in play, like O(3), +the parameters that are explicitly or implicitly components +of vectors can only be combined into the scalar objective +through the Euclidean norm. Otherwise the scalar objective +isn’t scalar in the geometric sense of “invariant to O(3)”, + +The passive symmetries of machine learning +8 +and the optimization won’t return a result that is invariant +(or equivariant) to O(3). Similarly, if the components of the +vector are normalized differently before they are summed +in quadrature, the objective won’t be invariant to O(3). And +similarly, if all the different contributions to the objective +aren’t converted to the same units before being combined +into the objective, then the model won’t be units covariant. +The common practices of making objectives with functional +forms other than Euclidean norm, normalizing features with +data ranges, and combining features with different units, all +make common ML methods, by construction, inconsistent +with the passive symmetries in play. +Neural nets, in their current form, violate many rules. +For example: Transcendental functions like exp() and +arctanh() and most other nonlinear functions can only +be applied to scalars—that is, not components of vectors +or tensors but only scalars—and only dimensionless. That +means that the nonlinearities in neural networks are predi- +cated on the weights removing the units of the input features, +and the linear combinations performing some kind of dot +products on the inputs. That, in turn, means that the internal +weights in the bottom and top layers of a neural network +implicitly have geometric properties and units. They have +the geometric properties and units such that the latent vari- +ables passed into the nonlinear functions are dimensionless +scalars. Because they have these properties, a trained neural +network cannot be covariant in the end, unless the inputs +and outputs are already covariant scalars. +There are exceptions to the restrictions on nonlinear func- +tions: If nonlinearities are mathematically homogeneous, as +it is for a pure monomial, or for the RELU function, dimen- +sional scalars (but not vector or tensor components) can be +taken as inputs. It’s interesting to ask whether the success +of RELU in ML might be related to its homogeneity. +L1 and L∞ norms are almost always inconsistent with the +passive symmetries. This is because the sum of absolute +values of input components, and the maximum of inputs, +are rarely either geometrically, or from a units perspective, +covariant. There is a rare exception if all features have the +same units, and none of the features are components of +geometric objects. +Similarly, regularizers such as those favoring flat loss min- +ima (Hochreiter & Schmidhuber, 1997; Dinh et al., 2017; +Petzka et al., 2021) are often not units covariant, changing +their values under certain weight transformations that leave +the overall function invariant. If reformulated as a regular- +izer that is a covariant function of the training points, this +problem vanishes (von Luxburg et al., 2004). +Finally, we mention that passive symmetries play a cru- +cial role also when it comes to latent variable models and +ICA, since unobserved latent factors usually come with a +large class of allowed gauge transformations (permutations, +coordinate-wise nonlinear transformations) which should +be incorporated correctly when studying notions of identifi- +ability (Khemakhem et al., 2020; Buchholz et al., 2022). +9. Discussion +In this conceptual contribution, we argue that passive sym- +metries are in play in essentially all ML or data-analysis +tasks. They are exact, and true by definition, since they +emerge from the redundancies or freedom in coordinate +systems, units, or data representation. Enforcement of these +symmetries should improve enormously the generalization +capabilities of ML methods. We demonstrate this with toy +examples. +In practice, implementation of the passive symmetries in an +ML problem might be very difficult. One reason is that the +symmetries are only exact when all relevant problem param- +eters (including often fundamental, unvaried constants) are +known and included in the learning problem. If the prob- +lem has a passive symmetry by a group G, but there are +missing elements K in the problem formulation (such as +Planck’s constant or the gravity vector in Section 5), then +the symmetry that is actually in play is the subgroup H +of G that fixes K. Naively there should be no difference +in the in-distribution performance between enforcing the +symmetry by H, or including K to the inputs and enforcing +the symmetry induced by G. However, using the full group +equivariance is conceptually more elegant and it allows for +out-of-distribution generalization (the model can generalize +to settings where K has changed). These unknown constants +or features K are pieces of essential contextual information +and can be hard to find or learn. In our toy examples we +show that with sufficient knowledge of the problem (rich +training data and knowledge of the group of passive sym- +metries) the relevant constant K can be learned from data, +including the Planck constant (for the blackbody-radiation +problem) and the gravitational acceleration vector (for the +double-pendulum example). Identifiability issues may arise +when more constants or non-constant features are missing. +Another difficulty is that some kinds of symmetries are +hard to enforce. For example, complete coordinate diffeo- +morphisms and problem reparameterizations involve enor- +mous groups which are hard to implement in a realistic +ML method. That said, many groups have been imple- +mented usefully, including translations, rotations, permuta- +tions, changes of units, and some coordinate transformations +(see Weiler et al. 2021 for a review of the latter). +In addition to the exact (and true by definition) passive +symmetries, and the observed active symmetries, there are +other kinds of approximate or weakly broken symmetries +we might call observer symmetries. These arise from the + +The passive symmetries of machine learning +9 +point that the content of a data record (an image, say) is +independent of the minor choices made by the observer in +taking that data record (shooting the image, say). The details +of the six-axis location and orientation of the camera, and +of the exposure time and focus, can be changed without +changing the semantic or label content of the image. These +symmetries are approximate, because these changes don’t +lead to invertible changes in the recorded data; there is no +group or groupoid in the space of the data. However, the +success of convolutional structure in image models might +have to do with the importance of these observer symmetries. +There is much more to do in this space. +Acknowledgement: It is a pleasure to thank Ben Blum- +Smith (JHU) for valuable discussions. This project was +started at the meeting Machine Learning for Science at +Schloss Dagstuhl, 2022 September 18–23. SV was partially +supported by ONR N00014-22-1-2126, the NSF–Simons +Research Collaboration on the Mathematical and Scien- +tific Foundations of Deep Learning (MoDL) (NSF DMS +2031985), NSF CISE 2212457, and an AI2AI Amazon re- +search award. +References +Bakarji, J., Callaham, J., Brunton, S. L., and Kutz, J. 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Springy double pendulum +We consider the dissipationless spherical double pendulum with springs, with a pivot o and two masses connected by springs. +The kinetic energy T and potential energy U of the system are given by +KE = |p1|2 +2m1 ++ |p2|2 +2m2 +, +(3) +PE = 1 +2k1(|q1 − qo| − l1)2 + 1 +2k2(|q2 − q1| − l2)2 − m1 g · (q1 − qo) − m2 g · (q2 − qo), +(4) +where q1, p1 are the position and momentum vectors for mass m1, similarly q2, p2 for mass m2, and a position qo for the +pivot. The springs have scalar spring constants k1, k2, and natural lengths l1, l2. The gravitational acceleration vector is g. +In this work, we fix qo with values (0, 0, 0) in base length units and g with (0, 0, −1) in base acceleration units, as well as +(m1, m2, k1, k2, l1, l2) set to (1, 1, 1, 1, 1, 1), but with each element of that list having appropriate base units. +The prediction task is to learn the positions and momenta over a set of T later times t given the initializations of the +pendulum positions and momenta at t0, +z(t) = (q1(t), q2(t), p1(t), p2(t)), +t ∈ {t0, t1, . . . , tT }. +(5) +The training inputs consist of N = 500 different initializations of the pendulum positions and momenta {z(i)(t(i) +0 )}N +i=1, and +the labels are the set of positions and momenta {z(i)(t(i) +1 ), z(i)(t(i) +2 ), . . . , z(i)(t(i) +T )}N +i=1 with T = 5. The model is evaluated +on a test data set with T = 150 and t0 = 0. +For the same prediction task, we consider three different O(3)-equivariant models, fKnown-g, fLearned-g and fNo-g, depending +how the gravitational acceleration vector g is involved. +Known-g +The model fKnown-g is a function that predicts the dynamics: +fKnown-g : (R3)4 × R3 × R3 × R → (R3)4 +(z(0), qo, g, ∆t) �→ ˆz(∆t) +(6) +where g is known as (0, 0, −1) in the base acceleration units and used with positions and momenta as input features. +Learned-g +The model fLearned-g is a function that predicts the dynamics: +fLearned-g : (R3)4 × R3 × R → (R3)4 +(z(0), qo, ∆t) �→ ˆz(∆t) +(7) +where g is unknown but set as an learnable variable and used with positions and momenta as input features. +No-g +The model fNo-g is a function that predicts the dynamics: +fNo-g : (R3)4 × R3 × R → (R3)4 +(z(0), qo, ∆t) �→ ˆz(∆t) +(8) +where g is unknown and not used as an input feature. +We evaluate the performance of the three predictive models based on the state relative error at a given time t in terms of the +positions and momenta of the masses, +State.RelErr(t) = +� +(ˆz(t) − z(t))⊤(ˆz(t) − z(t)) +� +ˆz(t)⊤ˆz(t) + +� +z(t)⊤z(t) +, +t ∈ {t1, . . . , tT }, +(9) +where ˆz(t) denotes the predicted positions and momenta at time t and z(t) the ground truth. + +The passive symmetries of machine learning +13 +B. Glossary +active symmetry: A symmetry is active when it is an observed or empirical regularity of the laws of physics. Examples +include the observation that the fundamental laws don’t depend on the location or time at which the experiment takes place. +conservation law: We say that a quantity obeys a conservation law if changes in that quantity (with time) inside some closed +volume can are quantitatively explained by fluxes of that quantity through the surface of that volume. Active symmetries can +lead to conservation laws in dynamical systems (Noether, 1918). +coordinate freedom: When physical quantities are measured, or represented in a computer, they must be expressed in some +coordinate system. The redundancy of this representation—the fact that the investigator had many choices for the coordinate +system—leads to the passive symmetry coordinate freedom: If the inputs to a physics problem are moved to a different +coordinate system (because of a change in the origin or orientation), the outputs of the problem must be correspondingly +moved. In much of the literature “coordinate freedom” is only used in relationship to general covariance, but it applies in all +contexts (including non-physics contexts) in which a coordinate system has been chosen. +covariance: When a physical law is written in a way that is equivariant with respect to all (or some) passive symmetries, +then the law is sometimes said to be covariant. +dimensional analysis: The technique in physics of deducing scalings by consideration of units covariance is dimensional +analysis. +equivariance: Let G be a group that acts on vector spaces X and Y as ρX and ρY respectively. We say that a function +f : X → Y is equivariant if for any group element g ∈ G and any possible input x, the function obeys f(ρX(g)x) = +ρY (g) · f(x). The actions of G in X and Y induce an action on the space of maps from X to Y . If f ∈ Maps(X,Y) then +g · f = ρY (g) ◦ f ◦ ρX(g)−1. The equivariant maps are the fixed points of this action. Equivariances define symmetries in +the space of maps. +gauge freedom: Some physical quantities in field theories (for example the vector potential in electromagnetism) have +additional degrees of freedom that go beyond the choice of coordinate system and units. These freedoms lead to additional +passive symmetries that are known as gauge freedom. +general covariance: The covariance of relevance in general relativity (Einstein, 1916) is known as general covariance. +Because general relativity is a metric theory in 3 + 1 spacetime dimensions with invariance with respect to arbitrary +diffeomorphisms, this is a very strong symmetry. General covariance is sometimes called “coordinate freedom”, but it is a +special case thereof. +invariance: An equivariance in which the action in the output space is trivial is called an invariance. Physicists sometimes +use invariant (gauge invariant, for example) for things we would call covariant. +passive symmetry: A symmetry is passive when it arises from a choice in the representation of the data. Examples include +coordinate freedom, gauge freedom, and units covariance. These symmetries are exact and true by definition. +scalar: A number (with or without units), whose value does not depend on the coordinate system in which it is represented, +is a scalar. Thus, say, the charge of a particle is a scalar, but the x coordinate of its velocity is not a scalar. +symmetry: Given a mathematical object X of any sort, (like a manifold, metric space, equation, etc), any mapping of the +object onto itself that preserves the corresponding structure is a symmetry. +tensor: A linear function of k − 1 vectors that outputs a vector, or a linear function of k vectors that outputs a scalar, is a +k-tensor. A rectangular array of data is not usually a tensor according to this definition. A vector can be seen as a 1-tensor, +and a scalar can be seen as a 0-tensor. +units: All physical quantities are measured with a system of what we call units. A quantity can be transformed from one +unit system to another by multiplication with a dimensionless number. Almost all quantities—including almost all scalars, +vectors, and tensors—have units. +units covariance: The left-hand side and the right-hand side of any equation must have the same units. This symmetry is +called (by us) units covariance (contra Villar et al. 2022 where it is called “units equivariance”). +vector: An ordered list of d numbers, all of which have the same units, that is subject to the passive O(d) symmetry +corresponding to coordinate-system rotations, is a vector in d dimensions. A generic list of d features is not usually a vector + +The passive symmetries of machine learning +14 +according to this definition. The inner (or dot) product of two vectors produces a scalar; for this reason a vector can be seen +as a 1-tensor. + diff --git a/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf b/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6a61664b37861ef0c9cb94d384648c714d9aae2e --- /dev/null +++ b/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7a9c049971dec30de3059c0a3b129c5d4f45874271d6a99b46218141aa669b59 +size 1386535 diff --git a/jtFMT4oBgHgl3EQf6DFB/vector_store/index.faiss b/jtFMT4oBgHgl3EQf6DFB/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..a021d6329a45a3183621d0d346a2705986b158ba --- /dev/null +++ b/jtFMT4oBgHgl3EQf6DFB/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:708186218d4db381d06b2982b7b008bf6026abd2d8a5142401ca1077dc2084dc +size 5111853 diff --git a/kdAyT4oBgHgl3EQfYPeA/content/tmp_files/2301.00199v1.pdf.txt b/kdAyT4oBgHgl3EQfYPeA/content/tmp_files/2301.00199v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..071b79b3f6e933ca69ec5118ca06666742a47e18 --- /dev/null +++ b/kdAyT4oBgHgl3EQfYPeA/content/tmp_files/2301.00199v1.pdf.txt @@ -0,0 +1,2669 @@ +Action Codes∗ +Frits Vaandrager1 and Thorsten Wißmann2 +1Radboud University, F.Vaandrager@cs.ru.nl +2Radboud University, Thorsten.Wissmann@ru.nl +January 3, 2023 +Abstract +We provide a new perspective on the problem how high-level state machine models +with abstract actions can be related to low-level models in which these actions are +refined by sequences of concrete actions. We describe the connection between high-level +and low-level actions using action codes, a variation of the prefix codes known from +coding theory. For each action code R, we introduce a contraction operator αR that +turns a low-level model M into a high-level model, and a refinement operator ϱR that +transforms a high-level model N into a low-level model. We establish a Galois connection +ϱR(N) ⊑ M ⇔ N ⊑ αR(M), where ⊑ denotes the well-known simulation preorder. In +practice, we typically want to obtain an overapproximation of model M. To this end, we +also introduce a concretization operator γR. This operator behaves like the refinement +operator, but adds arbitrary behavior at intermediate points during a refinement. We +establish a second Galois connection αR(M) ⊑ N ⇔ M ⊑ γR(N). We show how an +action code may be used to construct an adaptor that translates between concrete and +abstract inputs and outputs during learning and conformance testing of a black-box +system. If M models a black-box system then αR(M) describes the behavior that can +be observed by a tester/learner that interacts with this system via an adaptor derived +from code R. Whenever we have established that αR(M) implements (or conforms to) +N, we may conclude that M implements (or conforms to) γR(N). +1 +Introduction +Labeled transition systems (LTSs) constitute one of the most fundamental modeling mech- +anisms in Computer Science. An LTS consists of a directed graph whose nodes represent +states and whose edges are labeled with actions and represent state transitions. LTS-based +formalisms such as Finite Automata [19], Finite State Machines [22], I/O automata [23], +IOTSs [32], and process algebras [4] have been widely used to model and analyze a broad +∗Research supported by NWO TOP project 612.001.852 “Grey-box learning of Interfaces for Refactoring +Legacy Software (GIRLS)”. + +variety of software and hardware systems, and a rich body of theory has been developed for +them. +In order to manage the complexity of computer-based systems, designers structure such +systems into hierarchical layers. This makes it possible to describe and analyze systems at +different levels of abstraction. Many LTS-based frameworks have been proposed to model +systems at different hierarchical levels and to formally relate the resulting models, e.g. [4, +14, 24, 36]. In most of these frameworks, the states of a high-level LTS correspond to sets of +states of a low-level LTS via simulation or bisimulation-like relations. However, the actions +are fixed and considered to be atomic. +Actions used at a lower level of abstraction can +be hidden at a higher level, but higher-level actions will always be available at the lower +level. For this reason, Rensink & Gorrieri [16, 28] argue that these (bi)simulations relate +systems at the same conceptual level of abstraction, and therefore they call them horizontal +implementation relations. They contrast them with vertical implementation relations that +compare systems that belong to conceptually different abstraction levels, and have different +alphabets of actions they perform. +A prototypical example of a system with hierarchical layers is a computer network. To +reduce design complexity, such a network is organized as a stack of layers or levels, each one +built upon the one below it [31]. At the top is the application layer, with protocols such +as HTTP and SMTP, and at the bottom we find the physical layer that is concerned with +transmitting raw bits over a communication channel. Now consider a host that is in some +state s where it may receive an HTTP packet. If P is the set of possible HTTP packets then, +in an LTS model of the application layer, state s will contain outgoing transitions labeled +with action receive(p), for each packet p ∈ P. At the physical layer, however, receipt of +an HTTP packet will correspond to a sequence of receive(b) actions, with b a bit in {0, 1}. +Only after the final bits have been received it will be clear which HTTP packet was actually +received. Mechanisms for transforming high-level actions into sequences (or processes) of +low-level actions have been addressed extensively in work on action refinements [16]. This +work, however, is unable to describe the above scenario in a satisfactory manner. In existing +action refinement frameworks, it is somehow assumed that a host upfront correctly guesses the +HTTP packet that it will receive, even before the first bit has arrived. In order to illustrate +this problem, we consider the simplified example of an LTS, displayed in Figure 1, that accepts +either an input a or an input b. At a lower level of abstraction, input a may be implemented +start +a +b +Figure 1: A system that accepts an input a or b. +by three consecutive input actions 1; 4; 1, whereas input b is implemented by action sequence +1; 4; 2 (the ASCII encodings of a and b in octal format). An action refinement operator will +replace the a-transition in Figure 1 by a sequence of three consecutive transitions with labels +1, 4 and 1, respectively, and will handle the b-transition in an analogous manner. Thus, a +2 + +refinement operator will introduce a nondeterministic choice (Figure 2, left), rather than the +deterministic behavior that one would like to see (Figure 2, right). As a consequence of this +and other limitations, refinement operators have not found much practical use [16]. +start +1 +1 +4 +4 +1 +2 +start +1 +4 +1 +2 +Figure 2: Nondeterminism introduced by existing action refinement operators (left) vs desired +behavior (right). +Based on the observation that any action can be modeled as a state change, some authors +(e.g. [2, 10, 21]) prefer modeling formalisms in which the term “action” is only used infor- +mally, and Kripke structures rather than LTSs are used to model systems. These state-based +approaches have the advantage that a distinction between horizontal and vertical implemen- +tation relations is no longer needed, and a single implementation relation suffices. Purely +state-based approaches, however, are problematic in cases where we need to interact with a +black-box system and (by definition) we do not have access to the system state. Black-box +systems prominently occur in the areas of model based testing [33] and model learning [34]. +In these application areas, use of LTS-based models makes sense and there is a clear practical +need for formalisms that allow scientists and engineers to relate actions at different levels of +abstraction. +Van der Bijl et al [7], for instance, observe that in model based testing specification models +are usually more abstract than the System Under Test (SUT). This means that the generated +test cases may not have the required level of detail, and often a single abstract action has to +be translated (either manually or by an adaptor) to a sequence of concrete actions that are +applied to the SUT. Van der Bijl et al [7] study a very restricted type of action refinement in +which a single input is translated into a sequence of inputs, and implement this in a testing +tool. +Also in model learning, typically an adaptor is placed in between the SUT and the learner, +to take care of the translation between abstract and concrete actions. For example, in a case +study on reverse engineering of hand-held smartcard readers for Internet banking, Chalupar et +al [9] used abstract input symbols that combine several concrete inputs in order to accelerate +the learning process and reduce the size of the learned model. In particular, they introduced +a single abstract input COMBINED_PIN corresponding to a USB command, follow by a +4-digit PIN code, followed by an OK command. +Fiterău-Broştean et al [12] used model +learning for a comprehensive analysis of DTLS implementations. This work revealed four +serious security vulnerabilities, as well as several functional bugs and non-conformance issues. +Handshakes in (D)TLS are defined over flights of messages. Hence, (D)TLS entities are often +expected to produce multiple messages before expecting a response. During learning, Fiterău- +Broştean et al [12] used an adaptor that contracted multiple output messages from the SUT +into a single abstract output. Also in other case studies on the TLS [29], Wi-Fi [30] and +SSH [35, 13] protocols, multiple outputs from the SUT were contracted into a single output. +3 + +Verleg [35] used a single abstract input to execute the entire key re-exchange when learning +higher layers of SSH. +Suppose we have an SUT that can be described by an unknown, concrete model M, and +suppose a learner interacts with this SUT through an adaptor and learns an abstract model +N. What can we say about the relation between models M and N? This article provides +an answer to this fundamental question in the settings of LTSs and Mealy machines (Finite +State Machines, except that we do require finiteness). We formalize the concept of an adaptor +by introducing action codes, a variation of the prefix codes known from coding theory [5]. +Action codes describe how high-level actions may be converted into a sequence of low-level +actions, and vice versa. This makes them different from action refinements, which specify how +high-level actions can be translated into low-level processes, but to not address the reverse +translation. Our notion of an action code captures many adaptors that are used in practice, +and in particular those described in the case studies listed above. For each action code R, we +introduce a contraction operator αR that turns a low-level model M into a high-level model +by contracting concrete action sequences of M according to R. We also introduce the left +adjoint of αR, the refinement operator ϱR that turns a high-level model M into a low-level +model by refining abstract actions of N according to R. This new refinement operator does +relate the LTSs of Figure 1 and Figure 2 (right). Our first main result is a Galois connection +ϱR(N) ⊑ M +⇔ +N ⊑ αR(M), +where ⊑ denotes the simulation preorder. So if abstract model N implements contraction +αR(M), then refinement ϱR(N) implements concrete model M, and vice versa. +In practice, we typically want to obtain an overapproximation of concrete model M. To +this end, we introduce the right adjoint of αR, the concretization operator γR. This operator +behaves like the refinement operator, but adds arbitrary behavior at intermediate points +during a refinement (cf. the demonic completion of [6]). Our second main result is another +Galois connection: +αR(M) ⊑ N +⇔ +M ⊑ γR(N). +This connection is useful, because whenever we have established that αR(M) implements (or +conforms to) N, it allows us to conclude that M implements (or conforms to) γR(N). +Our third main result is that, in a setting of Mealy machines, an adaptor can be con- +structed for any action code for which a winning strategy exists in a certain 2 player game. +If a learner/tester interacts with an SUT via an adaptor generated from such an action code +R, and the SUT is modeled by Mealy machine M, then from the learner/tester perspective, +the composition of adaptor and SUT will behave like αR(M). Thus, if a learner succeeds to +learn an abstract model N such that N ≈ αR(M) then, by using our Galois connections, +the learner may conclude that ϱR(N) ⊑ M ⊑ γR(N). +The remainder of this article is structured as follows. We start with a preliminary Sec- +tion 2 that introduces basic notations and results for LTSs and their behavior. Next, action +codes and the contraction operator will be introduced in Section 3. After describing the re- +finement operator, we establish our first Galois connection in in Section 4. Next we define the +concretization operator and establish our second Galois connection in Sections 5. Section 6 +explains how action codes can be composed, and shows that contraction and refinement com- +mute with action code composition. Section 7 describes how adaptors can be constructed +4 + +from action codes, and identifies some technical restrictions on action codes that are required +to make this possible. Finally, Section 8 contains a discussion of our results and identifies +directions for future research. +2 +Preliminaries +If Σ is a set of symbols then Σ∗ denotes the set of all finite words over Σ, and Σ+ the set of all +non-empty words. We use ε to denote the empty word, so e.g. Σ∗ = Σ+ ∪{ε}. Concatenation +of words u, w ∈ Σ∗ is notated u · w (or simply u w). We write u ≤ w if u is a prefix of w, i.e. +if there is v ∈ Σ∗ with u v = w. We write |w| to denote the length of word w. +We use f : X⇀Y to denote a partial map f from X to Y and write dom(f) ⊆ X for its +domain, i.e. set of x ∈ X on which f is defined. The image im(f) of a partial map f : X⇀Y +is the set of elements of Y it can reach: +im(f) := {f(x) | x ∈ dom(f)} ⊆ Y. +Definition 2.1. For a set A of labels, a labelled transition system (LTS) is a tuple M = +⟨Q, q0, +⟩ where +• Q is a set of states; +• q0 ∈ Q is a starting state; +• +⊆ Q × A × Q is a transition relation. +We write LTS(A) for the class of all LTSs with action labels from A. We refer to the three +components of an LTS M as QM, qM +0 +and +M, respectively, and introduce the following +notation: +• q +a +q′ denotes (q, a, q′) ∈ +; +• q +a +denotes that there is a q′ such that q +a +q′; +• q +w +q′ for w ∈ A∗ denotes that there are finite sequences a1, . . . , an ∈ A, r0, . . . , rn ∈ Q +such that w = a1 · · · an, and r0 = q, rn = q′ and ri−1 +ai ri for all 1 ≤ i ≤ n; +• q +w +denotes that there is a q′ such that q +w +q′; +• q ∈ Q is reachable if there is a w ∈ A∗ such that q0 +w +q. +A special class of LTSs that is frequently used in conformance testing and model learning +are Mealy machines. Mealy machines with a finite number of states are commonly referred +to as Finite State Machines. +Definition 2.2. For a non-empty sets of inputs I and outputs O, a (non-deterministic) Mealy +machine M ∈ LTS(I × O) is an LTS where the labels are pairs of an input and an output. +We write q +i/o q′ to denote that (q, (i, o), q′) ∈ +. Whenever we omit a symbol in predicate +q +i/o +−→ q′ this is quantified existentially. Thus +i/o +if there are q and q′ such that q +i/o +q′, +q +i/ q′ if there is an o such that q +i/o q′, and q +i/ if there is a q′ such that q +i/ q′. +5 + +Example 2.3. Figure 3 visualizes a simple Mealy machine with inputs {a, b} and outputs +{0, 1}. The machine always outputs 0 in response to an input, except in one specific situation. +Output 1 is produced in response to input b if the previous input was a and the total number +of preceding inputs is odd. The machine has four states q0, q1, q2 and q3, with starting state q0 +marked by an incoming arrow. In states q0 and q2 the number of preceding inputs is always +even, whereas in states q1 and q3 it is always odd. In states q2 and q3 the previous input is +always a, whereas in states q0 and q1 either the previous input is b, or no input has occurred +yet. Thus, only in state q3 input b triggers an output 1. +q0 +start +q1 +q2 +q3 +b/0 +a/0 +a/0 +b/0 +b/0 +a/0 +a/0 +b/1 +Figure 3: A Mealy machine. +We need to introduce some formal notation and terminology for LTSs. +Definition 2.4. Let M = ⟨Q, q0, +⟩ ∈ LTS(A) be an LTS. We say that +• M is deterministic if, whenever q +a for some q and a, there is a unique q′ with q +a +q′. +• M is a tree-shaped if each state q ∈ Q can be reached via a unique sequence of transi- +tions from state q0. +• q ∈ Q is a leaf, notated q +, if there is no a ∈ A with q +a . +• M is grounded if every state q ∈ Q has a path to a leaf. +We can now define the set of traces of an LTS: +Definition 2.5. Let M = ⟨Q, q0, +⟩ ∈ LTS(A) be an LTS. A word w ∈ A∗ is a trace of +state q ∈ Q if q +w , and a trace of M is it is a trace of q0. We write trace(M) for the set of +all traces of M: +trace(M) += +{w ∈ A∗ | q0 +w } +A trace of M is complete if it is not a proper prefix of any other observation of M. Complete +traces correspond to sequences of transitions that end in a leaf state. +Definition 2.6 (Simulations). For M, N ∈ LTS(A), a simulation from M to N is a relation +S ⊆ QM × QN such that +1. qM +0 +S qN +0 and +6 + +2. if q1 S q2 and q1 +a +M q′ +1 then there exists a state q′ +2 such that q2 +a +N q′ +2 and q′ +1 S q′ +2. +We write M ⊑ N if there exists a simulation from M to N. +It is a classical result that trace inclusion coincides with the simulation preorder for +deterministic labeled transition systems (see e.g. [25]): +Lemma 2.7. Let M, N ∈ LTS(A) with N deterministic. Then trace(M) ⊆ trace(N) iff +M ⊑ N. +We will often consider LTSs up to isomorphism of their reachable parts: +Definition 2.8 (Isomorphism). For M, N ∈ LTS(A), an ismorphism from M to N is a +bijection f : QM +R → QN +R , where: +1. QM +R ⊆ QM and QN +R ⊆ QN are the subsets of reachable states in M and N, respectively; +2. f(qM +0 ) = qN +0 , and +3. q +a +M q′ iff f(q) +a +N f(q′), for all q, q′ ∈ QM +R , a ∈ A. +We write M ∼= N if there exists an isomorphism from M to N. +Note that ∼= is an equivalence relation on LTS(A), and that M ∼= N implies M ⊑ N, +since each isomorphism (when viewed as a relation) is trivially a simulation. +3 +Action Codes +Action codes describe how to translate between two action label alphabets, for example from +A to B. Intuitively, we understand the first alphabet A as the actions at the lower, concrete +level, and the second alphabet B as the actions at the higher, more abstract level. In an +action code, a single abstract action b ∈ B corresponds to a finite, non-empty sequence of +concrete actions a1 · · · an in A. Essentially, action codes are just a special type of prefix +codes [5]. We provide two equivalent definitions of action codes: one via tree-shaped LTSs +and one via a partial map. +Definition 3.1 (Action code). For sets of action labels A and B, a (tree-shaped) action code +R from A to B is a structure R = ⟨M, l⟩, with M = ⟨R, r0, +⟩ ∈ LTS(A) a deterministic, +tree-shaped LTS with L being the set of non-root leaves L ⊆ R\{r0} and an injective function +l: L → B. We write Code(A, B) for all action codes from A to B. +The injectivity of l and the tree-shape ensure that every abstract b ∈ B is represented by +at most one w ∈ A+. +Example 3.2. Figure 4 shows an action code for a fragment of the ASCII encoding in octal +format, e.g., sequence 1 1 5 encodes the letter “M”, sequence 1 4 5 encodes the letter “e”, etc. +7 + +r0 +start +r1 +r3 +r2 +r4 +r5 +r6 +M +r7 +e +r8 +a +r9 +l +r10 +y +1 +1 +4 +5 +7 +5 +5 +1 +4 +1 +Figure 4: Action code for a fragment of the ASCII encoding. +r0 +start +r1 +r3 +r4 +r5 +r6 +/0 +r7 +/0 +r8 +r9 +/1 +r10 +/1 +r11 +r12 +/2 +r13 +/2 +r14 +r15 +/3 +r16 +/3 +switch_on/fill_water_tank +switch_on/ready +add_water/fill_bean_container +add_water/ready +add_beans/empty_coffee_grounds_container_and_drip_tray +add_beans/ready +remove_waste/ready +/coffee +/espresso +/coffee +/espresso +/coffee +/espresso +/coffee +/espresso +Figure 5: Action code for a coffee machine. +Example 3.3. Figure 5 shows an action code for the activity of getting a cup of coffee or +espresso, in the special case of Mealy machines, i.e. where A = I × O and B = I′ × O′ are +sets of input/output-pairs. Rather than the full sequence of interventions that is required in +order to get a drink, the abstract input/output pair only reports on the type of drink that +was ordered and the number of interventions that occurred. +8 + +The definition of action codes as labelled transition system themselves allows an intuitive +visualization. For easier mathematical reasoning, we characterize action codes also in terms +of maps: +Definition 3.4. A (map-based) action code from A to B is a partial map f : B⇀A+ which +is prefix-free, by which we mean that for all b, b′ ∈ dom(f), +f(b) ≤ f(b′) +implies +b = b′. +(1) +In the following, we show that these prefix-free partial maps bijectively correspond to the +tree-shaped LTSs: +Lemma 3.5. Every tree-shaped action code R ∈ Code(A, B) induces a unique map-based +action code f : B⇀A+ with the property that for all b ∈ B, w ∈ A+: +f(b) = w +iff +∃r ∈ L: r0 +w +R r, +l(r) = b +(2) +Proof. Take equation (2) as the definition of f: +f(b) = +� +w +if there are r ∈ L, w ∈ A∗ with r0 +w +R r, l(r) = b, +undefined +otherwise. +Since r0 /∈ L, the range of f indeed restricts to A+ ⫋ A∗. For well-definedness of f, first +note that due to the injectivity of the labelling l: L → B, there is at most one r ∈ L with +l(r) = b, and by R being tree-shaped, there is precisely one path r0 +R r. Let us now prove +the required properties of this partial map: +• Prefix-freeness (1). Consider b, b′ ∈ B with f(b) ≤ f(b′). Hence, we have runs +r0 +w +R r +r ∈ L +l(r) = b +w = f(b) +and +r0 +w′ +R r′ +r′ ∈ L +l(r′) = b′ +w′ = f(b′). +By f(b) ≤ f(b′), there is some u ∈ A∗ such that wu = w′. Thus, the run for w′ = f(b′) +can be decomposed for some ¯r ∈ QR: +r0 +w +R ¯r +u +R r′. +The LTS R is required to be deterministic, so ¯r = r. Since r ∈ L is a leaf, i.e. a +dead-lock state, we necessarily have u = ε and r = r′. Finally, b = l(r) = l(r′) = b′. +• Characterizing property (2). Verification is immediate because both the property +(2) and the definition r use the same witnesses, and every witness w ∈ A∗ must be a +non-empty word. +• Uniqueness. Consider another prefix-free partial map g: B⇀A+ satisfying +for all b ∈ B, w ∈ A+ : +g(b) = w +iff +∃r ∈ L: r0 +w +R r, +l(r) = b. +Then, it is immediate that dom(g) = im(L) = dom(f) and that g(b) = f(b) for all +b ∈ im(L). +9 + +Lemma 3.6. For each map-based action code f : B⇀A+, there is (up to isomorphism) a +unique tree-shaped action code R ∈ Code(A, B) which is grounded and makes property (2) +true. +Proof. Define action code R as follows: +• R := {w ∈ A∗ | w = ε or ∃b ∈ dom(f): w ≤ f(b)}, +• r0 := ε, +• we put v +a +R w iff there is a ∈ A with va = w, +• l(w) is the unique b ∈ B with f(b) = w. +Let us first verify that this is a well-defined action code: +• There is nothing to be verified about the state set R (note that we do not require +finiteness in the definition of action code). +• By definition of R, the initial state r0 := ε is an element of R. +• For the transition structure we need to verify several properties: +– It is grounded because for every w ∈ A∗ with w ≤ f(b), b ∈ dom(f), the state +f(b) is the witnessing leaf (i.e. dead-lock state) below w. +– It is deterministic: whenever we have v +a +R w and v +a +R w′, we obtain w = +va = w′. +– It is tree-shaped: for each w ∈ R, there is a unique run to it, namely r0 +w +R w. +• Define set L ⊆ R by L = {f(b) | b ∈ B}. Then L does not contain the initial state r0, +but every leaf w ∈ Q \ {r0} is in L. Conversely, every w ∈ L is a leaf because f is +prefix-free (1). +Likewise, l: L → B is injective because f is prefix-free (1). +For uniqueness, consider another tree-shaped action code ¯R with set of non-root leaves +¯L satisfying (2), concretely +for all b ∈ B, w ∈ A+ : f(b) = w +iff +∃¯r ∈ ¯L: r0 +w +¯ +R ¯r +l +¯ +R(¯r) = b. +(3) +We now need to establish an isomorphism φ: ¯R → R. Define the map φ: R ¯ +R → RR by +φ(¯r) = the unique w ∈ A∗ with r +¯R +0 +w +¯ +R ¯r. +This map is well defined because ¯R is tree-shaped. To see that φ(¯r) ∈ A∗ is also in RR ⊆ A∗ +for every state ¯r of ¯R, let ¯r′ ∈ ¯L be an arbitrary leaf with ¯r +u +¯ +R ¯r′ in ¯R – such a leaf +exists because every action code ¯R is required to be grounded. Thus, for b = lR(¯r′) we have +φ(¯r) ≤ wu = f(b) by (3), and so φ(¯r) ∈ RR by above definition of R. With our definition of +φ and +R, it is mechanical to check that q +a +q′ iff φ(q) +a +φ(q′). Since ¯R is tree-shaped, φ +is injective. For surjectivity, consider w ∈ RR. If w = ε then we have φ(r ¯R +0 ) = ε. Otherwise, +10 + +pick any b ∈ dom(f) with w ≤ f(b). By (3), we have ¯r ∈ ¯L with q ¯ +R +0 +f(b) +¯ +R r in ¯R. The +intermediate state ¯r′ with +q +¯ +R +0 +w +¯ +R ¯r′ +u +¯ +R ¯r +w u = f(b) +does satisfy φ(¯r′) = w. Thus, φ is surjective and bijective in total. +Also, the labelling is preserved by φ: ¯R → R because for every leaf ¯r ∈ ¯L with +r +¯ +R +0 +w +¯ +R ¯r +we have f(l ¯ +R(¯r)) = w by (3), and so +l(φ(¯r)) = l(w) +def l += l +¯ +R(¯r). +because b := l ¯ +R(¯r) satisfies f(b) = w. +Example 3.7. Note that for the uniqueness in Lemma 3.6, we additionally need that the +tree-shaped action code is grounded. This essentially means that there is no infinite subtree +in which no leaf is. For example, consider A = {a}, arbitrary B and the action codes +R := +� +r0 +start +r1 +a +r2 +a +· · · +a +� +and +S := +� +q0 +start +� +. +Both action codes R and S have no non-root leaves, and so they both induce the empty +partial map f : B⇀A+ via Lemma 3.5, i.e. f is undefined for all b ∈ B. And indeed, R and +S are not isomorphic. The issue is that while the finite S is grounded, the infinite R is not +grounded. So R contains subtrees which do not contribute anything to the partial map f +but which hinder the existence of an isomorphism. +Having shown the correspondence between tree-shaped and map-based action codes Code(A, B), +we can switch between the two views in proofs. Mostly, we use the tree-shaped version for +visualization and the map-based version for mathematical reasoning. +Consider a concrete M ∈ LTS(A), together with an action code R from A to B. We can +construct an abstract LTS for the action labels B by walking through M with seven-league +boots, repeatedly choosing input sequences that correspond to complete observations of R, +and then contracting this sequence to a single abstract transition. +Notation 3.8. In the rest of the paper, we introduce operators αR, ϱR, γR on LTSs, involving +action codes R. Whenever the action code R is clear from the context, we omit the index +and simply speak of operators α, ϱ, γ for the sake of brevity. +Definition 3.9 (Contraction). For each action code R ∈ Code(A, B), the contraction oper- +ator αR : LTS(A) → LTS(B) is defined as follows. For M ∈ LTS(A), the LTS αR(M) has +states Qα(M) ⊆ QM and transitions +α(M) defined inductively by the next two rules, for all +q, q′ ∈ QM, b ∈ B: +qM +0 +∈ Qα(M) +(1α) +q ∈ Qα(M), +b ∈ dom(R), +q +R(b) +M q′ +q +b +α(M) q′, +q′ ∈ Qα(M) +(2α) +The initial state qα(M) +0 +:= qM +0 +is the same as in M. +11 + +Example 3.10. Figures 6 and 7 show two examples of action codes and the contractions +obtained when we apply them to the Mealy machine of Figure 3. The examples illustrate +that by choosing different codes we may obtain completely different abstractions of the same +Mealy machine. +r0 +start +r1 +r2 +r3 +A/0 +r4 +B/0 +a/0 +b/0 +a/0 +b/0 +q0 +start +q2 +A/0 +B/0 +B/0 +A/0 +Figure 6: Code for Mealy machine of Fig. 3 (left) and resulting contraction (right). +r0 +start +r1 +r2 +B/0 +r3 +C/0 +r4 +C/1 +a/0 +b/0 +b/0 +b/1 +q0 +start +q1 +B/0 +C/1 +B/0 +C/0 +Figure 7: Another code for Mealy machine of Fig. 3 (left) and resulting contraction (right). +The next proposition asserts that we can view αR as a monotone function αR : LTS(A) → +LTS(B) between preordered classes. +Proposition 3.11 (Monotonicity contraction). Let M, N ∈ LTS(A) with M ⊑ N, and let +R ∈ Code(A, B) be an action code. Then αR(M) ⊑ αR(N). +Proof. Given a simulation T from M to N, we will show that S := T ∩ (Qα(M) × Qα(N)) +is a simulation relation from αR(M) to αR(N); in other words S ⊆ Qα(M) × Qα(N) is the +restriction of T ⊆ QM × QN to the subsets Qα(M) ⊆ QM and Qα(N) ⊆ QN. +For the initial states, we directly have (qα(M) +0 +, qα(N) +0 +) ∈ S because +(qα(M) +0 +, qα(N) +0 +) = (qM +0 , qN +0 ) ∈ T. +For transitions, consider (q, p) ∈ S and q +b +q′ in α(M). By definition of α(M), we have +b ∈ dom(R) and q +R(b) q′ +in M. +12 + +In M and N, the states (q, p) ∈ S ⊆ T are also related by the simulation, and so we obtain +p +R(b) p′ +in N with (q′, p′) ∈ T. +By the definition of α, we have p′ ∈ Qα(N) and p +b +p′ in α(N). Thus, also (q′, p′) ∈ S as +desired. +The example below illustrates that the previously discussed trace language operator trace +can be viewed as an instance of contraction for an infinite action code: +Example 3.12 (Trace semantics). Define δ: LTS(A) → LTS(A + {$}) by δ((Q, q0, +)) = +(Q+{$}, q0, +∪{(q, $) | q ∈ Q}) and define the action code R ∈ Code(A+{$}, B) to B := A∗ +such that the concrete word a1 · · · an$ is related to the abstract symbol a1 · · · an ∈ B, then +αR ◦ δ: LTS(A) → LTS(A∗) sends every M ∈ LTS(A) to a system in LTS(A∗) with states +{q0, $} and transitions +q0 +w $ +⇐⇒ +w ∈ trace(M). +4 +Refinements +Now that we have introduced the contraction αR of an LTS for a code R, it is natural to +consider an operation in the other direction, which we call the refinement ϱR. Intuitively, +refinement replaces each abstract transition q +b +q′ by a sequence of concrete transitions, as +prescribed by R. +Definition 4.1 (Refinement). For each action code R ∈ Code(A, B), we define the refinement +operator ϱR : LTS(B) → LTS(A) as follows. For M ∈ LTS(B), the LTS ϱR(M) ∈ LTS(A) +has a set of states +Qϱ(M) := {(q, w) ∈ QM × A∗ | w = ε or (∃b: q +b +M ∧w ≨ R(b))} +and the initial state (qM +0 , ε). The transition relation +ϱ(M) is defined by the following rules: +(q, wa) ∈ Qϱ(M) +(q, w) +a +ϱ(M) (q, wa) +(1ϱ) +q +b +M q′ +wa = R(b) +(q, w) +a +ϱ(M) (q′, ε) +(2ϱ) +Intuitively, whenever ϱ(M) is in state (q, w), then this corresponds to being in state q +in the abstract automaton M ∈ LTS(B) and having observed the actions w ∈ A∗ so far. +However, we have insufficiently many actions for finding an abstract transition q +b +M q′ +with w = R(b) because w is still to short. Nevertheless, whenever ϱ(M) admits a transition +to a state (q, w) with w ̸= ε, then we know that we can eventually complete w to a sequence +corresponding to an abstract transition: there exist at least one q +b +M q′ for some b ∈ +dom(R) with w ≤ R(b). We will formally prove this in Lemma 4.6. If the abstract system +M is non-deterministic, then there may be multiple abstract transitions that match in the +final rule (2ϱ), but the transitions produced by rule (1ϱ) are deterministic. +13 + +Example 4.2. Figure 8 shows an example application of a refinement operator that replaces +the actions of the LTS M on the left by their ASCII encoding in octal format, as prescribed +by the action code from Figure 4. The initial state is (q0, ε), corresponding to q0 in M. +Since M contains abstract labels M and a, with R(M) = 1 1 5 and R(a) = 1 4 1, we need to +introduce additional states for having read 1, 1 1, and 1 4, because those are the sequences +of A-actions before we have observed a sequence R(b) ∈ A+ for some b ∈ B. +q0 +start +M +a +(q0, ε) +start +(q0, 1) +(q0, 11) +(q0, 14) +1 +1 +4 +5 +1 +Figure 8: LTS (left) and refinement obtained via action code from Figure 4 (right). +A more visual explanation of ϱR(M) is the following: for every state q ∈ QM, we consider +the outgoing transitions {q +b +M q′ | b ∈ B, q′ ∈ QM} and labels B′ ⊆ B that appear in it. +Then, this outgoing-transition structure is replaced with (a copy of) the minimal subgraph +of the tree R containing all leaves with labels in B′. +Like contraction, the refinement operation also preserves the simulation preorder. +Proposition 4.3 (Monotonicity). For all action codes R ∈ Code(A, B), if M ⊑ N in +LTS(B), then ϱR(M) ⊑ ϱR(N) in LTS(A). +Proof. Let S ⊆ QM × QN be the simulation witnessing M ⊑ N. Let us verify that +T := {((q, w), (p, w′)) ∈ Qϱ(M) × Qϱ(N) | w = w′, (q, p) ∈ S} +is a simulation from ϱ(M) to ϱ(N). Clearly, +qϱ(M) +0 += (qM +0 , ε) T (qN +0 , ε) = qϱ(N) +0 +. +Consider a pair in T, i.e. (q, w), (p, w) with q S p: +1. For transitions produced by rule (2ϱ), +(q, w) +a +ϱ(M) (q′, ε), +we have q +b +M q′ for some b ∈ dom(R) with R(b) = wa by rule assumption. Since +(q, p) ∈ S, the simulation S provides us with a transition +p +b +N p′ +in N +and +(q′, p′) ∈ S. +Thus, we can apply rule (2ϱ) in ϱ(N) to obtain +(p, w) +a +ϱ(N) (p′, ε) +in ϱ(N) +which satisfies (q′, ε) T (p′, ε) by definition of T. +14 + +2. For transitions produced by rule (1ϱ), +(q, w) +a +ϱ(M) (q, wa), +we have (q, wa) ∈ Qϱ(M). Obviously, wa ̸= ε, and so by the definition of Qϱ(M), there +must be at least one transition q +b +M q′ with wa ≨ R(b). By the definition of T, we +have (q, p) ∈ S and so the simulation S provides us with some p +b +N p′ in N with +(q′, p′) ∈ S. Hence, (p, wa) ∈ Qϱ(N) and so we have (p, w) +a +ϱ(N) (p, wa) by rule (1ϱ) +and (q, wa) T (p, wa) by definition of T. +Because R is deterministic, applying ϱR on a deterministic LTS results in a deterministic +LTS: +Proposition 4.4 (Refinement preserves determinism). For every action code R ∈ Code(A, B), +if M ∈ LTS(B) is deterministic, then ϱR(M) ∈ LTS(A) is deterministic, too. +Proof. Consider (q, w) ∈ Qϱ(M) and transitions +(q, w) +a +(q1, w1) +and +(q, w) +a +(q2, w2) +in ϱ(M). +We show (q1, w1) = (q2, w2) by case distinction: +1. Case: there is b ∈ dom(R) with R(b) = wa. Then there is a unique such b because +R is prefix-free. +Also because R is prefix-free, there is no b′ ∈ dom(R) such that +wa ≨ R(b′), and so (q, wa) /∈ Qϱ(M) and so neither of the transitions were produced by +rule (1ϱ). Thus, both transitions come from rule (2ϱ) and thus we have w1 = ε = w2 +and transitions: +q +b +M q1 +and +q +b +M q2. +By assumption, M is deterministic, so q1 = q2, and so (q1, w1) = (q2, w2). +2. Case: there is no b ∈ dom(R) with R(b) = wa. Thus, neither of the transitions can be +produced by rule (2ϱ) and so both were produced by rule (1ϱ). Then, we necessarily +have +(q1, w1) = (q, wa) = (q2, w2). +The next two technical lemmas are required to establish our first Galois connection. +Lemma 4.5. For all M ∈ LTS(B) and b ∈ dom(R) of an action code R ∈ Code(A, B), and +q ∈ QM: +q +b +q′ in M +iff +(q, ε) +R(b) (q′, ε) in ϱ(M). +Proof. For (⇒), R is defined for b and we yield R(b) = a1 · · · an with 1 ≤ n and ai ∈ A for +all 1 ≤ i ≤ n. Given q +b +q′ in M, we can construct a path in ϱR(M) by n − 1 applications +of rule (1ϱ) and one application of rule (2ϱ): +(q, ε) +a1 (q, a1) +a2 (q, a1a2) · · · +an−1 (q, a1 · · · an1) +an (q′, ε) +in ϱR(M), +15 + +with other notation for R(b) = a1 · · · an: +(q, ε) +R(b) (q′, ε) +in ϱR(M), +For (⇐), consider a generalized transition (q, ε) +R(b) +(q′, ε) in ϱ(M). Of course R(b) ∈ +A+ so we can write it as R(b) = wa for w ∈ A∗ and a ∈ A. Then, we can consider the +intermediate state of the given run for R(b): +(q, ε) +w +ϱ(M) (¯q, ¯w) +a +ϱ(M) (q′, ε). +All prefixes v ≤ w satisfy v ≨ R(b), so the first transitions for w must have been produced +by rule (1ϱ). Hence, (¯q, ¯w) = (q, w). The final a-transition must have been produced by +(2ϱ) because the second component of the tuple is ε. The assumption of this rule contains +q +b +M q′, as desired. +Lemma 4.6. For every transition (q, w) +a +(q′, w′) in ϱ(M), there is some transition q +b +q′′ +in M and u ∈ A∗ such that wau = R(b) and (q′, w′) +u +(q′′, ε) in ϱR(M). +Proof. For the transition (q, w) +a +(q′, w′), distinguish two cases: +• If w′ = ε, then the transition must have been produced by rule (2ϱ). Thus, the rule +assumption contains a transition q +b +M q′ with wa = R(b). This is the desired witness +q′′ := q′ and u = ε. +• If w′ ̸= ε, then the transition must have been produced by rule (1ϱ) and so q′ = q and +w′ = wa. The definition of Qϱ(M) unfolded for (q, wa) ∈ Qϱ(M) yields that there exists +a transition q +b +M q′′ with wa ≨ R(b). Let v ∈ A∗ and b ∈ A such that wavb = R(b). +Since wav ≨ R(b), we can produce further transitions using rule (1ϱ) for the letters in +v ∈ A∗ and can conclude using rule (2ϱ) for b: +(q′, w′) = (q, wa) +v +ϱ(M) (q, wau) +b +ϱ(M) (q′′, ε). +This is the desired transition (with u := vb). +Theorem 4.7 (Galois connection). Consider an action code R ∈ Code(A, B) and LTSs +N ∈ LTS(B) and M ∈ LTS(A): +1. If dom(R) = B, then ϱR(N) ⊑ M implies N ⊑ αR(M). +2. If M is deterministic, then N ⊑ αR(M) implies ϱR(N) ⊑ M. +ϱR(N) ⊑ M +N ⊑ αR(M) +If dom(R) = B +If M is deterministic +The condition dom(R) = B in the first direction means that the partial map R: B⇀A+ +is in fact a total map R: B → A+ because R(b) is defined for all b ∈ B. +Proof. Fix systems N ∈ LTS(B), M ∈ LTS(A), and as usual we omit index R from α and ϱ. +16 + +1. For direction (⇒), assume that dom(R) = B and ϱ(N) ⊑ M, witnessed by the simu- +lation S ⊆ Qϱ(N) × QM. We verify that we have a simulation T between N and α(M) +defined by: +T := {(p, q) ∈ QN × Qα(M) | +� +(p, ε), q +� +∈ S} +The definition of T is well-typed because Qα(M) ⊆ QM, and moreover, +{(p, ε) | p ∈ QN} ⊆ Qϱ(N). +The relation T relates the initial states qN +0 T qα(M) +0 +because +(qN +0 , ε) = qϱ(N) +0 +S qM +0 += qα(M) +0 +. +Now, suppose p T q and p +b +N p′. +By assumption dom(R) = B, we can apply +Lemma 4.5 and obtain +(p, ε) +R(b) +ϱ(N) (p′, ε) +in ϱ(N). +The simulation S from ϱ(N) to M transforms this into a path +q +R(b) +M q′ in M +with +(p′, ε) S q′. +Since q ∈ Qα(M) also q′ ∈ Qα(M) and thus +q +b +α(M) q′ in α(M) +and moreover p′ T q′. +2. For direction (⇐), assume N ⊑ α(M). Let S ⊆ QN × Qα(M) be a simulation relation +from N to α(M). We define the relations ¯S ⊆ Qϱ(N) × Qα(M) and T ⊆ Qϱ(N) × QM: +(p, ε) ¯S q +:⇔ +p S q +(p, w) T q +:⇔ +∃¯q ∈ Qα(M) : p S ¯q ∧ ¯q +w +M q +So visually, every related pair (p, w) T q entails states of the following form: +ϱ(N) +M +(p, ε) +¯q +(p, w) +q +w +w +¯S +T +17 + +We verify that T is a simulation from ϱ(N) to M. +Picking ¯q := q and w := ε shows that the related initial states qN +0 S qα(M) +0 +of N and +α(M) imply +qϱ(N) +0 += (qN +0 , ε) T qM +0 . +Suppose (p, w) T q and (p, w) +a +ϱ(N) (p′, w′). Lemma 4.6 shows that wa ∈ A+ sits +‘below’ some b ∈ B in the action code R: concretely, there are u ∈ A∗, and b ∈ B such +that: +(p′, w′) +u +ϱ(N) (p′′, ε) +and +wau = R(b) +and +p +b +N p′′. +We have the solid (i.e. non-dotted) part of the following picture: +ϱ(N) +M +N +α(M) +(p, ε) +(p, w) +(p′, w′) +(p′′, ε) +w +a +u +¯q +q +q′ +q′′ +w +a +u +¯S +T +¯S +p +p′′ +b +¯q +q′′ +b +S +S +Using the simulation property of p S ¯q, we obtain q′′ ∈ Qα(M) with ¯q +b +α(M) q′′ and +p′′ S q′′. The transition ¯q +b +q′′ in α(M) must have come from a path ¯q +R(b) +q′′ in +M. Denote the intermediate states for the decomposition R(b) = wau by qw, q′ ∈ QM: +¯q +w +M qw +a +M q′ +u +q′′ +The assumption that M is deterministic enforces that q = qw, so q +au +M q′′ as shown +in the picture. Also, p′′ S q′′ directly shows (p′′, ε) ¯S q′′. Now, the picture is complete. +For the final proof that T relates (p′, w′) and q′, we distinguish cases: +(a) Case u = ε: Then, p′ = p′′, w′ = ε and q′ = q′′. Thus, +� +(p′, w′), q′� +∈ ¯S ⊆ T. +(b) Case u ̸= ε: Then, wa ≨ R(b) and so wa ̸= R(b′) for all b′ ∈ B by prefix-freeness. +So the a-transition can only come from rule (1ϱ); hence p = p′ and w′ = wa. +Finally, (p′, w′) = (p, wa) T q′ by the definition of T and ¯q +wa q′. +18 + +Remark 4.8. In the above direction ⇐, we need determinism of M. If we also want to +support non-deterministic M, we can consider a less-pleasant ϱ′ +R that replaces every q +b +q′ +for R(a1 · · · an) = b with literally a sequence q +a1 +· · · +an +q′. Thus ϱ′ +R would rather create +a system on the left of Figure 2 whereas ϱR creates a system as on the right of Figure 2. +However, such an operator ϱ′ +R does not preserve determinism, but ϱR does as we prove next. +Remark 4.9. Another important assumption in the direction ⇐ is that for every abstract +b ∈ B there is at most one related word w ∈ A+ of concrete actions in the action code R. +And indeed the direction ⇐ fails if there are two symbols a1, a2 ∈ A, a1 ̸= a2 both related to +b ∈ B. Denote a system with two states and one transition directly by (• +b +•) (the initial +state is the left-hand one). Then, we have +α(• +a1 •) = (• +b +•) = α(• +a2 •). +Then, there exists no left adjoint ϱ. Such an adjoint, applied to (• +b +•) would need to +satisfy +ϱ(• +b +•) ⊑ (• +a1 •) +and +ϱ(• +b +•) ⊑ (• +a2 •) +Hence, the initial state in ϱ(• +b +•) is a leaf state, i.e. ϱ(• +b +•) is the bottom element for +the simulation order ⊑. This easily leads to a contradiction after using the Galois connection +in the other direction again. +5 +Concretizations +In this section, we consider another method of transforming an abstract system into a con- +crete one: the concretization operator. Whereas refinement is the lower adjoint of contraction +(Theorem 4.7), this section will establish that concretization is the upper adjoint (Theo- +rem 5.3) of contraction. Whereas for refinement we omitted transitions for which the action +code R was not defined, for concretization we add transitions to a new chaos state [18] in +which any action may occur. Essentially, this is the idea of demonic completion of [6]. +Definition 5.1 (Concretization). Let M ∈ LTS(B) be an LTS and R ∈ Code(A, B) an +action code R: B⇀A+. The concretization γR(M) ∈ LTS(A) consists of: +• Qγ(M) := QM × N ∪ {χ} where N ⊆ A∗ is defined by +N := {w ∈ A∗ | w = ε or ∃b ∈ dom(R): w ≨ R(b)} +• qγ(M) +0 +:= (qM +0 , ε) +• Transitions defined by the following rules, for all (q, w) ∈ QM × N, a ∈ A, and b ∈ B: +wa ∈ N +(q, w) +a +γ(M) (q, wa) +(1γ) +q +b +M q′, +R(b) = wa +(q, w) +a +γ(M) (q′, ε) +(2γ) +19 + +wa /∈ N, wa /∈ im(R) +(q, w) +a +γ(M) χ +(3γ) +χ +a +γ(M) χ +(4γ) +Intuitively, N represent the internal nodes of the tree-representation of the action code +R. The transitions then try to accumulate a word w ∈ A∗ know to the action code. As soon +as we reach w = R(b) for some b, we use a b-transition in the original M ∈ LTS(B) to a new +state. +The transition structure of γ is built in such a way that transitions for b ∈ B in M +correspond to runs of R(b) in γ(M) in the following sense: +Lemma 5.2. For every M ∈ LTS(B), q ∈ QM, R ∈ Code(A, B), and b ∈ dom(R): +1. Whenever q +b +M q′ then (q, ε) +R(b) +γ(M) (q′, ε). +2. Whenever (q, ε) +R(b) +γ(M) ¯q, then ¯q = (q′, ε) for some q′ ∈ QM with q +b +M q′. +Proof. +1. Given q +b +M q′ in M, write R(b) ∈ A+ as R(b) = wa for w ∈ A∗ and a ∈ A. +Write w = w1 · · · wn, with n ∈ N and wi ∈ A, for 1 ≤ i ≤ n. For every 1 ≤ i ≤ n, +the word w1 · · · wi ∈ A∗ is contained in N because w1 · · · wi ≨ R(b). Hence, rule (1γ) +provides us with transitions +(q, ε) +w1 +γ(M) (q, w1) +w2 +γ(M) (q, w1w2) · · · +γ(M) (q, w1 · · · wn) = (q, w) +Finally, rule (2γ) has the assumptions q +b +M q′ and wa = R(b) fulfilled, so we have +(q, ε) +w +γ(M) (q, w) +a +γ(M) (q′, ε), +i.e. (q, ε) +R(b) +γ(M) (q′, ε), as desired. +2. Assume (q, ε) +R(b) +γ(M) ¯q. As before, decompose R(b) into R(b) = wa for w ∈ A∗ and +a ∈ A. Write w = w1 · · · wn for wi ∈ A, n ∈ N. For every i, 1 ≤ i < n, any transition +(q, w1 · · · wi) +wi+1 +γ(M) p +can only be produced by rule (1γ), because w1 · · · wiwi+1 ∈ N. Hence, p = (q, w1 · · · wi+). +Thus, the given run for R(b) ∈ A+ is of the form +(q, ε) +w1 +γ(M) (q, w1) +w2 +γ(M) · · · +wn +γ(M) (q, w1 · · · wn) = (q, w) +a +γ(M) ¯q. +In order to see that the final a-transition is produced by rule (2γ), note that wa = R(b) +implies that wa /∈ N by prefix-freeness (1), and so rule (1γ) can not have produced this +a-transition to ¯q. Obviously, rule (3γ) does not match either, and so only rule (2γ) is +left. Thus, there is some q +b +M q′ and ¯q is of the form ¯q = (q′, ε), as desired. +20 + +Proposition 5.3 (Galois connection). For every action code R ∈ Code(A, B), M ∈ LTS(A), +and N ∈ LTS(B), we have +αR(M) ⊑ N (in LTS(B)) +⇐⇒ +M ⊑ γR(N) (in LTS(A)). +Proof. Since we have only one action code R at hand, we omit the index R for α and γ. We +prove both directions seperately: +(⇒) Let α(M) ⊑ N be witnessed by the simulation S ⊑ Qα(M) × QN. We show that +T ⊑ QM × Qγ(N) defined by +T := {(p′, (q, w)) | p′ ∈ QM, (q, w) ∈ Qγ(N), ∃p ∈ QM, p +w +M p′, (p, q) ∈ S} +∪ {(p, χ) | p ∈ QM} +is a simulation. Note that if χ is omitted from γ(N) if it is not reachable, and then we +also omit it from T. For the initial states, we immediately have +(qM +0 , qγ(N) +0 +) = (qM +0 , (qN +0 , ε)) ∈ T +by the definition of T, because qM +0 +ε +qM +0 +and (qM +0 , qN +0 ) = (qα(M) +0 +, qN +0 ) ∈ S. +Since T is defined as a union, we can verify the two parts seperately: +(a) Consider (p′, (q, w)) ∈ T and p′ +a +M p′′. By definition of T, there is some p ∈ QM +with +p +w +M p′ +and +(p, q) ∈ S. +We distinguish whether wa ∈ N and wa ∈ im(R): +• If wa ∈ N, then we have (q, w) +a +γ(N) (q, wa) by rule (1γ) and (p′′, (q, wa)) ∈ +T by definition of T. +• If wa /∈ N and wa ∈ im(R), then there is some b ∈ dom(R) with R(b) = wa. +We have p +w +M p′ +a +M p′′ and so p +b +α(M) p′′ by the definition of α(M). +Using that S is a simulation, we obtain a transition q +b +N q′ in N. +By +rule (2γ), this translates into a transition (q, w) +a +γ(N) (q′, ε) in γ(N). By +definition of T, we find (p′′, (q′, ε)) ∈ T. +• If wa /∈ N and wa /∈ im(R), then we have (q, w) +χ +γ(N) by rule (3γ). In this +case, Then, we also have (p′, χ) ∈ T. +(b) Consider (p, χ) ∈ T and p +a +M p′ in M. We have χ +a +γ(N) χ by rule (4γ) and +(p′, χ) ∈ T, again. +(⇐) Assume M ⊑ γR(N) in LTS(A), witnessed by a simulation S ⊆ QM × Qγ(N). Define +T ⊆ Qα(M) × QN by +T := {(p, q) ∈ Qα(M) × QN | (p, (q, ε)) ∈ S}. +Here, we use that Qα(M) ⊆ QM. For the initial states, note that (qα(M) +0 +, qN +0 ) ∈ T +because +(qM +0 , (qN +0 , ε)) = (qM +0 , qγ(N) +0 +) ∈ S. +21 + +For the remaining verification, consider (p, q) ∈ T and a transition p +b +α(M) p′ in +α(M). By the definition of α, we have b ∈ dom(R) and a run +p +R(b) +M p′ +in M. +Using that S is a simulation and that (p, (q, ε)) ∈ S, this yields a run +(q, ε) +R(b) +γ(N) q′ +in γ(N) +with +(p′, q′) ∈ S. +We do not know yet in which of the two components of Qγ(N) = (QN × N) ∪ {chi} the +state q′ is. We investigate by decomposing the run for R(b) ∈ A+ into wa = R(b) for +w ∈ A∗ and a ∈ A, calling the intermediate state ¯q ∈ Qγ(N): +(q, ε) +w +γ(N) ¯q +a +γ(N) q′ +in γ(N). +Since w ≨ R(b), we have w ∈ N. Looking at the rules for the transitions of γ(N), we +see that the only option for the transitions in (q, ε) +w +γ(N) ¯q with w ∈ N is via rule +(1γ), so we necessarily obtain ¯q = (q, w): +(q, ε) +w +γ(N) (q, w) +a +γ(N) q′ +Using that wa = R(b), only rule (2γ) can have produced the transition (q, w) +a +q′, +hence q′ = (q′′, ε) for some q′′ ∈ QN with q +b +N q′′ in N. In total, we have (p′, (q′′, ε)) = +(p′, q′) ∈ S and by the definition of T: +(p′, q′′) ∈ T +q +b +N q′′ +in N. +This shows that T is indeed a simulation. +Corollary 5.4 (Monotonicity of concretization). For every action code R ∈ Code(A, B), +M ⊑ N in LTS(B) implies γR(M) ⊑ γR(N) in LTS(A). +Proof. It is a standard result that the operators in a Galois connections are monotone. We +recall the concrete proof for the convenience of the reader. Consider M ⊑ N in LTS(B). By +the reflexivity, we have +γR(M) ⊑ γR(M) +in LTS(A). +Applying the Galois connection (Proposition 5.3) from right to left yields +αR(γR(M)) ⊑ M +in LTS(B). +By transitivity of ⊑ and M ⊑ N, we obtain +αR(γR(M)) ⊑ N +in LTS(B). +Applying the Galois connection (Proposition 5.3) conversely from left to right yields the +desired +γR(M) ⊑ γR(N) +in LTS(A). +22 + +Remark 5.5. Monotonicity of concretization also follows by observing that the rules in +Definition 5.1 all fit the tyft format of [17] if we view (·, w) as a unary operator for each +sequence w ∈ N. +Monotonicity then follows from the result of [17] that the simulation +preorder is a congruence for any operator defined using the tyft format. Since contraction +also can be defined using the tyft format, also monotonicity of contraction (Proposition 3.11) +follows from the result of [17]. +One may think that the many transitions to the chaos state χ, would make the construc- +tion γR trivial. However, only those paths lead to the chaos for which the action code is not +defined: +Lemma 5.6. For R ∈ Code(A, B), w ∈ A∗, and M ∈ LTS(B), if (q, ε) +w +χ in γR(M), +then w ̸≤ R(b) for all b ∈ B. +Proof. Let ua ≤ w, for u ∈ A∗, a ∈ A, be the shortest prefix such that (q, ε) +ua +χ in +γR(M). (Note that we can assume this shortest prefix to be non-empty because (q, ε) ̸= χ). +Thus, the last transition (for label a) must have been produced by rule (3γ): +(q, ε) +u +(q′, s) +a +χ +• If u ̸= s, then there is some b′ ∈ B with R(b′) ≤ u. If we also had w ≤ R(b), this +would lead to a contradiction: R(b′) ≤ u ≨ ua ≤ w ≤ R(b) so b = b′ by prefix-freeness +(1) and on the other hand R(b′) ≨ R(b) = R(b′), a contradiction. +• If u = s, then by the assumptions of rule (3γ), we obtain ua = sa /∈ N and ua = sa /∈ +im(R). Hence, we have ua ̸≤ R(b) and so also w ̸≤ R(b). +Corollary 5.7. Assume that for every w ∈ A∗, there is some b ∈ dom(R) with w ≤ R(b) or +R(b) ≤ w. Then for all M ∈ LTS(B), χ is not reachable in γR(M). +Like refinement, concretization preserves determinism. +Proposition 5.8 (Concretization preserves determinism). For every R ∈ Code(A, B), if +M ∈ LTS(B) is a deterministic LTS, then γR(M) is deterministic, too. +Proof. We verify the determinacy seperately for the disjoint components of Qγ(M) := (QM × +N) ∪ {χ} +1. For (q, w) ∈ QM × N and two transitions +(q, w) +a +γ(M) ¯q1 +(q, w) +a +γ(M) ¯q2 +we distinguish cases like in the assumptions of the rules for +γ(M): +• If wa ∈ N, then both transitions have been produced by rule (1γ) and so ¯q1 = +(q, wa) = ¯q2. +23 + +• If wa /∈ N and wa ∈ im(R), then there are b1, b2 ∈ dom(R) with +q +b1 +M q′ +1 +¯q1 = (q′ +1, ε) +R(b1) = wa +q +b2 +M q′ +2 +¯q2 = (q′ +2, ε) +R(b2) = wa +Since R: B⇀A+ is prefix-free (1), it is in particular injective and so b1 = b2. +The LTS M was assumed to be deterministic, thus q′ +1 = q′ +2 and so ¯q1 = (q′ +1, ε) = +(q′ +2, ε) = ¯q2. +• If wa /∈ N and wa /∈ im(R), then both transitions have been produced by rule +(3γ) and so ¯q1 = χ = ¯q2. +2. Any two outgoing transitions of χ +χ +a +γ(M) ¯q1 +and +χ +a +γ(M) ¯q2 +have necessarily been created by (4γ), and so ¯q1 = χ = ¯q2. +If we are willing to make an extra assumption then γR is even the right inverse of αR, that +is, we have a Galois insertion. The assumption is that every b ∈ B that labels a transition +in the LTS also occurs in the action code R: +Theorem 5.9 (Galois insertion). For every R ∈ Code(A, B) and every M ∈ LTS(B), if +M ∈ LTS(dom(R)), then M ∼= αR(γR(M)). +Note that dom(R) ⊆ B, and so LTS(dom(R)) ⊆ LTS(B). +Proof. Since we have only one action code R at hand, we omit the index R in α and γ in +this proof. The LTS α(γ(M)) has precisely the states +Qα(γ(M)) = {(q, ε) | q ∈ QM}. +In order to see that, we find: +(⊆) If ¯q ∈ Qα(γ(M)), then there is a word b1 · · · bn ∈ dom(R)∗ and are states ¯q1, . . . , ¯qn with +qγ(M) +0 += (qM +0 , ε) +b1 +α(γ(M)) ¯q1 +b2 +α(γ(M)) · · · +bn +α(γ(M)) ¯qn = ¯q. +in α(γ(M)). +By definition of α, this corresponds to transitions +qγ(M) +0 += (qM +0 , ε) +R(b1) +γ(M) ¯q1 +R(b2) +γ(M) · · · +R(bn) +γ(M) ¯qn = ¯q. +in γ(M). +Applying Lemma 5.2 to every ¯qi, we obtain that ¯q = (q, ε) for some q ∈ QM. Note in +particular, ¯q ̸= χ and so χ /∈ Qα(γ(M)). +(⊇) The converse inclusion iterates the other direction of Lemma 5.2: we assume M ∈ +LTS(B) to be reachable, hence every state q ∈ QM is reachable via some word b1 · · · bn ∈ +B∗ by iterating Lemma 5.2: +qM +0 +b1 · · · bn +M q. +24 + +The assumption that M ∈ LTS(dom(R)) implies that R: B⇀A+ is defined for every +bi, and thus (q, ε) is reachable in Qγ(M): +qγ(M) +0 += (qM +0 , ε) +R(b1) · · · R(bn) +γ(M) (q, ε). +And hence, the definition of α then sends this run to +qα(γ(M)) +0 += qγ(M) +0 += (qM +0 , ε) +b1 · · · bn +α(γ(M)) (q, ε). +and so (q, ε) ∈ Qα(γ(M)). +The witnessing bijective bisimulation is +φ: QM −→ Qα(γ(M)) +φ(q) = (q, ε) +∈ Qα(γ(M)) ⊆ Qγ(M). +By our above characterization of Qα(γ(M)), φ is a bijection. It remains to verify that φ is a +bisimulation: +• For every transition in α(γ(M)), concretely (q, ε) +b +(q′, ε), we have (q, ε) +R(b) (q′, ε) +in γ(M) by the definition of α. By Lemma 5.2, this implies q +b +q′ in M; and indeed +φ(q) = (q, ε) and φ(q′) = (q′, ε). +• Conversely, for every transition q +b +q′ in M, we have a transition +(q, ε) +R(b) (q′, ε) +in γ(M) +by Lemma 5.2 and by b ∈ dom(R) provided by the assumption M ∈ LTS(dom(R)). +By the definition of α, we thus have +φ(q) = (q, ε) +b +(q′, ε) = φ(q′) +in α(γ(M)). +In total, φ is an isomorphism in LTS(B). +Since we may add the chaos state χ in the concretization, which introduces nondetermin- +ism, it is clear that γR is not a left inverse of αR in general. +6 +Action Code Composition +Since notions of abstraction can be stacked up, it is natural to consider multiple adaptors +for multiple action codes. +Assume an action code R ∈ Code(A, B) and an action code +S ∈ Code(B, C). Then the composition of R and S should be an action code from A to C. +Definition 6.1. Given two map-based action codes R: B⇀A+ and S : C⇀B+, we define +their composition (R ∗ S): C⇀A+ by +(R ∗ S)(c) = +� +R(b1) · · · R(bn) +if S(c) = b1 · · · bn with ∀i: bi ∈ dom(R) +undefined +otherwise +25 + +The composed action code R ∗ S is only defined for c ∈ C if S is defined for c and +additionally R is defined for every letter bi ∈ B that appears in the word S(c) ∈ B+. +Remark 6.2. The defined composition is an instance of Kleisli composition for a monad, +which is a standard concept in functional programming and category theory. Kleisli compo- +sition is a recipe to compose maps of the form C → T(B) and B → T(A) to a map of type +C → T(A), where T is a monad. In our case, the monad is T(X) = X+ + 1 where 1 is an +arbitrary singleton and + denotes disjoint union. This monad T itself is a combination of two +monads: S(X) = X+ is the free semigroup-monad. Monads corresponds to algebraic theories +and the algebraic theory corresponding to S is that of semigroups. The monad P(X) = X +1 +is called the maybe monad (or sometimes called optional in programming), which allows to +model partial maps. The algebraic theory corresponding to P is that of pointed sets (the +theory consists of one nullary operation). Warning: even though T(X) = P(S(X)) = X+ +1 +and M(X) = X∗ are naturally isomorphic, they are different monads, because M is the list +monad, whose corresponding algebraic theory is that of monoids. +Lemma 6.3. Action codes are closed under composition. +Concretely, given two map-based action codes R: B⇀A+ and S : C⇀B+, their Kleisli +composition (R ∗ S): C⇀A+ is again a prefix-free partial map. +Proof. We need to show that (R ∗ S): C⇀A+ is prefix-free. To this end, consider c, c′ ∈ +dom(R ∗ S) with +(R ∗ S)(c) ≤ (R ∗ S)(c′). +Since R ∗ S is defined for both c and c′ we can spell out the words as +(R ∗ S)(c) = R(b1) · · · R(bn) +for n ∈ N and S(c) = b1 · · · bn +(R ∗ S)(c′) = R(b′ +1) · · · R(b′ +m) +for m ∈ N and S(c′) = b′ +1 · · · b′ +m. +Note that we do not know yet whether n or m is bigger! We only know that +R(b1) · · · R(bn) ≤ R(b′ +1) · · · R(b′ +m). +(4) +We now show by induction that +for all i with 0 ≤ i ≤ min(n, m): +bi = b′ +i. +• In the base case i = 0, there is nothing to be shown. +• In the step for i, assume that we have ∀0 ≤ j < i: bi = b′ +i as the induction hypothesis. +Thus, we also have R(bj) = R(b′ +j) for all j < i and so the words +R(b1) · · · R(bi) · · · R(bn) and R(b′ +1) · · · R(b′ +i) · · · R(b′ +m). +have a common prefix R(b1) · · · R(bi−1) = R(b′ +1) · · · R(b′ +i−1). For general u, v, w ∈ C∗, +if uv ≤ uw, then v ≤ w. So after removing the common prefix from both sides of (4), +obtain +R(bi) · · · R(bn) ≤ R(b′ +i) · · · R(b′ +m). +In such a scenario, we either have R(bi) ≤ R(b′ +i) or R(bi) ≥ R(b′ +i). Since R is prefix-free +(1), we obtain bi = b′ +i in either case. +26 + +We can now use the inductively proven statement to show that b1 · · · bn ≤ b′ +1 · · · b′ +m by case +distinction: +• If min(n, m) = m, i.e. n ≥ m, then we can remove the common prefix R(b1) · · · R(bm) = +R(b′ +1) · · · R(b′ +m) from both sides of (4) in order to obtain +R(bm+1) · · · R(bn) ≤ ε +Since all R(bi) ∈ A+ for all 1 ≤ i ≤ n, we necessarily have m = n. This implies +b1 · · · bn ≤ b′ +1 · · · b′ +m (both sides are identical). +• If min(n, m) = n, i.e. n ≤ m, then we directly have b1 · · · bn ≤ b′ +1 · · · b′ +m. +So in any case, S(c) = b1 · · · bn ≤ b′ +1 · · · b′ +m = S(c′). +Using that S is prefix-free (1), we +conclude c = c′, as desired. +Theorem 6.4. Contraction commutes with action code composition: for action codes R ∈ +Code(A, B), S ∈ Code(B, C), we have +αR∗S(M) = αS(αR(M)) +for all M ∈ LTS(A). +In other words, we have a commutative diagram: +LTS(A) +LTS(C) +LTS(B) +αR +αR∗S +αS +Proof. We show that the systems αR∗S(M) and αS(αR(M)) are even identical. Note that +we have state sets: +QαR∗S(M) ⊆ QM and QαR(αS(M)) ⊆ QαS(M) ⊆ QM +which are both subsets of QM. Their initial states are identical, because they are both qM +0 . +We establish the isomorphism by simultaneously showing that the state sets match and that +the transitions match: +(⊆) Consider a transition +q +c +q′ +in αR∗S(M) +for which we already assume that q ∈ QαR(αS(M)). Thus, we have +q +(R ∗ S)(c) q′ +in M. +By the definition of R ∗ S, we have c ∈ dom(S) and b1, . . . , bn ∈ dom(R) with S(c) = +b1 · · · bn, so above sequence can be rewritten as +q +R(b1) · · · R(bn) q′ +in M +27 + +or equivalently +q +R(b1) · · · +R(bn) q′ +in M +By definition of αR, we have +q +b1 · · · +bn q′ +in αR(M) +or equivalently +q +S(c) q′ +in αR(M). +Finally, by the definition of αS, this yields +q +c +q′ +in αS(αR(M)). +This first shows that all states of αR∗S(M) are also contained in αR(αS(M)) and +secondly that the transitions are included, too. +(⊇) The converse direction is analogous, starting with a transition +q +c +q′ +in αS(αR(M)) +for which we know q ∈ QαR∗S(M) already. Thus, c ∈ dom(S) and we obtain +q +S(c) q′ +in αR(M). +With b1 · · · bn = S(c), we have +q +b1 · · · +bn q′ +in αR(M). +This implies that bi ∈ dom(R) for every 1 ≤ i ≤ n and moreover +� +q +R(b1) · · · +R(bn) q′� += +� +q +R(b1) · · · R(bn) q′� +in M. +Since all bi ∈ dom(R) and c ∈ dom(S), we find that c ∈ dom(R∗S) and so (R∗S)(c) = +R(b1) · · · R(bn) and +q +(R ∗ S)(c) q′ +in M. +Finally, by the definition of αR∗S, we conclude +q +c +q′ +in αR∗S(M). +It is a standard result about Galois connections (and adjunctions in general) that they +are compatible with composition: the right-adjoint of the composition of two functions is +equal to the composition of the respective right-adjoints. One only needs to be warned that +‘equal’ here refers to the equality induced by the order ⊑, which means mutual simulation: +For all action codes R ∈ Code(A, B), S ∈ Code(B, C) and M ∈ LTS(C): +γR∗S(M) ⊑ γR(γS(M)) +and +γR∗S(M) ⊒ γR(γS(M)). +This is however weaker than the notion of isomorphism we consider (Definition 2.8). Con- +cretely, we even have the following counterexample with γR∗S(M) ̸∼= γR(γS(M)). +28 + +Example 6.5. Concretization does not commute with action code composition: Consider +sets A = {a}, B = {b}, C = {c} and the action codes +R ∈ Code(A, B) +R: B⇀A+ +b �→ a a +S ∈ Code(B, C) +S : C⇀B+ +undefined everywhere +Start with a singleton system that has no transitions: +M := +q0 +start +in LTS(C) +For the empty S, the concretization γS : LTS(C) → LTS(B) sends this into the system +γS(M) = +q0, ε +start +χ +b +b +in LTS(B) +We have a b-transition from q0 to χ because S(b) is undefined. For the next action code R, +the concretization γR : LTS(B) → LTS(A) treats χ as an ordinary state, so it produces the +following: +γR(γS(M)) ∼= +(q0, ε), ε +start +(q0, ε), a +χ +χ, a +a +a +a +a +in LTS(A) +Here, we omitted the unreachable chaos state χ introduced by γR, because the unreachable +parts are not relevant for our notion of isomorphism ∼=. +On the other hand, the composed action code R∗S ∈ Code(A, C) is undefined everywhere, +so analogously to γS, concretization for R ∗ S sends above M ∈ LTS(C) to +γR∗S(M) = +q0, ε +start +χ +a +a +in LTS(A). +Obviously, γR∗S(M) is not isomorphic to γR(γS(M)), but there are canonical simulations in +either direction, induced by the Galois connection between α and γ, using that α commutes +with action code composition. +Theorem 6.6. Refinement commutes with action code composition: for action codes R ∈ +Code(A, B), S ∈ Code(B, C), we have +ϱR∗S(M) = ϱR(αS(M)) +for all M ∈ LTS(C). +In other words, we have a commutative diagram: +LTS(A) +LTS(C) +LTS(B) +ϱR +ϱR∗S +ϱS +29 + +Proof. For the map-based action code R: B⇀A+, define the partial map +R∗ : B∗⇀A∗ +R∗(ε) = ε +R∗(b w) = R(b) R∗(w) +(if both defined). +By this, we mean that the inductive case R∗(b w) is only defined if both R(b) and R∗(w) are +defined.1 With this definition, we have that +(R ∗ S)(c) = R∗(S(c)) for all c ∈ C. +For the isomorphism h: ϱR(ϱS(M)) → ϱR∗S∗(M), the involved state sets are by Defini- +tion 4.1 of the form: +QϱS(M) ⊆ QM × B∗ +QϱR(ϱS(M)) ⊆ (QM × B∗) × A∗ +QϱR∗S(M) ⊆ QM × A∗ +Define a partial map h: QϱR(ϱS(M)) → QϱR∗S∗(M) by +h((q, u), v) = +� +(q, R∗(u) v) +if R∗(u) is defined +undefined +otherwise. +Such a partial map is sufficient to establish an isomorphism between the reachable parts +(cf. Definition 2.8) of ϱR(ϱS(M)) and ϱR∗S∗(M), because we can show that if ((q, u), v) is +reachable, then R∗(u) is defined: if ((q, u), v) is reachable, then the shortest path from the +initial state must end with a path of the form: +((q, ε), ε) +w +((q, u), ε) +v′ +((q, u), v) +in ϱR(ϱS(M)). +If we require that v′ is the shortest path from ((q, u), ε) to ((q, u), v), then all transitions of +v′ must come from rule (1ϱ), and so v = v′. If we require w to be the shortest path, then by +an iterated application of Lemma 4.6, we find that w = R∗(u). +In order to show that h is a simulation, consider a reachable transition +((q, u), v) +a +((q′, u′), v′) +in ϱR(ϱS(M)). +The transition being reachable implies that R∗(u) is defined. By Lemma 4.6, there exists a +transition +(q, u) +b +(q′′, u′′) +in ϱS(M) +with R(b) = var and some r ∈ A∗ such that +((q, u), ε) +v +((q, u), v) +a +((q′, u′), v′) +r +((q′′, u′′), ε) +in ϱR(ϱS(M)). +Applying Lemma 4.6 to the above b-transition in ϱS(M) provides us with some c ∈ C, +q′′′ ∈ QM, and s ∈ B∗ with S(c) = ubs such that +q +c +q′′′ +in M +and +(q, ε) +u +(q, u) +b +(q′′, u′′) +s +(q′′′, ε) +in ϱS(M). +1R∗ is also called the Kleisli extension of R for the monad (−)∗ on partial maps +30 + +Since all involved states are reachable, R∗(u) and R∗(s) are defined. In total, we have that +(R ∗ S)(c) = R∗(ubs) = R∗(u) R(b) R∗(s) = R∗(u) var R∗(s) +and in particular +R∗(u) va ≤ (R ∗ S)(c). +Thus, the state +h((q, u), v) = (q, R∗(u) v) in ϱR∗S(M) +has an a-transition to +h((q′, u′), v′) = (q′, R∗(u′) v′), +as desired. +For the verification that h is a simulation in the converse direction, i.e. from ϱR∗S(M) to +ϱR(ϱS(M)), consider a transition +(q, u) +a +(q′, u′) +in ϱR∗S(M). +Again, using Lemma 4.6, we obtain c ∈ C with +(R ∗ S)(c) = uav and q +c +q′′ and (q′, u′) +v +(q′′, ε). +By the definition of R ∗ S, we thus obtain that u ∈ A∗ must be of the shape +R∗(w) r = u +for some w ∈ dom(R)∗ +By the definition of R ∗ S, we have +w b ≤ S(c) with u = R∗(w) r and ra ≤ R(b). +Then, h((q, w), r) = (q, u) and we distinguish: +• If ua = (R∗S)(c), then the above a-transition in ϱR∗S(M) is produced by rule (2ϱ), and +we can use the same rule in ϱS(M) and ϱR(ϱS(M)) to establish the desired transition +a-transition to +((q, w), r) +a +((q, ε), ε) +in ϱR(ϱS(M)). +• If ua ≨ (R∗S)(c) but ra = R(b), then we use the rule (1ϱ) in ϱS(M) and but rule (2ϱ) +in ϱR(ϱS(M)): +((q, w), r) +a +((q, w b), ε) +in ϱR(ϱS(M)). +• If ua ≨ (R∗S)(c) and ra ≨ R(b), then we use rules (1ϱ) in both ϱS(M) and ϱR(ϱS(M)): +((q, w), r) +a +((q, w), r a) +in ϱR(ϱS(M)). +Hence, in any of the above cases, we have a corresponding a-transition in ϱR(ϱS(M)). +31 + +Learner / +Tester +Adaptor +R +SUT +M +x ∈ X +i ∈ I +o ∈ O +y ∈ Y +Figure 9: Using action codes for learning/testing. +7 +Adaptors +In this section, we describe how action codes may be used for learning and testing of black- +box systems that can be modelled as Mealy machines. The general architecture is shown in +Figure 9. On the right we see the system under test (SUT), some piece of hardware/software +whose behavior can be modeled by a Mealy machine M with inputs I and outputs O. On the +left we see the learner/tester, an agent which either tries to construct an abstract model N +of M by performing experiments, or already has such a model N and performs experiments +(tests) to find counterexamples which demonstrate that M and N behave differently. The +learner/tester uses abstract inputs X and outputs Y . In between the learner/tester and the +SUT we place an adaptor, which uses an action code R to translate between the abstract +world of the learner/tester and the concrete world of the SUT. In order to enable the adaptor +to do its job, we need to make four (reasonable) assumptions. +Our first assumption is that the SUT will accept any input from I in any state, that is, +we require that M is input enabled: +∀q ∈ QM ∀i ∈ I : q +i/ +M +Our second assumption ensures that whenever the adaptor sends a concrete symbol i ∈ I to +the SUT, the adaptor will accept any output o ∈ O that the SUT may possibly produce in +response. We require that R is output enabled for M: +∀r ∈ QR ∀i ∈ I ∀o ∈ O: r +i/ +R ∧ +i/o +M +⇒ +r +i/o +R +Our third assumption ensures that when the adaptor receives an abstract input x ∈ X from +the learner/tester, the adaptor can choose concrete inputs from I that drive R from its initial +state r0 to a leaf with label (x, y), for some y ∈ Y . The output y can then be returned as a +response to the learner/tester. Reaching a leaf with label x is nontrivial since the transitions +taken in R are also determined by the outputs provided by the SUT. We may think of the +situation in terms of a 2 player game where the adaptor wins if the game ends in an x leaf, +and the SUT wins otherwise. We require that R is finite (has a finite number of states) and +has a winning strategy for every input x ∈ X. The notion of winning is formalized in the +following inductive definition. +Definition 7.1 (Winning). Let R = ⟨R, r0, +, l⟩ ∈ Code(I × O, X × Y ) be a finite action +code and let x ∈ X. Then +32 + +1. A leaf r ∈ R is winning for x if π1(l(r)) = x.2 +2. An internal state r ∈ R is winning for x with input i ∈ I if r +i/ and, for each transition +of the form r +i/o r′, r′ is winning for x. +3. An internal state r ∈ R is winning for x if it is winning for x with some i ∈ I. +4. R has a winning strategy for x if r0 is winning for x. +Example 7.2. The examples of action codes for Mealy machines that we have seen thus far +(Figures 5, 6 and 7) are winning for all the abstract inputs that label their leafs. The action +code of Figure 5 is not winning for the abstract input +(latte macchiato), for the simple +reason that this input does not label any of the leafs. If we remove the incoming transition of +state r13 in Figure 5, then the resulting code is no longer winning for +(espresso), although +it is still winning for +(coffee). +Our fourth and final assumption is that action code R is determinate. If an action code is +determinate then, for each state r and abstract input x, there is at most one concrete input +i such that r is winning for x with i. +Definition 7.3 (Determinate). An action code R is determinate if, for each state r, whenever +r +i1/ +r1, r +i2/ +r2 and from both r1 and r2 there is a path to a leaf labeled with input x, +then i1 = i2. +Example 7.4. All the examples of action codes for Mealy machines that we have seen thus +far (Figures 5, 6 and 7) are determinate. Figure 10 gives an example of an action code that +is not determinate: in state r0 two different concrete inputs a and b are enabled that lead to +leafs with the same abstract input label 0. Note that (trivially) this action code does have a +winning strategy for abstract input 0. +r0 +start +r1 +0/A +r2 +0/B +a/0 +b/0 +Figure 10: An action code that is not determinate. +2We use projections functions π1 and π2 to denote the first and second element of a pair, respectively. So +π1(x, y) = x and π2(x, y) = y. +33 + +Algorithm 1 Pseudocode for an adaptor that implements action code R. +1: function Adaptor(R) +2: +while true do +3: +x ← Receive-from-learner() +4: +r ← r0 +5: +while r is internal do +▷ loop invariant: r is winning for x +6: +i ← unique input such that r is winning for x with i +7: +Send-to-SUT(i) +8: +o ← Receive-from-SUT() +9: +r ← unique state r′ such that r +i/o r′ +▷ R output enabled for M +10: +end while +11: +Send-to-learner(π2(l(r))) +12: +end while +13: end function +Algorithm 1 shows pseudocode for an adaptor that implements action code R. During +learning/testing, the adaptor records the current state of the action code in a variable r. +When an abstract input x arrives, it first sets r to r0. As long as current state r is internal, +the adaptor chooses an input i that is winning for x, and forwards it to the SUT. When the +SUT replies with an output o, the adaptor sets r to a state r′ with r +i/o +−→ r′. When the new +r is internal the adaptor chooses again a winning input, and updates its current state after +interacting with the SUT, etc. When the new r is a leaf with label (x, y) then the adaptor +returns symbol y to the learner/tester and waits for the next abstract input to arrive. +From the perspective of the learner/tester, the combination of the adaptor and SUT be- +haves the same as the contraction αR(M). Below we will formalize this statement by model- +ing both the combination of adaptor and SUT, as well as contraction αR(M) as expressions in +the process calculus CCS [27], and then establish the existence of delay simulations between +these expressions. This implies that both expressions have the same traces if we remove all +occurrences of the synchronizations between adaptor and SUT, which are invisible from the +perspective of the learner. +Definition 7.5. Let M = ⟨Q, q0, +⟩ ∈ LTS(A) be an LTS, where A is a set of labels that +contains the hidden action τ. Let q +q′ denote that there is finite sequence of states +r0, . . . , rn ∈ Q such that r0 = q, rn = q′ and ri−1 +τ +ri for all 1 ≤ i ≤ n. A relation +S ⊆ Q × Q is called a delay simulation if it satisfies the following transfer property: +• If (q, r) ∈ S and q +a +q′ then either a = τ and q = q′, or ∃r′, r′′ such that r +r′ +a +r′′ +and (q′, r′′) ∈ S. +We write q ⊑d r if there exists a weak delay simulation that relates q and r. We say that q +and r are delay simulation equivalent, notation q ≡d r, if both q ⊑d r and r ⊑d q. +Theorem 7.6. Let M ∈ LTS(I × O) be an input enabled Mealy machine and let R ∈ +Code(I ×O, X ×Y ) be a finite, determinate action code that has a winning strategy for every +input in X and that is output enabled for M. Then the composition of an implementation for +M and an adaptor for R is delay simulation equivalent to an implementation for αR(M). +34 + +Proof. We describe the behavior of an implementation for M and an adaptor for R formally +as expressions in Milner’s Calculus of Communicating Systems (CCS) [27]. The semantics of +CCS is defined in terms of an infinite LTS in which the states are CCS expressions, and the +transitions between states are defined by structural operational semantics rules given in [27]. +In the rest of this proof, we will assume that the reader is familiar with the CCS calculus. +In our CCS expressions we use action names taken from I, O, X and Y , and without loss of +generality we assume these four sets to be disjoint. Process Impl(M) describes the behavior of +an implementation for M in which inputs and outputs are separated and occur sequentially. +We define Impl(M) as the CCS expression M(qM +0 ), where for q ∈ QM and i ∈ I, +M(q) += +� +i∈I +i · M(q, i) +M(q, i) += +� +o∈O,q′∈QM | q +i/o +q′ +¯o · M(q′) +Similarly, we introduce a process Adaptor(R) that describes the behavior of an adaptor for +action code R. Following the pseudocode of Algorithm 1, we define Adaptor(R) as the CCS +expression P(r0), where for r ∈ R and x ∈ X, +P(r) += +� +x∈X +x · Q(r, x) +Q(r, x) += ¯i · R(r, x) +if r is internal and i winning for x in r +R(r, x) += +� +o∈O,r′∈R | +i/o +M∧r +i/o +r′∧i winning for x in r +o · Q(r′, x) +Q(r, x) += +π2(l(r)) · P(r0) +if r is a leaf +Processes Adaptor(R) and Impl(M) may synchronize via actions taken from I ∪O. If we com- +pose these processes using the CCS composition operator |, and apply the CCS restriction +operator \ to hide all communications, we obtain a CCS expression that describes the behav- +ior of the parallel composition of the adaptor and the SUT. We claim that this composition +is delay simulation equivalent to the expression Impl(αR(M)) that describes the behavior of +an implementation of αR(M): +(Adaptor(R) | Impl(M)) \ (I ∪ O) +≡d +Impl(αR(M)) +(5) +Here we define Impl(αR(M)) as the CCS expression N(qα(M) +0 +), where for q ∈ Qα(M) and +x ∈ X, +N(q) += +� +x∈X +x · N(q, x) +N(q, x) += +� +y∈Y,q′∈Qα(M) | q +x/y +α(M)q′ +¯y · N(q′) +35 + +Consider the following relation S between CCS expressions: +S += +{((P(r0) | M(q)) \ (I ∪ O), N(q)) | q ∈ Qα(M)} +∪ {((Q(r, x) | M(q′)) \ (I ∪ O), N(q, x)) | q ∈ Qα(M), q′ ∈ QM, +r ∈ R winning for x ∈ X ∧ ∃σ ∈ (I × O)∗ : r0 +σ +R r ∧ q +σ +M q′} +∪ {((R(r, x) | M(q′, i)) \ (I ∪ O), N(q, x)) | q ∈ Qα(M), q′ ∈ QM, +r ∈ R winning for x ∈ X with i ∈ I ∧ ∃σ ∈ (I × O)∗ : r0 +σ +R r ∧ q +σ +M q′} +We claim that S is a delay simulation relation. In order to prove this, we check that the +transfer property holds for all pairs of related states and enabled transitions: +1. Assume ((P(r0) | M(q))\(I ∪O), N(q)) ∈ S and (P(r0) | M(q))\(I ∪O) +x (Q(r0, x) | +M(q)) \ (I ∪ O), for some x ∈ X. We observe that N(q) +x +N(q, x) and note that +((Q(r0, x) | M(q)) \ (I ∪ O), N(q, x)) ∈ S since r0 is winning for x, r0 +σ +R r0 and +q +σ +M q. +2. Assume ((Q(r, x) | M(q′)) \ (I ∪ O), N(q, x)) ∈ S and (Q(r, x) | M(q′)) \ (I ∪ O) +τ +(R(r, x) | M(q′, i)) \ (I ∪ O), for r internal and i the unique input that is winning for x +in r. By the assumption, q ∈ Qα(M), q′ ∈ QM, and there exists σ ∈ (I × O)∗ such that +r0 +σ +R r and q +σ +M q′. Then ((R(r, x) | M(q′, i))\(I ∪O), N(q, x)) ∈ S, as required. +3. Assume ((R(r, x) | M(q′, i))\(I ∪O), N(q, x)) ∈ S and (R(r, x) | M(q′, i))\(I ∪O) +τ +(Q(r′, x) | M(q′′))\(I ∪O), with r +i/o r′ and q′ +i/o q′′. By the assumption, q ∈ Qα(M), +q′ ∈ QM, r is winning for x with i, and there exists σ ∈ (I × O)∗ such that r0 +σ +R r +and q +σ +M q′. Then q′′ ∈ QM and, by definition of winning, r′ is winning for x. +Moreover, if we take σ′ = σ · (i, o), then r0 +σ′ +R r′ and q +σ′ +M q′′. This implies that +((Q(r′, x) | M(q′′)) \ (I ∪ O), N(q, x)) ∈ S, as required. +4. Assume ((Q(r, x) | M(q′))\(I ∪O), N(q, x)) ∈ S and (Q(r, x) | M(q′))\(I ∪O) +π2(l(r)) +(P(r0) | M(q′)) \ (I ∪ O), for r a leaf. By the assumption, q ∈ Qα(M), q′ ∈ QM, r is +winning for x, and there exists σ ∈ (I × O)∗ such that r0 +σ +R r and q +σ +M q′. By +definition of the contraction operator, q +l(r) +α(M) q′ and q′ ∈ Qα(M). But this means +N(q, x) +π2(l(r)) +N(q′). Now observe that ((P(r0) | M(q′)) \ (I ∪ O), N(q′)) ∈ S, as +required. +Next consider the following relation T between CCS expressions: +T += +{(N(q), (P(r0) | M(q)) \ (I ∪ O)) | q ∈ Qα(M)} +∪ {(N(q, x), (Q(r0, x) | M(q)) \ (I ∪ O)) | q ∈ Qα(M)} +We claim that T is a delay simulation relation, and check that the transfer property holds +for all pairs of related states and enabled transitions: +1. Assume (N(q), (P(r0) | M(q)) \ (I ∪ O)) ∈ T and N(q) +x +N(q, x), for some x ∈ X. +We observe that (P(r0) | M(q)) \ (I ∪ O) +x (Q(r0, x) | M(q)) \ (I ∪ O) and note that +(N(q, x), (Q(r0, x) | M(q)) \ (I ∪ O)) ∈ T. +36 + +2. Assume (N(q, x), (Q(r0, x) | M(q)) \ (I ∪ O)) ∈ T and N(q, x) +¯y +N(q′). Then α(M) +has a transition q +x/y q′ and q′ ∈ Qα(M). By definition of α(M), R has a leaf r with +l(r) = (x, y) and there exists a sequence σ such that r0 +σ +R r and q +σ +M q′. Let +σ += +(i1, o1)(i2, o2) · · · (in, on) +Then R has states r1, . . . , rn and M has states s0, . . . , sn such that: +r0 +i1/o1 +R r1 +i2/o2 +R r2 · · · +in/on +R rn +q = s0 +i1/o1 +M s1 +i2/o2 +M s2 · · · +in/on +M sn = q′ +From these runs in R and M we may construct a sequence of τ-transitions: +(Q(r0, x) | M(s0)) \ (I ∪ O) +τ +(R(r0, x) | M(s0, i1)) \ (I ∪ O) +τ +(Q(r1, x) | M(s1)) \ (I ∪ O) +... +τ +(R(rn−1, x) | M(sn−1, in)) \ (I ∪ O) +τ +(Q(rn, x) | M(sn)) \ (I ∪ O) +Note that our assumptions that R is determinate and has a winning strategy for every +input x ∈ X impy that the inputs that occur in σ are always winning. From the above +sequence of τ-transitions we may conclude +(Q(r0, x) | M(q)) \ (I ∪ O) +(Q(r, x) | M(q′)) \ (I ∪ O) +Since (Q(r, x) | M(q′)) \ (I ∪ O) +¯y +(P(r0) | M(q′)) \ (I ∪ O) and (N(q′), (P(r0) | +M(q′)) \ (I ∪ O)) ∈ T, the transfer property follows. +Because S is a delay simulation from (Adaptor(R) | Impl(M)) \ (I ∪ O) to Impl(αR(M)), and +T is a delay simulation from Impl(αR(M)) to (Adaptor(R) | Impl(M)) \ (I ∪ O), identity (5) +follows, and thereby the theorem. +Active automata learning algorithms and tools for Mealy machines typically assume that +the system under learning is output deterministic3: the output and target state of a transition +are uniquely determined by its source state and input. +Definition 7.7. Mealy machine M is output deterministic if, for each state q and input i, +q +i/o r ∧ q +i/o′ +r′ +⇒ +o = o′ ∧ r = r′. +The proposition below asserts that for action codes that are determinate, contraction pre- +serves output determinism. This property makes it possible to use existing automata learning +tools to learn models of systems that consist of an output deterministic SUT composed with +a determinate adaptor. +3The notion of deterministic that we use in this article is the standard one for LTSs. In the literature on +Mealy machines and FSMs, machines that we call output deterministic are called deterministic, and machines +that we call deterministic are called observable. +37 + +Proposition 7.8. Suppose M is a Mealy machine and R is an action code. If M is output +deterministic and R is determinate then αR(M) is output deterministic. +Proof. Assume M is output deterministic and R is determinate. Let N = αR(M). Suppose +that N has transitions q +x/y′ +N q′ and q +x/y′′ +N q′′. We need to show y′ = y′′ and q′ = q′′. +The transitions have been derived using rule (2α), and formulated for tree-shaped action +codes R, we know there are generalized transitions +q +u′/s′ +M q′ +r0 +u′/s′ +R r′ +r′ ∈ L l(r′) = (x, y′) +q +u′′/s′′ +M q′′ +r0 +u′′/s′′ +R r′′ +r′′ ∈ L l(r′′) = (x, y′′) +Now since R is determinate, the first inputs in u′ and u′′ must be identical. But since M is +output deterministic, this implies that the first outputs in s′ and s′′ must also be identical. +Moreover, the paths from q to q′ and q′′ in M share the same initial transition. Since action +codes are deterministic, the paths from r0 to r′ and r′′ in R also share the same initial +transition. By repeating this line of reasoning, we can “zip together” the paths from q to q′ +and q′′ in M, and the paths from r0 to r′ and r′′ in R, and obtain u′ = u′′, s′ = s′′, q′ = q′′, +r′ = r′′ and y′ = y′′, as required. +8 +Discussion +By introducing the notion of action codes, we provided a new perspective on the problem of +how high-level state machine models with abstract actions can be related to low-level models +in which these actions are refined by sequences of concrete actions. This perspective may +help with the systematic design of adaptors during learning and testing, and the subsequent +interpretation of obtained results. Our theory allows for action codes (such as in Figure 5) +that are adaptive in the sense that outputs which occur in response to inputs at the concrete +level may determine the sequence of concrete inputs that refines an abstract input. We are +not aware of case studies in which such adaptive codes are used, but believe they may be +of practical interest. One may, for instance, consider a scenario in which an abstract action +AUTHENTICATE is refined by a protocol in which a user is either asked to authenticate by +entering a PIN code, or by providing a fingerprint. +Close to our work are the results of Rensink and Gorrieri [28], who investigate vertical +implementation relations to link specifications and implementations belonging to conceptu- +ally different levels of abstraction. These relations are indexed by a refinement function that +maps abstract actions into concrete processes. +Within a setting of a CCS-like language, +Rensink & Gorrieri list a number of proof rules that should hold for any vertical imple- +mentation relation, and propose vertical bisimulation as a candidate vertical implementation +relation for which these proof rules hold. In the setting of our paper, we can define two +vertical implementation relations ⊑R +γ and ⊑R +ϱ , for any action code R, by +M ⊑R +γ N +⇔ +M ⊑ γR(N), +M ⊑R +ϱ N +⇔ +M ⊑ ϱR(N). +38 + +Then ⊑R +ϱ ⊆ ⊑R +γ and both relations satisfy all language-independent proof rules of [28]. For +instance +M ⊑ M′ +M′ ⊑R +γ N ′ +N ′ ⊑ N +M ⊑R +γ N +(since γR is monotone and ⊑ is transitive). +With the action code R of Figure 4, both +implementation relations will relate the LTS of Figure 2 (right) with the LTS of Figure 1. +However, if we consider the vertical bisimulation preorder of Rensink and Gorrieri [28], the +LTS of Figure 2(right) does not implement the LTS of Figure 1, when indexed by a refinement +that maps a to 1 4 1, and b to 1 4 2. This example suggests that bisimulations may not be +suitable as vertical implementation relations. +Also close to our work are results of Burton et al [8], who propose a vertical implementa- +tion relation in the context of the CSP language. Instead of action codes, they use extraction +patterns, a strict monotonic map extr : Dom → B∗, where Dom is the prefix closure of a +set dom ⊆ A∗ of concrete action sequences that may be regarded as complete. By providing +a mapping from sequences of concrete actions to sequences of abstract actions, extraction +patterns are more general than our action codes. However, since extraction mappings are +not required to have an inverse, establishing interesting Galois connections in this general +setting may be difficult. With an extraction pattern defined in the obvious way, the LTS of +Figure 2(right) does implement the LTS of Figure 1 according to the implementation relation +proposed by Burton et al [8]. +We developed our theory for LTSs and Mealy machines, using the simulation preorder as +the implementation relation. It would be interesting to transfer our results to other modeling +frameworks (such as IOTSs [32] and timed automata [3]) and other preorders and equivalences +in the linear-time branching-time spectrum for LTSs [15] and IOTSs [20]. +Our theory is orthogonal to the work of Aarts et al [1], which advocates the use of so- +called mappers to formalize adaptors that abstract the large action alphabets of realistic +network protocols into small sets of actions that can be handled by a learning tool. Aarts +et al [1] also describe the relation between abstract and concrete models using a Galois +connection. In practical applications of model learning, it can make sense to construct an +adaptor that combines a mapper in the sense of [1] with an action code as introduced in +this paper. Fiterău-Broştean et al [11] describe a small domain specific language which they +used to describe mapper components, and from which adaptor software can be generated +automatically. It would be interesting to extend this domain specific language so that it may +also be used to specify action codes for practical applications. +Different action codes lead to different contractions, and thereby to different abstract +views of the same system, see for instance Figure 6 and Figure 7. We may try to exploit +this fact during learning and testing. For instance, in case of a system M that is too big for +state-of-the-art learning algorithms, we may still succeed to learn partial views using cleverly +selected action codes. Using our Galois connections we then could obtain various upper and +lower bounds for M. Ideally, such an approach may even succeed to uniquely identify M. +Maarse [26] quantified the quality of a contraction αR(M) in terms of the graph-theoretic +concept of eccentricity. If q and q′ are states in an LTS M then d(q, q′) is defined as the +number of transitions in the shortest path from q to q′ (or ∞ if no such path exists). For +39 + +any set of states Q ⊆ QM, the eccentricity ε(Q) is defined as +max +q′∈QM min +q∈Q d(q, q′) +that is, the maximal distance one needs to travel to visit a state of M, starting from a +state of Q. A good contraction has a small set of states Q and a low eccentricity ε(Q): the +contraction only covers a small subset Q of the states of M, but any state from M can be +reached via a short sequences of transitions from a Q-state. +Acknowledgements +As part of a MSc thesis project under supervision of the first author, +Timo Maarse studied a different and more restricted type of action codes (called action +refinements) [26]. It turned out, however, that for these action codes, the concretization +operator is not monotone. The present paper was inspired by [26] and arose from our efforts +to fix this problem. We thank Paul Fiterău-Broştean for examples of the use of action codes +in model learning. 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Ito, editors, Theoretical Computer Science: Explor- +ing New Frontiers of Theoretical Informatics, pages 315–330, Berlin, Heidelberg, 2000. +Springer Berlin Heidelberg. +43 + diff --git a/kdAyT4oBgHgl3EQfYPeA/content/tmp_files/load_file.txt b/kdAyT4oBgHgl3EQfYPeA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8671773dac9a395ca9527781d3938edec283e760 --- /dev/null +++ b/kdAyT4oBgHgl3EQfYPeA/content/tmp_files/load_file.txt @@ -0,0 +1,1186 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf,len=1185 +page_content='Action Codes∗ Frits Vaandrager1 and Thorsten Wißmann2 1Radboud University, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='Vaandrager@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='nl 2Radboud University, Thorsten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='Wissmann@ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='nl January 3, 2023 Abstract We provide a new perspective on the problem how high-level state machine models with abstract actions can be related to low-level models in which these actions are refined by sequences of concrete actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We describe the connection between high-level and low-level actions using action codes, a variation of the prefix codes known from coding theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For each action code R, we introduce a contraction operator αR that turns a low-level model M into a high-level model, and a refinement operator ϱR that transforms a high-level model N into a low-level model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We establish a Galois connection ϱR(N) ⊑ M ⇔ N ⊑ αR(M), where ⊑ denotes the well-known simulation preorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In practice, we typically want to obtain an overapproximation of model M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' To this end, we also introduce a concretization operator γR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This operator behaves like the refinement operator, but adds arbitrary behavior at intermediate points during a refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We establish a second Galois connection αR(M) ⊑ N ⇔ M ⊑ γR(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We show how an action code may be used to construct an adaptor that translates between concrete and abstract inputs and outputs during learning and conformance testing of a black-box system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If M models a black-box system then αR(M) describes the behavior that can be observed by a tester/learner that interacts with this system via an adaptor derived from code R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Whenever we have established that αR(M) implements (or conforms to) N, we may conclude that M implements (or conforms to) γR(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 1 Introduction Labeled transition systems (LTSs) constitute one of the most fundamental modeling mech- anisms in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' An LTS consists of a directed graph whose nodes represent states and whose edges are labeled with actions and represent state transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' LTS-based formalisms such as Finite Automata [19], Finite State Machines [22], I/O automata [23], IOTSs [32], and process algebras [4] have been widely used to model and analyze a broad ∗Research supported by NWO TOP project 612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='852 “Grey-box learning of Interfaces for Refactoring Legacy Software (GIRLS)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' variety of software and hardware systems, and a rich body of theory has been developed for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In order to manage the complexity of computer-based systems, designers structure such systems into hierarchical layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This makes it possible to describe and analyze systems at different levels of abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Many LTS-based frameworks have been proposed to model systems at different hierarchical levels and to formally relate the resulting models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' [4, 14, 24, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In most of these frameworks, the states of a high-level LTS correspond to sets of states of a low-level LTS via simulation or bisimulation-like relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' However, the actions are fixed and considered to be atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Actions used at a lower level of abstraction can be hidden at a higher level, but higher-level actions will always be available at the lower level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For this reason, Rensink & Gorrieri [16, 28] argue that these (bi)simulations relate systems at the same conceptual level of abstraction, and therefore they call them horizontal implementation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' They contrast them with vertical implementation relations that compare systems that belong to conceptually different abstraction levels, and have different alphabets of actions they perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' A prototypical example of a system with hierarchical layers is a computer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' To reduce design complexity, such a network is organized as a stack of layers or levels, each one built upon the one below it [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' At the top is the application layer, with protocols such as HTTP and SMTP, and at the bottom we find the physical layer that is concerned with transmitting raw bits over a communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Now consider a host that is in some state s where it may receive an HTTP packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If P is the set of possible HTTP packets then, in an LTS model of the application layer, state s will contain outgoing transitions labeled with action receive(p), for each packet p ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' At the physical layer, however, receipt of an HTTP packet will correspond to a sequence of receive(b) actions, with b a bit in {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Only after the final bits have been received it will be clear which HTTP packet was actually received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Mechanisms for transforming high-level actions into sequences (or processes) of low-level actions have been addressed extensively in work on action refinements [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This work, however, is unable to describe the above scenario in a satisfactory manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In existing action refinement frameworks, it is somehow assumed that a host upfront correctly guesses the HTTP packet that it will receive, even before the first bit has arrived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In order to illustrate this problem, we consider the simplified example of an LTS, displayed in Figure 1, that accepts either an input a or an input b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' At a lower level of abstraction, input a may be implemented start a b Figure 1: A system that accepts an input a or b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' by three consecutive input actions 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 1, whereas input b is implemented by action sequence 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 2 (the ASCII encodings of a and b in octal format).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' An action refinement operator will replace the a-transition in Figure 1 by a sequence of three consecutive transitions with labels 1, 4 and 1, respectively, and will handle the b-transition in an analogous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, a 2 refinement operator will introduce a nondeterministic choice (Figure 2, left), rather than the deterministic behavior that one would like to see (Figure 2, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' As a consequence of this and other limitations, refinement operators have not found much practical use [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' start 1 1 4 4 1 2 start 1 4 1 2 Figure 2: Nondeterminism introduced by existing action refinement operators (left) vs desired behavior (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Based on the observation that any action can be modeled as a state change, some authors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' [2, 10, 21]) prefer modeling formalisms in which the term “action” is only used infor- mally, and Kripke structures rather than LTSs are used to model systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' These state-based approaches have the advantage that a distinction between horizontal and vertical implemen- tation relations is no longer needed, and a single implementation relation suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Purely state-based approaches, however, are problematic in cases where we need to interact with a black-box system and (by definition) we do not have access to the system state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Black-box systems prominently occur in the areas of model based testing [33] and model learning [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In these application areas, use of LTS-based models makes sense and there is a clear practical need for formalisms that allow scientists and engineers to relate actions at different levels of abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Van der Bijl et al [7], for instance, observe that in model based testing specification models are usually more abstract than the System Under Test (SUT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This means that the generated test cases may not have the required level of detail, and often a single abstract action has to be translated (either manually or by an adaptor) to a sequence of concrete actions that are applied to the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Van der Bijl et al [7] study a very restricted type of action refinement in which a single input is translated into a sequence of inputs, and implement this in a testing tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Also in model learning, typically an adaptor is placed in between the SUT and the learner, to take care of the translation between abstract and concrete actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For example, in a case study on reverse engineering of hand-held smartcard readers for Internet banking, Chalupar et al [9] used abstract input symbols that combine several concrete inputs in order to accelerate the learning process and reduce the size of the learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In particular, they introduced a single abstract input COMBINED_PIN corresponding to a USB command, follow by a 4-digit PIN code, followed by an OK command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Fiterău-Broştean et al [12] used model learning for a comprehensive analysis of DTLS implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This work revealed four serious security vulnerabilities, as well as several functional bugs and non-conformance issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Handshakes in (D)TLS are defined over flights of messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Hence, (D)TLS entities are often expected to produce multiple messages before expecting a response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' During learning, Fiterău- Broştean et al [12] used an adaptor that contracted multiple output messages from the SUT into a single abstract output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Also in other case studies on the TLS [29], Wi-Fi [30] and SSH [35, 13] protocols, multiple outputs from the SUT were contracted into a single output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 3 Verleg [35] used a single abstract input to execute the entire key re-exchange when learning higher layers of SSH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Suppose we have an SUT that can be described by an unknown, concrete model M, and suppose a learner interacts with this SUT through an adaptor and learns an abstract model N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' What can we say about the relation between models M and N?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This article provides an answer to this fundamental question in the settings of LTSs and Mealy machines (Finite State Machines, except that we do require finiteness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We formalize the concept of an adaptor by introducing action codes, a variation of the prefix codes known from coding theory [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Action codes describe how high-level actions may be converted into a sequence of low-level actions, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This makes them different from action refinements, which specify how high-level actions can be translated into low-level processes, but to not address the reverse translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Our notion of an action code captures many adaptors that are used in practice, and in particular those described in the case studies listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For each action code R, we introduce a contraction operator αR that turns a low-level model M into a high-level model by contracting concrete action sequences of M according to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We also introduce the left adjoint of αR, the refinement operator ϱR that turns a high-level model M into a low-level model by refining abstract actions of N according to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This new refinement operator does relate the LTSs of Figure 1 and Figure 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Our first main result is a Galois connection ϱR(N) ⊑ M ⇔ N ⊑ αR(M), where ⊑ denotes the simulation preorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' So if abstract model N implements contraction αR(M), then refinement ϱR(N) implements concrete model M, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In practice, we typically want to obtain an overapproximation of concrete model M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' To this end, we introduce the right adjoint of αR, the concretization operator γR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This operator behaves like the refinement operator, but adds arbitrary behavior at intermediate points during a refinement (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' the demonic completion of [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Our second main result is another Galois connection: αR(M) ⊑ N ⇔ M ⊑ γR(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This connection is useful, because whenever we have established that αR(M) implements (or conforms to) N, it allows us to conclude that M implements (or conforms to) γR(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Our third main result is that, in a setting of Mealy machines, an adaptor can be con- structed for any action code for which a winning strategy exists in a certain 2 player game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If a learner/tester interacts with an SUT via an adaptor generated from such an action code R, and the SUT is modeled by Mealy machine M, then from the learner/tester perspective, the composition of adaptor and SUT will behave like αR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, if a learner succeeds to learn an abstract model N such that N ≈ αR(M) then, by using our Galois connections, the learner may conclude that ϱR(N) ⊑ M ⊑ γR(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The remainder of this article is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We start with a preliminary Sec- tion 2 that introduces basic notations and results for LTSs and their behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Next, action codes and the contraction operator will be introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' After describing the re- finement operator, we establish our first Galois connection in in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Next we define the concretization operator and establish our second Galois connection in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Section 6 explains how action codes can be composed, and shows that contraction and refinement com- mute with action code composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Section 7 describes how adaptors can be constructed 4 from action codes, and identifies some technical restrictions on action codes that are required to make this possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Finally, Section 8 contains a discussion of our results and identifies directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 2 Preliminaries If Σ is a set of symbols then Σ∗ denotes the set of all finite words over Σ, and Σ+ the set of all non-empty words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We use ε to denote the empty word, so e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Σ∗ = Σ+ ∪{ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Concatenation of words u, w ∈ Σ∗ is notated u · w (or simply u w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We write u ≤ w if u is a prefix of w, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' if there is v ∈ Σ∗ with u v = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We write |w| to denote the length of word w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We use f : X⇀Y to denote a partial map f from X to Y and write dom(f) ⊆ X for its domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' set of x ∈ X on which f is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The image im(f) of a partial map f : X⇀Y is the set of elements of Y it can reach: im(f) := {f(x) | x ∈ dom(f)} ⊆ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For a set A of labels, a labelled transition system (LTS) is a tuple M = ⟨Q, q0, ⟩ where Q is a set of states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' q0 ∈ Q is a starting state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' ⊆ Q × A × Q is a transition relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We write LTS(A) for the class of all LTSs with action labels from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We refer to the three components of an LTS M as QM, qM 0 and M, respectively, and introduce the following notation: q a q′ denotes (q, a, q′) ∈ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' q a denotes that there is a q′ such that q a q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' q w q′ for w ∈ A∗ denotes that there are finite sequences a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' , an ∈ A, r0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' , rn ∈ Q such that w = a1 · · · an, and r0 = q, rn = q′ and ri−1 ai ri for all 1 ≤ i ≤ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' q w denotes that there is a q′ such that q w q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' q ∈ Q is reachable if there is a w ∈ A∗ such that q0 w q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' A special class of LTSs that is frequently used in conformance testing and model learning are Mealy machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Mealy machines with a finite number of states are commonly referred to as Finite State Machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For a non-empty sets of inputs I and outputs O, a (non-deterministic) Mealy machine M ∈ LTS(I × O) is an LTS where the labels are pairs of an input and an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We write q i/o q′ to denote that (q, (i, o), q′) ∈ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Whenever we omit a symbol in predicate q i/o −→ q′ this is quantified existentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus i/o if there are q and q′ such that q i/o q′, q i/ q′ if there is an o such that q i/o q′, and q i/ if there is a q′ such that q i/ q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 5 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Figure 3 visualizes a simple Mealy machine with inputs {a, b} and outputs {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The machine always outputs 0 in response to an input, except in one specific situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Output 1 is produced in response to input b if the previous input was a and the total number of preceding inputs is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The machine has four states q0, q1, q2 and q3, with starting state q0 marked by an incoming arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In states q0 and q2 the number of preceding inputs is always even, whereas in states q1 and q3 it is always odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In states q2 and q3 the previous input is always a, whereas in states q0 and q1 either the previous input is b, or no input has occurred yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, only in state q3 input b triggers an output 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' q0 start q1 q2 q3 b/0 a/0 a/0 b/0 b/0 a/0 a/0 b/1 Figure 3: A Mealy machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We need to introduce some formal notation and terminology for LTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let M = ⟨Q, q0, ⟩ ∈ LTS(A) be an LTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We say that M is deterministic if, whenever q a for some q and a, there is a unique q′ with q a q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' M is a tree-shaped if each state q ∈ Q can be reached via a unique sequence of transi- tions from state q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' q ∈ Q is a leaf, notated q , if there is no a ∈ A with q a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' M is grounded if every state q ∈ Q has a path to a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We can now define the set of traces of an LTS: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let M = ⟨Q, q0, ⟩ ∈ LTS(A) be an LTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' A word w ∈ A∗ is a trace of state q ∈ Q if q w , and a trace of M is it is a trace of q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We write trace(M) for the set of all traces of M: trace(M) = {w ∈ A∗ | q0 w } A trace of M is complete if it is not a proper prefix of any other observation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Complete traces correspond to sequences of transitions that end in a leaf state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6 (Simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For M, N ∈ LTS(A), a simulation from M to N is a relation S ⊆ QM × QN such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' qM 0 S qN 0 and 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' if q1 S q2 and q1 a M q′ 1 then there exists a state q′ 2 such that q2 a N q′ 2 and q′ 1 S q′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We write M ⊑ N if there exists a simulation from M to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' It is a classical result that trace inclusion coincides with the simulation preorder for deterministic labeled transition systems (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' [25]): Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let M, N ∈ LTS(A) with N deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then trace(M) ⊆ trace(N) iff M ⊑ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We will often consider LTSs up to isomorphism of their reachable parts: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='8 (Isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For M, N ∈ LTS(A), an ismorphism from M to N is a bijection f : QM R → QN R , where: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' QM R ⊆ QM and QN R ⊆ QN are the subsets of reachable states in M and N, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' f(qM 0 ) = qN 0 , and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' q a M q′ iff f(q) a N f(q′), for all q, q′ ∈ QM R , a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We write M ∼= N if there exists an isomorphism from M to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Note that ∼= is an equivalence relation on LTS(A), and that M ∼= N implies M ⊑ N, since each isomorphism (when viewed as a relation) is trivially a simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 3 Action Codes Action codes describe how to translate between two action label alphabets, for example from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Intuitively, we understand the first alphabet A as the actions at the lower, concrete level, and the second alphabet B as the actions at the higher, more abstract level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In an action code, a single abstract action b ∈ B corresponds to a finite, non-empty sequence of concrete actions a1 · · · an in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Essentially, action codes are just a special type of prefix codes [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We provide two equivalent definitions of action codes: one via tree-shaped LTSs and one via a partial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='1 (Action code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For sets of action labels A and B, a (tree-shaped) action code R from A to B is a structure R = ⟨M, l⟩, with M = ⟨R, r0, ⟩ ∈ LTS(A) a deterministic, tree-shaped LTS with L being the set of non-root leaves L ⊆ R\\{r0} and an injective function l: L → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We write Code(A, B) for all action codes from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The injectivity of l and the tree-shape ensure that every abstract b ∈ B is represented by at most one w ∈ A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Figure 4 shows an action code for a fragment of the ASCII encoding in octal format, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=', sequence 1 1 5 encodes the letter “M”, sequence 1 4 5 encodes the letter “e”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 7 r0 start r1 r3 r2 r4 r5 r6 M r7 e r8 a r9 l r10 y 1 1 4 5 7 5 5 1 4 1 Figure 4: Action code for a fragment of the ASCII encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' r0 start r1 r3 r4 r5 r6 /0 r7 /0 r8 r9 /1 r10 /1 r11 r12 /2 r13 /2 r14 r15 /3 r16 /3 switch_on/fill_water_tank switch_on/ready add_water/fill_bean_container add_water/ready add_beans/empty_coffee_grounds_container_and_drip_tray add_beans/ready remove_waste/ready /coffee /espresso /coffee /espresso /coffee /espresso /coffee /espresso Figure 5: Action code for a coffee machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Figure 5 shows an action code for the activity of getting a cup of coffee or espresso, in the special case of Mealy machines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' where A = I × O and B = I′ × O′ are sets of input/output-pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Rather than the full sequence of interventions that is required in order to get a drink, the abstract input/output pair only reports on the type of drink that was ordered and the number of interventions that occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 8 The definition of action codes as labelled transition system themselves allows an intuitive visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For easier mathematical reasoning, we characterize action codes also in terms of maps: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' A (map-based) action code from A to B is a partial map f : B⇀A+ which is prefix-free, by which we mean that for all b, b′ ∈ dom(f), f(b) ≤ f(b′) implies b = b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' (1) In the following, we show that these prefix-free partial maps bijectively correspond to the tree-shaped LTSs: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Every tree-shaped action code R ∈ Code(A, B) induces a unique map-based action code f : B⇀A+ with the property that for all b ∈ B, w ∈ A+: f(b) = w iff ∃r ∈ L: r0 w R r, l(r) = b (2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Take equation (2) as the definition of f: f(b) = � w if there are r ∈ L, w ∈ A∗ with r0 w R r, l(r) = b, undefined otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since r0 /∈ L, the range of f indeed restricts to A+ ⫋ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For well-definedness of f, first note that due to the injectivity of the labelling l: L → B, there is at most one r ∈ L with l(r) = b, and by R being tree-shaped, there is precisely one path r0 R r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let us now prove the required properties of this partial map: Prefix-freeness (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Consider b, b′ ∈ B with f(b) ≤ f(b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Hence, we have runs r0 w R r r ∈ L l(r) = b w = f(b) and r0 w′ R r′ r′ ∈ L l(r′) = b′ w′ = f(b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By f(b) ≤ f(b′), there is some u ∈ A∗ such that wu = w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, the run for w′ = f(b′) can be decomposed for some ¯r ∈ QR: r0 w R ¯r u R r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The LTS R is required to be deterministic, so ¯r = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since r ∈ L is a leaf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' a dead-lock state, we necessarily have u = ε and r = r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Finally, b = l(r) = l(r′) = b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Characterizing property (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Verification is immediate because both the property (2) and the definition r use the same witnesses, and every witness w ∈ A∗ must be a non-empty word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Consider another prefix-free partial map g: B⇀A+ satisfying for all b ∈ B, w ∈ A+ : g(b) = w iff ∃r ∈ L: r0 w R r, l(r) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then, it is immediate that dom(g) = im(L) = dom(f) and that g(b) = f(b) for all b ∈ im(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 9 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For each map-based action code f : B⇀A+, there is (up to isomorphism) a unique tree-shaped action code R ∈ Code(A, B) which is grounded and makes property (2) true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Define action code R as follows: R := {w ∈ A∗ | w = ε or ∃b ∈ dom(f): w ≤ f(b)}, r0 := ε, we put v a R w iff there is a ∈ A with va = w, l(w) is the unique b ∈ B with f(b) = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let us first verify that this is a well-defined action code: There is nothing to be verified about the state set R (note that we do not require finiteness in the definition of action code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By definition of R, the initial state r0 := ε is an element of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For the transition structure we need to verify several properties: – It is grounded because for every w ∈ A∗ with w ≤ f(b), b ∈ dom(f), the state f(b) is the witnessing leaf (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' dead-lock state) below w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' – It is deterministic: whenever we have v a R w and v a R w′, we obtain w = va = w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' – It is tree-shaped: for each w ∈ R, there is a unique run to it, namely r0 w R w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Define set L ⊆ R by L = {f(b) | b ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then L does not contain the initial state r0, but every leaf w ∈ Q \\ {r0} is in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Conversely, every w ∈ L is a leaf because f is prefix-free (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Likewise, l: L → B is injective because f is prefix-free (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For uniqueness, consider another tree-shaped action code ¯R with set of non-root leaves ¯L satisfying (2), concretely for all b ∈ B, w ∈ A+ : f(b) = w iff ∃¯r ∈ ¯L: r0 w ¯ R ¯r l ¯ R(¯r) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' (3) We now need to establish an isomorphism φ: ¯R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Define the map φ: R ¯ R → RR by φ(¯r) = the unique w ∈ A∗ with r ¯R 0 w ¯ R ¯r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This map is well defined because ¯R is tree-shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' To see that φ(¯r) ∈ A∗ is also in RR ⊆ A∗ for every state ¯r of ¯R, let ¯r′ ∈ ¯L be an arbitrary leaf with ¯r u ¯ R ¯r′ in ¯R – such a leaf exists because every action code ¯R is required to be grounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, for b = lR(¯r′) we have φ(¯r) ≤ wu = f(b) by (3), and so φ(¯r) ∈ RR by above definition of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' With our definition of φ and R, it is mechanical to check that q a q′ iff φ(q) a φ(q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since ¯R is tree-shaped, φ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For surjectivity, consider w ∈ RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If w = ε then we have φ(r ¯R 0 ) = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Otherwise, 10 pick any b ∈ dom(f) with w ≤ f(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By (3), we have ¯r ∈ ¯L with q ¯ R 0 f(b) ¯ R r in ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The intermediate state ¯r′ with q ¯ R 0 w ¯ R ¯r′ u ¯ R ¯r w u = f(b) does satisfy φ(¯r′) = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, φ is surjective and bijective in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Also, the labelling is preserved by φ: ¯R → R because for every leaf ¯r ∈ ¯L with r ¯ R 0 w ¯ R ¯r we have f(l ¯ R(¯r)) = w by (3), and so l(φ(¯r)) = l(w) def l = l ¯ R(¯r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' because b := l ¯ R(¯r) satisfies f(b) = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Note that for the uniqueness in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6, we additionally need that the tree-shaped action code is grounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This essentially means that there is no infinite subtree in which no leaf is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For example, consider A = {a}, arbitrary B and the action codes R := � r0 start r1 a r2 a · · a � and S := � q0 start � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Both action codes R and S have no non-root leaves, and so they both induce the empty partial map f : B⇀A+ via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' f is undefined for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' And indeed, R and S are not isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The issue is that while the finite S is grounded, the infinite R is not grounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' So R contains subtrees which do not contribute anything to the partial map f but which hinder the existence of an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Having shown the correspondence between tree-shaped and map-based action codes Code(A, B), we can switch between the two views in proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Mostly, we use the tree-shaped version for visualization and the map-based version for mathematical reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Consider a concrete M ∈ LTS(A), together with an action code R from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We can construct an abstract LTS for the action labels B by walking through M with seven-league boots, repeatedly choosing input sequences that correspond to complete observations of R, and then contracting this sequence to a single abstract transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In the rest of the paper, we introduce operators αR, ϱR, γR on LTSs, involving action codes R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Whenever the action code R is clear from the context, we omit the index and simply speak of operators α, ϱ, γ for the sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='9 (Contraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For each action code R ∈ Code(A, B), the contraction oper- ator αR : LTS(A) → LTS(B) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For M ∈ LTS(A), the LTS αR(M) has states Qα(M) ⊆ QM and transitions α(M) defined inductively by the next two rules, for all q, q′ ∈ QM, b ∈ B: qM 0 ∈ Qα(M) (1α) q ∈ Qα(M), b ∈ dom(R), q R(b) M q′ q b α(M) q′, q′ ∈ Qα(M) (2α) The initial state qα(M) 0 := qM 0 is the same as in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 11 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Figures 6 and 7 show two examples of action codes and the contractions obtained when we apply them to the Mealy machine of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The examples illustrate that by choosing different codes we may obtain completely different abstractions of the same Mealy machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' r0 start r1 r2 r3 A/0 r4 B/0 a/0 b/0 a/0 b/0 q0 start q2 A/0 B/0 B/0 A/0 Figure 6: Code for Mealy machine of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 3 (left) and resulting contraction (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' r0 start r1 r2 B/0 r3 C/0 r4 C/1 a/0 b/0 b/0 b/1 q0 start q1 B/0 C/1 B/0 C/0 Figure 7: Another code for Mealy machine of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 3 (left) and resulting contraction (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The next proposition asserts that we can view αR as a monotone function αR : LTS(A) → LTS(B) between preordered classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='11 (Monotonicity contraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let M, N ∈ LTS(A) with M ⊑ N, and let R ∈ Code(A, B) be an action code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then αR(M) ⊑ αR(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Given a simulation T from M to N, we will show that S := T ∩ (Qα(M) × Qα(N)) is a simulation relation from αR(M) to αR(N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' in other words S ⊆ Qα(M) × Qα(N) is the restriction of T ⊆ QM × QN to the subsets Qα(M) ⊆ QM and Qα(N) ⊆ QN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For the initial states, we directly have (qα(M) 0 , qα(N) 0 ) ∈ S because (qα(M) 0 , qα(N) 0 ) = (qM 0 , qN 0 ) ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For transitions, consider (q, p) ∈ S and q b q′ in α(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By definition of α(M), we have b ∈ dom(R) and q R(b) q′ in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 12 In M and N, the states (q, p) ∈ S ⊆ T are also related by the simulation, and so we obtain p R(b) p′ in N with (q′, p′) ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By the definition of α, we have p′ ∈ Qα(N) and p b p′ in α(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, also (q′, p′) ∈ S as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The example below illustrates that the previously discussed trace language operator trace can be viewed as an instance of contraction for an infinite action code: Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='12 (Trace semantics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Define δ: LTS(A) → LTS(A + {$}) by δ((Q, q0, )) = (Q+{$}, q0, ∪{(q, $) | q ∈ Q}) and define the action code R ∈ Code(A+{$}, B) to B := A∗ such that the concrete word a1 · · · an$ is related to the abstract symbol a1 · · · an ∈ B, then αR ◦ δ: LTS(A) → LTS(A∗) sends every M ∈ LTS(A) to a system in LTS(A∗) with states {q0, $} and transitions q0 w $ ⇐⇒ w ∈ trace(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 4 Refinements Now that we have introduced the contraction αR of an LTS for a code R, it is natural to consider an operation in the other direction, which we call the refinement ϱR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Intuitively, refinement replaces each abstract transition q b q′ by a sequence of concrete transitions, as prescribed by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='1 (Refinement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For each action code R ∈ Code(A, B), we define the refinement operator ϱR : LTS(B) → LTS(A) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For M ∈ LTS(B), the LTS ϱR(M) ∈ LTS(A) has a set of states Qϱ(M) := {(q, w) ∈ QM × A∗ | w = ε or (∃b: q b M ∧w ≨ R(b))} and the initial state (qM 0 , ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The transition relation ϱ(M) is defined by the following rules: (q, wa) ∈ Qϱ(M) (q, w) a ϱ(M) (q, wa) (1ϱ) q b M q′ wa = R(b) (q, w) a ϱ(M) (q′, ε) (2ϱ) Intuitively, whenever ϱ(M) is in state (q, w), then this corresponds to being in state q in the abstract automaton M ∈ LTS(B) and having observed the actions w ∈ A∗ so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' However, we have insufficiently many actions for finding an abstract transition q b M q′ with w = R(b) because w is still to short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Nevertheless, whenever ϱ(M) admits a transition to a state (q, w) with w ̸= ε, then we know that we can eventually complete w to a sequence corresponding to an abstract transition: there exist at least one q b M q′ for some b ∈ dom(R) with w ≤ R(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We will formally prove this in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If the abstract system M is non-deterministic, then there may be multiple abstract transitions that match in the final rule (2ϱ), but the transitions produced by rule (1ϱ) are deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 13 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Figure 8 shows an example application of a refinement operator that replaces the actions of the LTS M on the left by their ASCII encoding in octal format, as prescribed by the action code from Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The initial state is (q0, ε), corresponding to q0 in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since M contains abstract labels M and a, with R(M) = 1 1 5 and R(a) = 1 4 1, we need to introduce additional states for having read 1, 1 1, and 1 4, because those are the sequences of A-actions before we have observed a sequence R(b) ∈ A+ for some b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' q0 start M a (q0, ε) start (q0, 1) (q0, 11) (q0, 14) 1 1 4 5 1 Figure 8: LTS (left) and refinement obtained via action code from Figure 4 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' A more visual explanation of ϱR(M) is the following: for every state q ∈ QM, we consider the outgoing transitions {q b M q′ | b ∈ B, q′ ∈ QM} and labels B′ ⊆ B that appear in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then, this outgoing-transition structure is replaced with (a copy of) the minimal subgraph of the tree R containing all leaves with labels in B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Like contraction, the refinement operation also preserves the simulation preorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='3 (Monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For all action codes R ∈ Code(A, B), if M ⊑ N in LTS(B), then ϱR(M) ⊑ ϱR(N) in LTS(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let S ⊆ QM × QN be the simulation witnessing M ⊑ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let us verify that T := {((q, w), (p, w′)) ∈ Qϱ(M) × Qϱ(N) | w = w′, (q, p) ∈ S} is a simulation from ϱ(M) to ϱ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Clearly, qϱ(M) 0 = (qM 0 , ε) T (qN 0 , ε) = qϱ(N) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Consider a pair in T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' (q, w), (p, w) with q S p: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For transitions produced by rule (2ϱ), (q, w) a ϱ(M) (q′, ε), we have q b M q′ for some b ∈ dom(R) with R(b) = wa by rule assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since (q, p) ∈ S, the simulation S provides us with a transition p b N p′ in N and (q′, p′) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, we can apply rule (2ϱ) in ϱ(N) to obtain (p, w) a ϱ(N) (p′, ε) in ϱ(N) which satisfies (q′, ε) T (p′, ε) by definition of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For transitions produced by rule (1ϱ), (q, w) a ϱ(M) (q, wa), we have (q, wa) ∈ Qϱ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Obviously, wa ̸= ε, and so by the definition of Qϱ(M), there must be at least one transition q b M q′ with wa ≨ R(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By the definition of T, we have (q, p) ∈ S and so the simulation S provides us with some p b N p′ in N with (q′, p′) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Hence, (p, wa) ∈ Qϱ(N) and so we have (p, w) a ϱ(N) (p, wa) by rule (1ϱ) and (q, wa) T (p, wa) by definition of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Because R is deterministic, applying ϱR on a deterministic LTS results in a deterministic LTS: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='4 (Refinement preserves determinism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For every action code R ∈ Code(A, B), if M ∈ LTS(B) is deterministic, then ϱR(M) ∈ LTS(A) is deterministic, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Consider (q, w) ∈ Qϱ(M) and transitions (q, w) a (q1, w1) and (q, w) a (q2, w2) in ϱ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We show (q1, w1) = (q2, w2) by case distinction: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Case: there is b ∈ dom(R) with R(b) = wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then there is a unique such b because R is prefix-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Also because R is prefix-free, there is no b′ ∈ dom(R) such that wa ≨ R(b′), and so (q, wa) /∈ Qϱ(M) and so neither of the transitions were produced by rule (1ϱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, both transitions come from rule (2ϱ) and thus we have w1 = ε = w2 and transitions: q b M q1 and q b M q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By assumption, M is deterministic, so q1 = q2, and so (q1, w1) = (q2, w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Case: there is no b ∈ dom(R) with R(b) = wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, neither of the transitions can be produced by rule (2ϱ) and so both were produced by rule (1ϱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then, we necessarily have (q1, w1) = (q, wa) = (q2, w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The next two technical lemmas are required to establish our first Galois connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For all M ∈ LTS(B) and b ∈ dom(R) of an action code R ∈ Code(A, B), and q ∈ QM: q b q′ in M iff (q, ε) R(b) (q′, ε) in ϱ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For (⇒), R is defined for b and we yield R(b) = a1 · · · an with 1 ≤ n and ai ∈ A for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Given q b q′ in M, we can construct a path in ϱR(M) by n − 1 applications of rule (1ϱ) and one application of rule (2ϱ): (q, ε) a1 (q, a1) a2 (q, a1a2) · · · an−1 (q, a1 · · · an1) an (q′, ε) in ϱR(M), 15 with other notation for R(b) = a1 · · · an: (q, ε) R(b) (q′, ε) in ϱR(M), For (⇐), consider a generalized transition (q, ε) R(b) (q′, ε) in ϱ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Of course R(b) ∈ A+ so we can write it as R(b) = wa for w ∈ A∗ and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then, we can consider the intermediate state of the given run for R(b): (q, ε) w ϱ(M) (¯q, ¯w) a ϱ(M) (q′, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' All prefixes v ≤ w satisfy v ≨ R(b), so the first transitions for w must have been produced by rule (1ϱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Hence, (¯q, ¯w) = (q, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The final a-transition must have been produced by (2ϱ) because the second component of the tuple is ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The assumption of this rule contains q b M q′, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For every transition (q, w) a (q′, w′) in ϱ(M), there is some transition q b q′′ in M and u ∈ A∗ such that wau = R(b) and (q′, w′) u (q′′, ε) in ϱR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For the transition (q, w) a (q′, w′), distinguish two cases: If w′ = ε, then the transition must have been produced by rule (2ϱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, the rule assumption contains a transition q b M q′ with wa = R(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This is the desired witness q′′ := q′ and u = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If w′ ̸= ε, then the transition must have been produced by rule (1ϱ) and so q′ = q and w′ = wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The definition of Qϱ(M) unfolded for (q, wa) ∈ Qϱ(M) yields that there exists a transition q b M q′′ with wa ≨ R(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let v ∈ A∗ and b ∈ A such that wavb = R(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since wav ≨ R(b), we can produce further transitions using rule (1ϱ) for the letters in v ∈ A∗ and can conclude using rule (2ϱ) for b: (q′, w′) = (q, wa) v ϱ(M) (q, wau) b ϱ(M) (q′′, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This is the desired transition (with u := vb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='7 (Galois connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Consider an action code R ∈ Code(A, B) and LTSs N ∈ LTS(B) and M ∈ LTS(A): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If dom(R) = B, then ϱR(N) ⊑ M implies N ⊑ αR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If M is deterministic, then N ⊑ αR(M) implies ϱR(N) ⊑ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' ϱR(N) ⊑ M N ⊑ αR(M) If dom(R) = B If M is deterministic The condition dom(R) = B in the first direction means that the partial map R: B⇀A+ is in fact a total map R: B → A+ because R(b) is defined for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Fix systems N ∈ LTS(B), M ∈ LTS(A), and as usual we omit index R from α and ϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For direction (⇒), assume that dom(R) = B and ϱ(N) ⊑ M, witnessed by the simu- lation S ⊆ Qϱ(N) × QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We verify that we have a simulation T between N and α(M) defined by: T := {(p, q) ∈ QN × Qα(M) | � (p, ε), q � ∈ S} The definition of T is well-typed because Qα(M) ⊆ QM, and moreover, {(p, ε) | p ∈ QN} ⊆ Qϱ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The relation T relates the initial states qN 0 T qα(M) 0 because (qN 0 , ε) = qϱ(N) 0 S qM 0 = qα(M) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Now, suppose p T q and p b N p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By assumption dom(R) = B, we can apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='5 and obtain (p, ε) R(b) ϱ(N) (p′, ε) in ϱ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The simulation S from ϱ(N) to M transforms this into a path q R(b) M q′ in M with (p′, ε) S q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since q ∈ Qα(M) also q′ ∈ Qα(M) and thus q b α(M) q′ in α(M) and moreover p′ T q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For direction (⇐), assume N ⊑ α(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let S ⊆ QN × Qα(M) be a simulation relation from N to α(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We define the relations ¯S ⊆ Qϱ(N) × Qα(M) and T ⊆ Qϱ(N) × QM: (p, ε) ¯S q :⇔ p S q (p, w) T q :⇔ ∃¯q ∈ Qα(M) : p S ¯q ∧ ¯q w M q So visually, every related pair (p, w) T q entails states of the following form: ϱ(N) M (p, ε) ¯q (p, w) q w w ¯S T 17 We verify that T is a simulation from ϱ(N) to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Picking ¯q := q and w := ε shows that the related initial states qN 0 S qα(M) 0 of N and α(M) imply qϱ(N) 0 = (qN 0 , ε) T qM 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Suppose (p, w) T q and (p, w) a ϱ(N) (p′, w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6 shows that wa ∈ A+ sits ‘below’ some b ∈ B in the action code R: concretely, there are u ∈ A∗, and b ∈ B such that: (p′, w′) u ϱ(N) (p′′, ε) and wau = R(b) and p b N p′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We have the solid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' non-dotted) part of the following picture: ϱ(N) M N α(M) (p, ε) (p, w) (p′, w′) (p′′, ε) w a u ¯q q q′ q′′ w a u ¯S T ¯S p p′′ b ¯q q′′ b S S Using the simulation property of p S ¯q, we obtain q′′ ∈ Qα(M) with ¯q b α(M) q′′ and p′′ S q′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The transition ¯q b q′′ in α(M) must have come from a path ¯q R(b) q′′ in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Denote the intermediate states for the decomposition R(b) = wau by qw, q′ ∈ QM: ¯q w M qw a M q′ u q′′ The assumption that M is deterministic enforces that q = qw, so q au M q′′ as shown in the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Also, p′′ S q′′ directly shows (p′′, ε) ¯S q′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Now, the picture is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For the final proof that T relates (p′, w′) and q′, we distinguish cases: (a) Case u = ε: Then, p′ = p′′, w′ = ε and q′ = q′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, � (p′, w′), q′� ∈ ¯S ⊆ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' (b) Case u ̸= ε: Then, wa ≨ R(b) and so wa ̸= R(b′) for all b′ ∈ B by prefix-freeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' So the a-transition can only come from rule (1ϱ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' hence p = p′ and w′ = wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Finally, (p′, w′) = (p, wa) T q′ by the definition of T and ¯q wa q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 18 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In the above direction ⇐, we need determinism of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If we also want to support non-deterministic M, we can consider a less-pleasant ϱ′ R that replaces every q b q′ for R(a1 · · · an) = b with literally a sequence q a1 · · an q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus ϱ′ R would rather create a system on the left of Figure 2 whereas ϱR creates a system as on the right of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' However, such an operator ϱ′ R does not preserve determinism, but ϱR does as we prove next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Another important assumption in the direction ⇐ is that for every abstract b ∈ B there is at most one related word w ∈ A+ of concrete actions in the action code R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' And indeed the direction ⇐ fails if there are two symbols a1, a2 ∈ A, a1 ̸= a2 both related to b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Denote a system with two states and one transition directly by (• b ) (the initial state is the left-hand one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then, we have α(• a1 •) = (• b ) = α(• a2 •).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then, there exists no left adjoint ϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Such an adjoint, applied to (• b ) would need to satisfy ϱ(• b ) ⊑ (• a1 •) and ϱ(• b ) ⊑ (• a2 •) Hence, the initial state in ϱ(• b ) is a leaf state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' ϱ(• b ) is the bottom element for the simulation order ⊑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This easily leads to a contradiction after using the Galois connection in the other direction again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 5 Concretizations In this section, we consider another method of transforming an abstract system into a con- crete one: the concretization operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Whereas refinement is the lower adjoint of contraction (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='7), this section will establish that concretization is the upper adjoint (Theo- rem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='3) of contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Whereas for refinement we omitted transitions for which the action code R was not defined, for concretization we add transitions to a new chaos state [18] in which any action may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Essentially, this is the idea of demonic completion of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='1 (Concretization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let M ∈ LTS(B) be an LTS and R ∈ Code(A, B) an action code R: B⇀A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The concretization γR(M) ∈ LTS(A) consists of: Qγ(M) := QM × N ∪ {χ} where N ⊆ A∗ is defined by N := {w ∈ A∗ | w = ε or ∃b ∈ dom(R): w ≨ R(b)} qγ(M) 0 := (qM 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' ε) Transitions defined by the following rules,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' for all (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' w) ∈ QM × N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' a ∈ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' and b ∈ B: wa ∈ N (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' w) a γ(M) (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' wa) (1γ) q b M q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' R(b) = wa (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' w) a γ(M) (q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' ε) (2γ) 19 wa /∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' wa /∈ im(R) (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' w) a γ(M) χ (3γ) χ a γ(M) χ (4γ) Intuitively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' N represent the internal nodes of the tree-representation of the action code R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The transitions then try to accumulate a word w ∈ A∗ know to the action code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' As soon as we reach w = R(b) for some b, we use a b-transition in the original M ∈ LTS(B) to a new state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The transition structure of γ is built in such a way that transitions for b ∈ B in M correspond to runs of R(b) in γ(M) in the following sense: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For every M ∈ LTS(B), q ∈ QM, R ∈ Code(A, B), and b ∈ dom(R): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Whenever q b M q′ then (q, ε) R(b) γ(M) (q′, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Whenever (q, ε) R(b) γ(M) ¯q, then ¯q = (q′, ε) for some q′ ∈ QM with q b M q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Given q b M q′ in M, write R(b) ∈ A+ as R(b) = wa for w ∈ A∗ and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Write w = w1 · · · wn, with n ∈ N and wi ∈ A, for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For every 1 ≤ i ≤ n, the word w1 · · · wi ∈ A∗ is contained in N because w1 · · · wi ≨ R(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Hence, rule (1γ) provides us with transitions (q, ε) w1 γ(M) (q, w1) w2 γ(M) (q, w1w2) · · · γ(M) (q, w1 · · · wn) = (q, w) Finally, rule (2γ) has the assumptions q b M q′ and wa = R(b) fulfilled, so we have (q, ε) w γ(M) (q, w) a γ(M) (q′, ε), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' (q, ε) R(b) γ(M) (q′, ε), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Assume (q, ε) R(b) γ(M) ¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' As before, decompose R(b) into R(b) = wa for w ∈ A∗ and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Write w = w1 · · · wn for wi ∈ A, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For every i, 1 ≤ i < n, any transition (q, w1 · · · wi) wi+1 γ(M) p can only be produced by rule (1γ), because w1 · · · wiwi+1 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Hence, p = (q, w1 · · · wi+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, the given run for R(b) ∈ A+ is of the form (q, ε) w1 γ(M) (q, w1) w2 γ(M) · · · wn γ(M) (q, w1 · · · wn) = (q, w) a γ(M) ¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In order to see that the final a-transition is produced by rule (2γ), note that wa = R(b) implies that wa /∈ N by prefix-freeness (1), and so rule (1γ) can not have produced this a-transition to ¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Obviously, rule (3γ) does not match either, and so only rule (2γ) is left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, there is some q b M q′ and ¯q is of the form ¯q = (q′, ε), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 20 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='3 (Galois connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For every action code R ∈ Code(A, B), M ∈ LTS(A), and N ∈ LTS(B), we have αR(M) ⊑ N (in LTS(B)) ⇐⇒ M ⊑ γR(N) (in LTS(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since we have only one action code R at hand, we omit the index R for α and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We prove both directions seperately: (⇒) Let α(M) ⊑ N be witnessed by the simulation S ⊑ Qα(M) × QN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We show that T ⊑ QM × Qγ(N) defined by T := {(p′, (q, w)) | p′ ∈ QM, (q, w) ∈ Qγ(N), ∃p ∈ QM, p w M p′, (p, q) ∈ S} ∪ {(p, χ) | p ∈ QM} is a simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Note that if χ is omitted from γ(N) if it is not reachable, and then we also omit it from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For the initial states, we immediately have (qM 0 , qγ(N) 0 ) = (qM 0 , (qN 0 , ε)) ∈ T by the definition of T, because qM 0 ε qM 0 and (qM 0 , qN 0 ) = (qα(M) 0 , qN 0 ) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since T is defined as a union, we can verify the two parts seperately: (a) Consider (p′, (q, w)) ∈ T and p′ a M p′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By definition of T, there is some p ∈ QM with p w M p′ and (p, q) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We distinguish whether wa ∈ N and wa ∈ im(R): If wa ∈ N, then we have (q, w) a γ(N) (q, wa) by rule (1γ) and (p′′, (q, wa)) ∈ T by definition of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If wa /∈ N and wa ∈ im(R), then there is some b ∈ dom(R) with R(b) = wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We have p w M p′ a M p′′ and so p b α(M) p′′ by the definition of α(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Using that S is a simulation, we obtain a transition q b N q′ in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By rule (2γ), this translates into a transition (q, w) a γ(N) (q′, ε) in γ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By definition of T, we find (p′′, (q′, ε)) ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If wa /∈ N and wa /∈ im(R), then we have (q, w) χ γ(N) by rule (3γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In this case, Then, we also have (p′, χ) ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' (b) Consider (p, χ) ∈ T and p a M p′ in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We have χ a γ(N) χ by rule (4γ) and (p′, χ) ∈ T, again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' (⇐) Assume M ⊑ γR(N) in LTS(A), witnessed by a simulation S ⊆ QM × Qγ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Define T ⊆ Qα(M) × QN by T := {(p, q) ∈ Qα(M) × QN | (p, (q, ε)) ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Here, we use that Qα(M) ⊆ QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For the initial states, note that (qα(M) 0 , qN 0 ) ∈ T because (qM 0 , (qN 0 , ε)) = (qM 0 , qγ(N) 0 ) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 21 For the remaining verification, consider (p, q) ∈ T and a transition p b α(M) p′ in α(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By the definition of α, we have b ∈ dom(R) and a run p R(b) M p′ in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Using that S is a simulation and that (p, (q, ε)) ∈ S, this yields a run (q, ε) R(b) γ(N) q′ in γ(N) with (p′, q′) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We do not know yet in which of the two components of Qγ(N) = (QN × N) ∪ {chi} the state q′ is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We investigate by decomposing the run for R(b) ∈ A+ into wa = R(b) for w ∈ A∗ and a ∈ A, calling the intermediate state ¯q ∈ Qγ(N): (q, ε) w γ(N) ¯q a γ(N) q′ in γ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since w ≨ R(b), we have w ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Looking at the rules for the transitions of γ(N), we see that the only option for the transitions in (q, ε) w γ(N) ¯q with w ∈ N is via rule (1γ), so we necessarily obtain ¯q = (q, w): (q, ε) w γ(N) (q, w) a γ(N) q′ Using that wa = R(b), only rule (2γ) can have produced the transition (q, w) a q′, hence q′ = (q′′, ε) for some q′′ ∈ QN with q b N q′′ in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In total, we have (p′, (q′′, ε)) = (p′, q′) ∈ S and by the definition of T: (p′, q′′) ∈ T q b N q′′ in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This shows that T is indeed a simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='4 (Monotonicity of concretization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For every action code R ∈ Code(A, B), M ⊑ N in LTS(B) implies γR(M) ⊑ γR(N) in LTS(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' It is a standard result that the operators in a Galois connections are monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We recall the concrete proof for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Consider M ⊑ N in LTS(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By the reflexivity, we have γR(M) ⊑ γR(M) in LTS(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Applying the Galois connection (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='3) from right to left yields αR(γR(M)) ⊑ M in LTS(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By transitivity of ⊑ and M ⊑ N, we obtain αR(γR(M)) ⊑ N in LTS(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Applying the Galois connection (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='3) conversely from left to right yields the desired γR(M) ⊑ γR(N) in LTS(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 22 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Monotonicity of concretization also follows by observing that the rules in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='1 all fit the tyft format of [17] if we view (·, w) as a unary operator for each sequence w ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Monotonicity then follows from the result of [17] that the simulation preorder is a congruence for any operator defined using the tyft format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since contraction also can be defined using the tyft format, also monotonicity of contraction (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='11) follows from the result of [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' One may think that the many transitions to the chaos state χ, would make the construc- tion γR trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' However, only those paths lead to the chaos for which the action code is not defined: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For R ∈ Code(A, B), w ∈ A∗, and M ∈ LTS(B), if (q, ε) w χ in γR(M), then w ̸≤ R(b) for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let ua ≤ w, for u ∈ A∗, a ∈ A, be the shortest prefix such that (q, ε) ua χ in γR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' (Note that we can assume this shortest prefix to be non-empty because (q, ε) ̸= χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, the last transition (for label a) must have been produced by rule (3γ): (q, ε) u (q′, s) a χ If u ̸= s, then there is some b′ ∈ B with R(b′) ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If we also had w ≤ R(b), this would lead to a contradiction: R(b′) ≤ u ≨ ua ≤ w ≤ R(b) so b = b′ by prefix-freeness (1) and on the other hand R(b′) ≨ R(b) = R(b′), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If u = s, then by the assumptions of rule (3γ), we obtain ua = sa /∈ N and ua = sa /∈ im(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Hence, we have ua ̸≤ R(b) and so also w ̸≤ R(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Assume that for every w ∈ A∗, there is some b ∈ dom(R) with w ≤ R(b) or R(b) ≤ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then for all M ∈ LTS(B), χ is not reachable in γR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Like refinement, concretization preserves determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='8 (Concretization preserves determinism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For every R ∈ Code(A, B), if M ∈ LTS(B) is a deterministic LTS, then γR(M) is deterministic, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We verify the determinacy seperately for the disjoint components of Qγ(M) := (QM × N) ∪ {χ} 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For (q, w) ∈ QM × N and two transitions (q, w) a γ(M) ¯q1 (q, w) a γ(M) ¯q2 we distinguish cases like in the assumptions of the rules for γ(M): If wa ∈ N, then both transitions have been produced by rule (1γ) and so ¯q1 = (q, wa) = ¯q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 23 If wa /∈ N and wa ∈ im(R), then there are b1, b2 ∈ dom(R) with q b1 M q′ 1 ¯q1 = (q′ 1, ε) R(b1) = wa q b2 M q′ 2 ¯q2 = (q′ 2, ε) R(b2) = wa Since R: B⇀A+ is prefix-free (1), it is in particular injective and so b1 = b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The LTS M was assumed to be deterministic, thus q′ 1 = q′ 2 and so ¯q1 = (q′ 1, ε) = (q′ 2, ε) = ¯q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If wa /∈ N and wa /∈ im(R), then both transitions have been produced by rule (3γ) and so ¯q1 = χ = ¯q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Any two outgoing transitions of χ χ a γ(M) ¯q1 and χ a γ(M) ¯q2 have necessarily been created by (4γ), and so ¯q1 = χ = ¯q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If we are willing to make an extra assumption then γR is even the right inverse of αR, that is, we have a Galois insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The assumption is that every b ∈ B that labels a transition in the LTS also occurs in the action code R: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='9 (Galois insertion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For every R ∈ Code(A, B) and every M ∈ LTS(B), if M ∈ LTS(dom(R)), then M ∼= αR(γR(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Note that dom(R) ⊆ B, and so LTS(dom(R)) ⊆ LTS(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since we have only one action code R at hand, we omit the index R in α and γ in this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The LTS α(γ(M)) has precisely the states Qα(γ(M)) = {(q, ε) | q ∈ QM}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In order to see that, we find: (⊆) If ¯q ∈ Qα(γ(M)), then there is a word b1 · · · bn ∈ dom(R)∗ and are states ¯q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' , ¯qn with qγ(M) 0 = (qM 0 , ε) b1 α(γ(M)) ¯q1 b2 α(γ(M)) · · · bn α(γ(M)) ¯qn = ¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' in α(γ(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By definition of α, this corresponds to transitions qγ(M) 0 = (qM 0 , ε) R(b1) γ(M) ¯q1 R(b2) γ(M) · · · R(bn) γ(M) ¯qn = ¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' in γ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2 to every ¯qi, we obtain that ¯q = (q, ε) for some q ∈ QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Note in particular, ¯q ̸= χ and so χ /∈ Qα(γ(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' (⊇) The converse inclusion iterates the other direction of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2: we assume M ∈ LTS(B) to be reachable, hence every state q ∈ QM is reachable via some word b1 · · · bn ∈ B∗ by iterating Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2: qM 0 b1 · · · bn M q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 24 The assumption that M ∈ LTS(dom(R)) implies that R: B⇀A+ is defined for every bi, and thus (q, ε) is reachable in Qγ(M): qγ(M) 0 = (qM 0 , ε) R(b1) · · · R(bn) γ(M) (q, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' And hence, the definition of α then sends this run to qα(γ(M)) 0 = qγ(M) 0 = (qM 0 , ε) b1 · · · bn α(γ(M)) (q, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' and so (q, ε) ∈ Qα(γ(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The witnessing bijective bisimulation is φ: QM −→ Qα(γ(M)) φ(q) = (q, ε) ∈ Qα(γ(M)) ⊆ Qγ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By our above characterization of Qα(γ(M)), φ is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' It remains to verify that φ is a bisimulation: For every transition in α(γ(M)), concretely (q, ε) b (q′, ε), we have (q, ε) R(b) (q′, ε) in γ(M) by the definition of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2, this implies q b q′ in M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' and indeed φ(q) = (q, ε) and φ(q′) = (q′, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Conversely, for every transition q b q′ in M, we have a transition (q, ε) R(b) (q′, ε) in γ(M) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2 and by b ∈ dom(R) provided by the assumption M ∈ LTS(dom(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By the definition of α, we thus have φ(q) = (q, ε) b (q′, ε) = φ(q′) in α(γ(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In total, φ is an isomorphism in LTS(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since we may add the chaos state χ in the concretization, which introduces nondetermin- ism, it is clear that γR is not a left inverse of αR in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 6 Action Code Composition Since notions of abstraction can be stacked up, it is natural to consider multiple adaptors for multiple action codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Assume an action code R ∈ Code(A, B) and an action code S ∈ Code(B, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then the composition of R and S should be an action code from A to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Given two map-based action codes R: B⇀A+ and S : C⇀B+, we define their composition (R ∗ S): C⇀A+ by (R ∗ S)(c) = � R(b1) · · · R(bn) if S(c) = b1 · · · bn with ∀i: bi ∈ dom(R) undefined otherwise 25 The composed action code R ∗ S is only defined for c ∈ C if S is defined for c and additionally R is defined for every letter bi ∈ B that appears in the word S(c) ∈ B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The defined composition is an instance of Kleisli composition for a monad, which is a standard concept in functional programming and category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Kleisli compo- sition is a recipe to compose maps of the form C → T(B) and B → T(A) to a map of type C → T(A), where T is a monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In our case, the monad is T(X) = X+ + 1 where 1 is an arbitrary singleton and + denotes disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This monad T itself is a combination of two monads: S(X) = X+ is the free semigroup-monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Monads corresponds to algebraic theories and the algebraic theory corresponding to S is that of semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The monad P(X) = X +1 is called the maybe monad (or sometimes called optional in programming), which allows to model partial maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The algebraic theory corresponding to P is that of pointed sets (the theory consists of one nullary operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Warning: even though T(X) = P(S(X)) = X+ +1 and M(X) = X∗ are naturally isomorphic, they are different monads, because M is the list monad, whose corresponding algebraic theory is that of monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Action codes are closed under composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Concretely, given two map-based action codes R: B⇀A+ and S : C⇀B+, their Kleisli composition (R ∗ S): C⇀A+ is again a prefix-free partial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We need to show that (R ∗ S): C⇀A+ is prefix-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' To this end, consider c, c′ ∈ dom(R ∗ S) with (R ∗ S)(c) ≤ (R ∗ S)(c′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since R ∗ S is defined for both c and c′ we can spell out the words as (R ∗ S)(c) = R(b1) · · · R(bn) for n ∈ N and S(c) = b1 · · · bn (R ∗ S)(c′) = R(b′ 1) · · · R(b′ m) for m ∈ N and S(c′) = b′ 1 · · · b′ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Note that we do not know yet whether n or m is bigger!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We only know that R(b1) · · · R(bn) ≤ R(b′ 1) · · · R(b′ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' (4) We now show by induction that for all i with 0 ≤ i ≤ min(n, m): bi = b′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In the base case i = 0, there is nothing to be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In the step for i, assume that we have ∀0 ≤ j < i: bi = b′ i as the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, we also have R(bj) = R(b′ j) for all j < i and so the words R(b1) · · · R(bi) · · · R(bn) and R(b′ 1) · · · R(b′ i) · · · R(b′ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' have a common prefix R(b1) · · · R(bi−1) = R(b′ 1) · · · R(b′ i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For general u, v, w ∈ C∗, if uv ≤ uw, then v ≤ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' So after removing the common prefix from both sides of (4), obtain R(bi) · · · R(bn) ≤ R(b′ i) · · · R(b′ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In such a scenario, we either have R(bi) ≤ R(b′ i) or R(bi) ≥ R(b′ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since R is prefix-free (1), we obtain bi = b′ i in either case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 26 We can now use the inductively proven statement to show that b1 · · · bn ≤ b′ 1 · · · b′ m by case distinction: If min(n, m) = m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' n ≥ m, then we can remove the common prefix R(b1) · · · R(bm) = R(b′ 1) · · · R(b′ m) from both sides of (4) in order to obtain R(bm+1) · · · R(bn) ≤ ε Since all R(bi) ∈ A+ for all 1 ≤ i ≤ n, we necessarily have m = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This implies b1 · · · bn ≤ b′ 1 · · · b′ m (both sides are identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If min(n, m) = n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' n ≤ m, then we directly have b1 · · · bn ≤ b′ 1 · · · b′ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' So in any case, S(c) = b1 · · · bn ≤ b′ 1 · · · b′ m = S(c′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Using that S is prefix-free (1), we conclude c = c′, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Contraction commutes with action code composition: for action codes R ∈ Code(A, B), S ∈ Code(B, C), we have αR∗S(M) = αS(αR(M)) for all M ∈ LTS(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In other words, we have a commutative diagram: LTS(A) LTS(C) LTS(B) αR αR∗S αS Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We show that the systems αR∗S(M) and αS(αR(M)) are even identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Note that we have state sets: QαR∗S(M) ⊆ QM and QαR(αS(M)) ⊆ QαS(M) ⊆ QM which are both subsets of QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Their initial states are identical, because they are both qM 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We establish the isomorphism by simultaneously showing that the state sets match and that the transitions match: (⊆) Consider a transition q c q′ in αR∗S(M) for which we already assume that q ∈ QαR(αS(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, we have q (R ∗ S)(c) q′ in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By the definition of R ∗ S, we have c ∈ dom(S) and b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' , bn ∈ dom(R) with S(c) = b1 · · · bn, so above sequence can be rewritten as q R(b1) · · · R(bn) q′ in M 27 or equivalently q R(b1) · · · R(bn) q′ in M By definition of αR, we have q b1 · · · bn q′ in αR(M) or equivalently q S(c) q′ in αR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Finally, by the definition of αS, this yields q c q′ in αS(αR(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This first shows that all states of αR∗S(M) are also contained in αR(αS(M)) and secondly that the transitions are included, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' (⊇) The converse direction is analogous, starting with a transition q c q′ in αS(αR(M)) for which we know q ∈ QαR∗S(M) already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, c ∈ dom(S) and we obtain q S(c) q′ in αR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' With b1 · · · bn = S(c), we have q b1 · · · bn q′ in αR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This implies that bi ∈ dom(R) for every 1 ≤ i ≤ n and moreover � q R(b1) · · · R(bn) q′� = � q R(b1) · · · R(bn) q′� in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since all bi ∈ dom(R) and c ∈ dom(S), we find that c ∈ dom(R∗S) and so (R∗S)(c) = R(b1) · · · R(bn) and q (R ∗ S)(c) q′ in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Finally, by the definition of αR∗S, we conclude q c q′ in αR∗S(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' It is a standard result about Galois connections (and adjunctions in general) that they are compatible with composition: the right-adjoint of the composition of two functions is equal to the composition of the respective right-adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' One only needs to be warned that ‘equal’ here refers to the equality induced by the order ⊑, which means mutual simulation: For all action codes R ∈ Code(A, B), S ∈ Code(B, C) and M ∈ LTS(C): γR∗S(M) ⊑ γR(γS(M)) and γR∗S(M) ⊒ γR(γS(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This is however weaker than the notion of isomorphism we consider (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Con- cretely, we even have the following counterexample with γR∗S(M) ̸∼= γR(γS(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 28 Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Concretization does not commute with action code composition: Consider sets A = {a}, B = {b}, C = {c} and the action codes R ∈ Code(A, B) R: B⇀A+ b �→ a a S ∈ Code(B, C) S : C⇀B+ undefined everywhere Start with a singleton system that has no transitions: M := q0 start in LTS(C) For the empty S, the concretization γS : LTS(C) → LTS(B) sends this into the system γS(M) = q0, ε start χ b b in LTS(B) We have a b-transition from q0 to χ because S(b) is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For the next action code R, the concretization γR : LTS(B) → LTS(A) treats χ as an ordinary state, so it produces the following: γR(γS(M)) ∼= (q0, ε), ε start (q0, ε), a χ χ, a a a a a in LTS(A) Here, we omitted the unreachable chaos state χ introduced by γR, because the unreachable parts are not relevant for our notion of isomorphism ∼=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' On the other hand, the composed action code R∗S ∈ Code(A, C) is undefined everywhere, so analogously to γS, concretization for R ∗ S sends above M ∈ LTS(C) to γR∗S(M) = q0, ε start χ a a in LTS(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Obviously, γR∗S(M) is not isomorphic to γR(γS(M)), but there are canonical simulations in either direction, induced by the Galois connection between α and γ, using that α commutes with action code composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Refinement commutes with action code composition: for action codes R ∈ Code(A, B), S ∈ Code(B, C), we have ϱR∗S(M) = ϱR(αS(M)) for all M ∈ LTS(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In other words, we have a commutative diagram: LTS(A) LTS(C) LTS(B) ϱR ϱR∗S ϱS 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For the map-based action code R: B⇀A+, define the partial map R∗ : B∗⇀A∗ R∗(ε) = ε R∗(b w) = R(b) R∗(w) (if both defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By this, we mean that the inductive case R∗(b w) is only defined if both R(b) and R∗(w) are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='1 With this definition, we have that (R ∗ S)(c) = R∗(S(c)) for all c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For the isomorphism h: ϱR(ϱS(M)) → ϱR∗S∗(M), the involved state sets are by Defini- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='1 of the form: QϱS(M) ⊆ QM × B∗ QϱR(ϱS(M)) ⊆ (QM × B∗) × A∗ QϱR∗S(M) ⊆ QM × A∗ Define a partial map h: QϱR(ϱS(M)) → QϱR∗S∗(M) by h((q, u), v) = � (q, R∗(u) v) if R∗(u) is defined undefined otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Such a partial map is sufficient to establish an isomorphism between the reachable parts (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='8) of ϱR(ϱS(M)) and ϱR∗S∗(M), because we can show that if ((q, u), v) is reachable, then R∗(u) is defined: if ((q, u), v) is reachable, then the shortest path from the initial state must end with a path of the form: ((q, ε), ε) w ((q, u), ε) v′ ((q, u), v) in ϱR(ϱS(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If we require that v′ is the shortest path from ((q, u), ε) to ((q, u), v), then all transitions of v′ must come from rule (1ϱ), and so v = v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If we require w to be the shortest path, then by an iterated application of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6, we find that w = R∗(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In order to show that h is a simulation, consider a reachable transition ((q, u), v) a ((q′, u′), v′) in ϱR(ϱS(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The transition being reachable implies that R∗(u) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6, there exists a transition (q, u) b (q′′, u′′) in ϱS(M) with R(b) = var and some r ∈ A∗ such that ((q, u), ε) v ((q, u), v) a ((q′, u′), v′) r ((q′′, u′′), ε) in ϱR(ϱS(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6 to the above b-transition in ϱS(M) provides us with some c ∈ C, q′′′ ∈ QM, and s ∈ B∗ with S(c) = ubs such that q c q′′′ in M and (q, ε) u (q, u) b (q′′, u′′) s (q′′′, ε) in ϱS(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 1R∗ is also called the Kleisli extension of R for the monad (−)∗ on partial maps 30 Since all involved states are reachable, R∗(u) and R∗(s) are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In total, we have that (R ∗ S)(c) = R∗(ubs) = R∗(u) R(b) R∗(s) = R∗(u) var R∗(s) and in particular R∗(u) va ≤ (R ∗ S)(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Thus, the state h((q, u), v) = (q, R∗(u) v) in ϱR∗S(M) has an a-transition to h((q′, u′), v′) = (q′, R∗(u′) v′), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For the verification that h is a simulation in the converse direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' from ϱR∗S(M) to ϱR(ϱS(M)), consider a transition (q, u) a (q′, u′) in ϱR∗S(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Again, using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6, we obtain c ∈ C with (R ∗ S)(c) = uav and q c q′′ and (q′, u′) v (q′′, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By the definition of R ∗ S, we thus obtain that u ∈ A∗ must be of the shape R∗(w) r = u for some w ∈ dom(R)∗ By the definition of R ∗ S, we have w b ≤ S(c) with u = R∗(w) r and ra ≤ R(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then, h((q, w), r) = (q, u) and we distinguish: If ua = (R∗S)(c), then the above a-transition in ϱR∗S(M) is produced by rule (2ϱ), and we can use the same rule in ϱS(M) and ϱR(ϱS(M)) to establish the desired transition a-transition to ((q, w), r) a ((q, ε), ε) in ϱR(ϱS(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If ua ≨ (R∗S)(c) but ra = R(b), then we use the rule (1ϱ) in ϱS(M) and but rule (2ϱ) in ϱR(ϱS(M)): ((q, w), r) a ((q, w b), ε) in ϱR(ϱS(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If ua ≨ (R∗S)(c) and ra ≨ R(b), then we use rules (1ϱ) in both ϱS(M) and ϱR(ϱS(M)): ((q, w), r) a ((q, w), r a) in ϱR(ϱS(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Hence, in any of the above cases, we have a corresponding a-transition in ϱR(ϱS(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 31 Learner / Tester Adaptor R SUT M x ∈ X i ∈ I o ∈ O y ∈ Y Figure 9: Using action codes for learning/testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 7 Adaptors In this section, we describe how action codes may be used for learning and testing of black- box systems that can be modelled as Mealy machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The general architecture is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' On the right we see the system under test (SUT), some piece of hardware/software whose behavior can be modeled by a Mealy machine M with inputs I and outputs O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' On the left we see the learner/tester, an agent which either tries to construct an abstract model N of M by performing experiments, or already has such a model N and performs experiments (tests) to find counterexamples which demonstrate that M and N behave differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The learner/tester uses abstract inputs X and outputs Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In between the learner/tester and the SUT we place an adaptor, which uses an action code R to translate between the abstract world of the learner/tester and the concrete world of the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In order to enable the adaptor to do its job, we need to make four (reasonable) assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Our first assumption is that the SUT will accept any input from I in any state, that is, we require that M is input enabled: ∀q ∈ QM ∀i ∈ I : q i/ M Our second assumption ensures that whenever the adaptor sends a concrete symbol i ∈ I to the SUT, the adaptor will accept any output o ∈ O that the SUT may possibly produce in response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We require that R is output enabled for M: ∀r ∈ QR ∀i ∈ I ∀o ∈ O: r i/ R ∧ i/o M ⇒ r i/o R Our third assumption ensures that when the adaptor receives an abstract input x ∈ X from the learner/tester, the adaptor can choose concrete inputs from I that drive R from its initial state r0 to a leaf with label (x, y), for some y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The output y can then be returned as a response to the learner/tester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Reaching a leaf with label x is nontrivial since the transitions taken in R are also determined by the outputs provided by the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We may think of the situation in terms of a 2 player game where the adaptor wins if the game ends in an x leaf, and the SUT wins otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We require that R is finite (has a finite number of states) and has a winning strategy for every input x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The notion of winning is formalized in the following inductive definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='1 (Winning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let R = ⟨R, r0, , l⟩ ∈ Code(I × O, X × Y ) be a finite action code and let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' A leaf r ∈ R is winning for x if π1(l(r)) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' An internal state r ∈ R is winning for x with input i ∈ I if r i/ and, for each transition of the form r i/o r′, r′ is winning for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' An internal state r ∈ R is winning for x if it is winning for x with some i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' R has a winning strategy for x if r0 is winning for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The examples of action codes for Mealy machines that we have seen thus far (Figures 5, 6 and 7) are winning for all the abstract inputs that label their leafs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The action code of Figure 5 is not winning for the abstract input (latte macchiato), for the simple reason that this input does not label any of the leafs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If we remove the incoming transition of state r13 in Figure 5, then the resulting code is no longer winning for (espresso), although it is still winning for (coffee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Our fourth and final assumption is that action code R is determinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If an action code is determinate then, for each state r and abstract input x, there is at most one concrete input i such that r is winning for x with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='3 (Determinate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' An action code R is determinate if, for each state r, whenever r i1/ r1, r i2/ r2 and from both r1 and r2 there is a path to a leaf labeled with input x, then i1 = i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' All the examples of action codes for Mealy machines that we have seen thus far (Figures 5, 6 and 7) are determinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Figure 10 gives an example of an action code that is not determinate: in state r0 two different concrete inputs a and b are enabled that lead to leafs with the same abstract input label 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Note that (trivially) this action code does have a winning strategy for abstract input 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' r0 start r1 0/A r2 0/B a/0 b/0 Figure 10: An action code that is not determinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 2We use projections functions π1 and π2 to denote the first and second element of a pair, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' So π1(x, y) = x and π2(x, y) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 33 Algorithm 1 Pseudocode for an adaptor that implements action code R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 1: function Adaptor(R) 2: while true do 3: x ← Receive-from-learner() 4: r ← r0 5: while r is internal do ▷ loop invariant: r is winning for x 6: i ← unique input such that r is winning for x with i 7: Send-to-SUT(i) 8: o ← Receive-from-SUT() 9: r ← unique state r′ such that r i/o r′ ▷ R output enabled for M 10: end while 11: Send-to-learner(π2(l(r))) 12: end while 13: end function Algorithm 1 shows pseudocode for an adaptor that implements action code R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' During learning/testing, the adaptor records the current state of the action code in a variable r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' When an abstract input x arrives, it first sets r to r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' As long as current state r is internal, the adaptor chooses an input i that is winning for x, and forwards it to the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' When the SUT replies with an output o, the adaptor sets r to a state r′ with r i/o −→ r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' When the new r is internal the adaptor chooses again a winning input, and updates its current state after interacting with the SUT, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' When the new r is a leaf with label (x, y) then the adaptor returns symbol y to the learner/tester and waits for the next abstract input to arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' From the perspective of the learner/tester, the combination of the adaptor and SUT be- haves the same as the contraction αR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Below we will formalize this statement by model- ing both the combination of adaptor and SUT, as well as contraction αR(M) as expressions in the process calculus CCS [27], and then establish the existence of delay simulations between these expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This implies that both expressions have the same traces if we remove all occurrences of the synchronizations between adaptor and SUT, which are invisible from the perspective of the learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let M = ⟨Q, q0, ⟩ ∈ LTS(A) be an LTS, where A is a set of labels that contains the hidden action τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let q q′ denote that there is finite sequence of states r0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' , rn ∈ Q such that r0 = q, rn = q′ and ri−1 τ ri for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' A relation S ⊆ Q × Q is called a delay simulation if it satisfies the following transfer property: If (q, r) ∈ S and q a q′ then either a = τ and q = q′, or ∃r′, r′′ such that r r′ a r′′ and (q′, r′′) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We write q ⊑d r if there exists a weak delay simulation that relates q and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We say that q and r are delay simulation equivalent, notation q ≡d r, if both q ⊑d r and r ⊑d q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let M ∈ LTS(I × O) be an input enabled Mealy machine and let R ∈ Code(I ×O, X ×Y ) be a finite, determinate action code that has a winning strategy for every input in X and that is output enabled for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then the composition of an implementation for M and an adaptor for R is delay simulation equivalent to an implementation for αR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 34 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We describe the behavior of an implementation for M and an adaptor for R formally as expressions in Milner’s Calculus of Communicating Systems (CCS) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The semantics of CCS is defined in terms of an infinite LTS in which the states are CCS expressions, and the transitions between states are defined by structural operational semantics rules given in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In the rest of this proof, we will assume that the reader is familiar with the CCS calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In our CCS expressions we use action names taken from I, O, X and Y , and without loss of generality we assume these four sets to be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Process Impl(M) describes the behavior of an implementation for M in which inputs and outputs are separated and occur sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We define Impl(M) as the CCS expression M(qM 0 ), where for q ∈ QM and i ∈ I, M(q) = � i∈I i · M(q, i) M(q, i) = � o∈O,q′∈QM | q i/o q′ ¯o · M(q′) Similarly, we introduce a process Adaptor(R) that describes the behavior of an adaptor for action code R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Following the pseudocode of Algorithm 1, we define Adaptor(R) as the CCS expression P(r0), where for r ∈ R and x ∈ X, P(r) = � x∈X x · Q(r, x) Q(r, x) = ¯i · R(r, x) if r is internal and i winning for x in r R(r, x) = � o∈O,r′∈R | i/o M∧r i/o r′∧i winning for x in r o · Q(r′, x) Q(r, x) = π2(l(r)) · P(r0) if r is a leaf Processes Adaptor(R) and Impl(M) may synchronize via actions taken from I ∪O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If we com- pose these processes using the CCS composition operator |, and apply the CCS restriction operator \\ to hide all communications, we obtain a CCS expression that describes the behav- ior of the parallel composition of the adaptor and the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We claim that this composition is delay simulation equivalent to the expression Impl(αR(M)) that describes the behavior of an implementation of αR(M): (Adaptor(R) | Impl(M)) \\ (I ∪ O) ≡d Impl(αR(M)) (5) Here we define Impl(αR(M)) as the CCS expression N(qα(M) 0 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' where for q ∈ Qα(M) and x ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' N(q) = � x∈X x · N(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' x) N(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' x) = � y∈Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='q′∈Qα(M) | q x/y α(M)q′ ¯y · N(q′) 35 Consider the following relation S between CCS expressions: S = {((P(r0) | M(q)) \\ (I ∪ O),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' N(q)) | q ∈ Qα(M)} ∪ {((Q(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' x) | M(q′)) \\ (I ∪ O),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' N(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' x)) | q ∈ Qα(M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' q′ ∈ QM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' r ∈ R winning for x ∈ X ∧ ∃σ ∈ (I × O)∗ : r0 σ R r ∧ q σ M q′} ∪ {((R(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' x) | M(q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' i)) \\ (I ∪ O),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' N(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' x)) | q ∈ Qα(M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' q′ ∈ QM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' r ∈ R winning for x ∈ X with i ∈ I ∧ ∃σ ∈ (I × O)∗ : r0 σ R r ∧ q σ M q′} We claim that S is a delay simulation relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In order to prove this, we check that the transfer property holds for all pairs of related states and enabled transitions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Assume ((P(r0) | M(q))\\(I ∪O), N(q)) ∈ S and (P(r0) | M(q))\\(I ∪O) x (Q(r0, x) | M(q)) \\ (I ∪ O), for some x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We observe that N(q) x N(q, x) and note that ((Q(r0, x) | M(q)) \\ (I ∪ O), N(q, x)) ∈ S since r0 is winning for x, r0 σ R r0 and q σ M q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Assume ((Q(r, x) | M(q′)) \\ (I ∪ O), N(q, x)) ∈ S and (Q(r, x) | M(q′)) \\ (I ∪ O) τ (R(r, x) | M(q′, i)) \\ (I ∪ O), for r internal and i the unique input that is winning for x in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By the assumption, q ∈ Qα(M), q′ ∈ QM, and there exists σ ∈ (I × O)∗ such that r0 σ R r and q σ M q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then ((R(r, x) | M(q′, i))\\(I ∪O), N(q, x)) ∈ S, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Assume ((R(r, x) | M(q′, i))\\(I ∪O), N(q, x)) ∈ S and (R(r, x) | M(q′, i))\\(I ∪O) τ (Q(r′, x) | M(q′′))\\(I ∪O), with r i/o r′ and q′ i/o q′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By the assumption, q ∈ Qα(M), q′ ∈ QM, r is winning for x with i, and there exists σ ∈ (I × O)∗ such that r0 σ R r and q σ M q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then q′′ ∈ QM and, by definition of winning, r′ is winning for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Moreover, if we take σ′ = σ · (i, o), then r0 σ′ R r′ and q σ′ M q′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This implies that ((Q(r′, x) | M(q′′)) \\ (I ∪ O), N(q, x)) ∈ S, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Assume ((Q(r, x) | M(q′))\\(I ∪O), N(q, x)) ∈ S and (Q(r, x) | M(q′))\\(I ∪O) π2(l(r)) (P(r0) | M(q′)) \\ (I ∪ O), for r a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By the assumption, q ∈ Qα(M), q′ ∈ QM, r is winning for x, and there exists σ ∈ (I × O)∗ such that r0 σ R r and q σ M q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By definition of the contraction operator, q l(r) α(M) q′ and q′ ∈ Qα(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' But this means N(q, x) π2(l(r)) N(q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Now observe that ((P(r0) | M(q′)) \\ (I ∪ O), N(q′)) ∈ S, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Next consider the following relation T between CCS expressions: T = {(N(q), (P(r0) | M(q)) \\ (I ∪ O)) | q ∈ Qα(M)} ∪ {(N(q, x), (Q(r0, x) | M(q)) \\ (I ∪ O)) | q ∈ Qα(M)} We claim that T is a delay simulation relation, and check that the transfer property holds for all pairs of related states and enabled transitions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Assume (N(q), (P(r0) | M(q)) \\ (I ∪ O)) ∈ T and N(q) x N(q, x), for some x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We observe that (P(r0) | M(q)) \\ (I ∪ O) x (Q(r0, x) | M(q)) \\ (I ∪ O) and note that (N(q, x), (Q(r0, x) | M(q)) \\ (I ∪ O)) ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Assume (N(q, x), (Q(r0, x) | M(q)) \\ (I ∪ O)) ∈ T and N(q, x) ¯y N(q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Then α(M) has a transition q x/y q′ and q′ ∈ Qα(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By definition of α(M), R has a leaf r with l(r) = (x, y) and there exists a sequence σ such that r0 σ R r and q σ M q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let σ = (i1, o1)(i2, o2) · · · (in, on) Then R has states r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' , rn and M has states s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' , sn such that: r0 i1/o1 R r1 i2/o2 R r2 · · · in/on R rn q = s0 i1/o1 M s1 i2/o2 M s2 · · · in/on M sn = q′ From these runs in R and M we may construct a sequence of τ-transitions: (Q(r0, x) | M(s0)) \\ (I ∪ O) τ (R(r0, x) | M(s0, i1)) \\ (I ∪ O) τ (Q(r1, x) | M(s1)) \\ (I ∪ O) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' τ (R(rn−1, x) | M(sn−1, in)) \\ (I ∪ O) τ (Q(rn, x) | M(sn)) \\ (I ∪ O) Note that our assumptions that R is determinate and has a winning strategy for every input x ∈ X impy that the inputs that occur in σ are always winning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' From the above sequence of τ-transitions we may conclude (Q(r0, x) | M(q)) \\ (I ∪ O) (Q(r, x) | M(q′)) \\ (I ∪ O) Since (Q(r, x) | M(q′)) \\ (I ∪ O) ¯y (P(r0) | M(q′)) \\ (I ∪ O) and (N(q′), (P(r0) | M(q′)) \\ (I ∪ O)) ∈ T, the transfer property follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Because S is a delay simulation from (Adaptor(R) | Impl(M)) \\ (I ∪ O) to Impl(αR(M)), and T is a delay simulation from Impl(αR(M)) to (Adaptor(R) | Impl(M)) \\ (I ∪ O), identity (5) follows, and thereby the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Active automata learning algorithms and tools for Mealy machines typically assume that the system under learning is output deterministic3: the output and target state of a transition are uniquely determined by its source state and input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Mealy machine M is output deterministic if, for each state q and input i, q i/o r ∧ q i/o′ r′ ⇒ o = o′ ∧ r = r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The proposition below asserts that for action codes that are determinate, contraction pre- serves output determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This property makes it possible to use existing automata learning tools to learn models of systems that consist of an output deterministic SUT composed with a determinate adaptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 3The notion of deterministic that we use in this article is the standard one for LTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In the literature on Mealy machines and FSMs, machines that we call output deterministic are called deterministic, and machines that we call deterministic are called observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 37 Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Suppose M is a Mealy machine and R is an action code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If M is output deterministic and R is determinate then αR(M) is output deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Assume M is output deterministic and R is determinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Let N = αR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Suppose that N has transitions q x/y′ N q′ and q x/y′′ N q′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We need to show y′ = y′′ and q′ = q′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The transitions have been derived using rule (2α), and formulated for tree-shaped action codes R, we know there are generalized transitions q u′/s′ M q′ r0 u′/s′ R r′ r′ ∈ L l(r′) = (x, y′) q u′′/s′′ M q′′ r0 u′′/s′′ R r′′ r′′ ∈ L l(r′′) = (x, y′′) Now since R is determinate, the first inputs in u′ and u′′ must be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' But since M is output deterministic, this implies that the first outputs in s′ and s′′ must also be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Moreover, the paths from q to q′ and q′′ in M share the same initial transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Since action codes are deterministic, the paths from r0 to r′ and r′′ in R also share the same initial transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By repeating this line of reasoning, we can “zip together” the paths from q to q′ and q′′ in M, and the paths from r0 to r′ and r′′ in R, and obtain u′ = u′′, s′ = s′′, q′ = q′′, r′ = r′′ and y′ = y′′, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 8 Discussion By introducing the notion of action codes, we provided a new perspective on the problem of how high-level state machine models with abstract actions can be related to low-level models in which these actions are refined by sequences of concrete actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This perspective may help with the systematic design of adaptors during learning and testing, and the subsequent interpretation of obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Our theory allows for action codes (such as in Figure 5) that are adaptive in the sense that outputs which occur in response to inputs at the concrete level may determine the sequence of concrete inputs that refines an abstract input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We are not aware of case studies in which such adaptive codes are used, but believe they may be of practical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' One may, for instance, consider a scenario in which an abstract action AUTHENTICATE is refined by a protocol in which a user is either asked to authenticate by entering a PIN code, or by providing a fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Close to our work are the results of Rensink and Gorrieri [28], who investigate vertical implementation relations to link specifications and implementations belonging to conceptu- ally different levels of abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' These relations are indexed by a refinement function that maps abstract actions into concrete processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Within a setting of a CCS-like language, Rensink & Gorrieri list a number of proof rules that should hold for any vertical imple- mentation relation, and propose vertical bisimulation as a candidate vertical implementation relation for which these proof rules hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In the setting of our paper, we can define two vertical implementation relations ⊑R γ and ⊑R ϱ , for any action code R, by M ⊑R γ N ⇔ M ⊑ γR(N), M ⊑R ϱ N ⇔ M ⊑ ϱR(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' 38 Then ⊑R ϱ ⊆ ⊑R γ and both relations satisfy all language-independent proof rules of [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For instance M ⊑ M′ M′ ⊑R γ N ′ N ′ ⊑ N M ⊑R γ N (since γR is monotone and ⊑ is transitive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' With the action code R of Figure 4, both implementation relations will relate the LTS of Figure 2 (right) with the LTS of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' However, if we consider the vertical bisimulation preorder of Rensink and Gorrieri [28], the LTS of Figure 2(right) does not implement the LTS of Figure 1, when indexed by a refinement that maps a to 1 4 1, and b to 1 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' This example suggests that bisimulations may not be suitable as vertical implementation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Also close to our work are results of Burton et al [8], who propose a vertical implementa- tion relation in the context of the CSP language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Instead of action codes, they use extraction patterns, a strict monotonic map extr : Dom → B∗, where Dom is the prefix closure of a set dom ⊆ A∗ of concrete action sequences that may be regarded as complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' By providing a mapping from sequences of concrete actions to sequences of abstract actions, extraction patterns are more general than our action codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' However, since extraction mappings are not required to have an inverse, establishing interesting Galois connections in this general setting may be difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' With an extraction pattern defined in the obvious way, the LTS of Figure 2(right) does implement the LTS of Figure 1 according to the implementation relation proposed by Burton et al [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We developed our theory for LTSs and Mealy machines, using the simulation preorder as the implementation relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' It would be interesting to transfer our results to other modeling frameworks (such as IOTSs [32] and timed automata [3]) and other preorders and equivalences in the linear-time branching-time spectrum for LTSs [15] and IOTSs [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Our theory is orthogonal to the work of Aarts et al [1], which advocates the use of so- called mappers to formalize adaptors that abstract the large action alphabets of realistic network protocols into small sets of actions that can be handled by a learning tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Aarts et al [1] also describe the relation between abstract and concrete models using a Galois connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' In practical applications of model learning, it can make sense to construct an adaptor that combines a mapper in the sense of [1] with an action code as introduced in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Fiterău-Broştean et al [11] describe a small domain specific language which they used to describe mapper components, and from which adaptor software can be generated automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' It would be interesting to extend this domain specific language so that it may also be used to specify action codes for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Different action codes lead to different contractions, and thereby to different abstract views of the same system, see for instance Figure 6 and Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We may try to exploit this fact during learning and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For instance, in case of a system M that is too big for state-of-the-art learning algorithms, we may still succeed to learn partial views using cleverly selected action codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Using our Galois connections we then could obtain various upper and lower bounds for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Ideally, such an approach may even succeed to uniquely identify M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Maarse [26] quantified the quality of a contraction αR(M) in terms of the graph-theoretic concept of eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' If q and q′ are states in an LTS M then d(q, q′) is defined as the number of transitions in the shortest path from q to q′ (or ∞ if no such path exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' For 39 any set of states Q ⊆ QM, the eccentricity ε(Q) is defined as max q′∈QM min q∈Q d(q, q′) that is, the maximal distance one needs to travel to visit a state of M, starting from a state of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' A good contraction has a small set of states Q and a low eccentricity ε(Q): the contraction only covers a small subset Q of the states of M, but any state from M can be reached via a short sequences of transitions from a Q-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' Acknowledgements As part of a MSc thesis project under supervision of the first author, Timo Maarse studied a different and more restricted type of action codes (called action refinements) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' It turned out, however, that for these action codes, the concretization operator is not monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The present paper was inspired by [26] and arose from our efforts to fix this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' We thank Paul Fiterău-Broştean for examples of the use of action codes in model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQfYPeA/content/2301.00199v1.pdf'} +page_content=' The first author would like to thank Rocco De Nicola and IMT Lucca for their hospitality during the work on this paper.' metadata={'source': 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b/lNAzT4oBgHgl3EQfNvsP/content/tmp_files/2301.01152v1.pdf.txt @@ -0,0 +1,1472 @@ +arXiv:2301.01152v1 [math.CO] 3 Jan 2023 +Snake Paths in King and Knight Graphs +Nikolai Beluhov +Abstract. A snake path in a graph G is a path in G which is also an induced subgraph +of G. For all n, we find the greatest length of a snake path in the n × n king graph and +we give a complete description of the paths which attain this greatest length. The even +and odd cases behave very differently. We also estimate the greatest length of a snake +path or cycle in the m × n knight graph, for all m and n. +1 +Introduction +Let G be a simple graph. A snake path in G is a path in G which is also an induced +subgraph of G. Equivalently, a path P in G is a snake path when, for all vertices u and +v of P, we have that uv is an edge of G if and only if it is an edge of P. Intuitively, a +snake path never comes into contact with itself. +Snake paths are also known as induced paths and chordless paths. +A snake cycle is defined similarly. Just like paths, snake cycles are alternatively called +induced cycles and chordless cycles. Our focus will be mostly on snake paths, though we +will touch upon snake cycles, too. +Given a graph G, some of the most natural questions we can ask about its snake +paths are as follows: +Question A. What is the greatest length of a snake path in G? +Note that we measure the length of a path by the number of edges that it traverses, +rather than the number of vertices that it visits. +The answer of Question A coincides with the greatest diameter of an induced sub- +graph of G. +The greatest length of a snake path in G is also known as the induced detour number +of G, and the greatest length of a snake cycle in G as the induced circumference of G. +In the special case when G is a hypercube graph, Question A is known as the snake- +in-the-box problem, and its analogue for cycles as the coil-in-the-box problem. Both of +these problems have been studied extensively. +Question B. What is the structure of the longest snake paths in G? +Question C. How many longest snake paths are there in G? +Note that we formalise paths as subgraphs, rather than as sequences of vertices. +Specifically, to us a “path” is a tree subgraph where all vertices are of degree at most +two. The distinction between the two formalisations does not matter in most situations, +but it does matter for enumeration. For example, to us abc and cba are the same path. +1 + +We study Questions A–C for certain graphs G associated with chess pieces. +Given a chess piece F and a board A, it is natural to consider the graph whose +vertices are the cells of A and whose edges correspond to all possible moves of F on A. +We proceed to formalise this notion for the king and the knight. Other chess pieces can +be handled similarly. +To us, a cell is an ordered pair of integers. Or, equivalently, an integer point in the +plane. +Let m and n be positive integers. A board A of size m×n, with m rows and n columns, +is a set of cells of the form I × J, where I and J are integer intervals with |I| = n and +|J| = m. The standard board of size m × n has I = [0; n − 1] and J = [0; m − 1]. Since +all boards of the same size are translation copies of one another, sometimes we refer to +“the” board of a certain size, meaning the standard board of that size. +Given a set of cells S, we define the king graph on S, denoted G(K, S), to be the +graph on vertex set S where two distinct cells a′ = (x′, y′) and a′′ = (x′′, y′′) are joined +by an edge if and only if |x′ − x′′| ≤ 1 and |y′ − y′′| ≤ 1. Since all king graphs on boards +of the same size are isomorphic, sometimes we refer to “the” king graph of a certain size, +meaning the king graph on the standard board of that size. For convenience, we also use +the notation G(K, m × n) for the king graph of size m × n. Of course, G(K, m × n) can +also be viewed as the strong product of two paths with m and n vertices, respectively. +The knight graph on S, denoted G(N, S), is defined similarly, except that the adja- +cency condition becomes {|x′ − x′′|, |y′ − y′′|} = {1, 2} instead. +Dawson, in problem 187 of [1], considers G(N, 8 × 8) and presents a snake path of +length 31 as well as a snake cycle of length 32. +Knuth, in exercise 172 of [5], discusses the longest snake paths and cycles of various +chess piece graphs in the context of algorithmic generation. In particular, he determines +that there are 16 essentially distinct snake paths of the greatest length 31 and 6 essentially +distinct snake cycles of the greatest length 31 in G(K, 8 × 8); as well as an essentially +unique snake path of the greatest length 33 and 4 essentially distinct snake cycles of the +greatest length 32 in G(N, 8 × 8). (Thus Dawson’s path was not optimal, but his cycle +was.) +Our main results are as follows: +Theorem 1. Let n be an even positive integer. Then the greatest length of a snake +path in the king graph of size n × n is n2/2 − 1. Furthermore, when n ≥ 6, there are +exactly 16n snake paths which attain this greatest length. +This count includes rotations and reflections. In Section 3, we will see that with n ≥ 6 +the number of essentially distinct longest snake paths is 2n when n/2 is even and 2n + 1 +when n/2 is odd. +Over the course of the proof of Theorem 1, we will also give a complete description +of these paths. Roughly speaking, each one of them is shaped like a spiral. +That the total number of paths is given by such a nice formula is most likely only a +happy coincidence. Because of the overall structure of the proof, we should expect to see +a linear function of n within each parity of n/2; however, there is no obvious reason a +priori to expect these two functions to coincide, or their constant terms to vanish. +2 + +Theorem 2. Let n be an odd positive integer. Then the greatest length of a snake +path in the king graph of size n × n is (n2 − 1)/2. +Remarkably, the odd and even cases behave very differently. Despite the surface simi- +larity between the upper bounds of Theorems 1 and 2, the former bound is straightforward +while the latter one poses considerable difficulties. +In Section 5, we will add to Theorem 2 a complete description of the paths which +attain the greatest length, stated in Theorem 4. However, the description is somewhat +complicated, and relies on a long series of preceding definitions. Thus we do not reproduce +it in the introduction. The gist is that each stamp-folding permutation of ⌈n/2⌉ elements +yields two families of longest snake paths which share the same overall shape but differ +from one another by some tiny aberrations. +Theorem 4 does not imply an exact answer to Question C. In fact, because of the +connection to the stamp-folding problem, it seems unlikely that such an answer would +be feasible. Still, the theorem does yield some loose bounds. In particular, we will see +(Proposition 2) that the logarithm of the number of longest snake paths grows as Θ(n), +in stark contrast to the even case. +Theorems 1, 2, and 4 together completely resolve the questions of the greatest length +of a snake path and the structure of the longest snake paths in king graphs on square +boards. +Theorem 3. Let m and n be positive integers. Then both the longest snake path and +the longest snake cycle in the knight graph of size m×n are of length mn/2+O(m+n). +Note that we do not specify the sign of the error term: Since O-notation only bounds +the absolute value of a function, the classes mn/2 + O(m + n) and mn/2 − O(m + n) +consist of the same functions of m and n. Same goes for the estimates in Section 8. +Compared to the treatment of Question A in Theorems 1 and 2, with Theorem 3 we +do not attempt to obtain an exact answer, and are content instead with an asymptotic +estimate. On the bright side, this asymptotic estimate applies to cycles as well as paths, +and it is valid on all rectangular boards. +It is worth noting that one step in our proof of Theorem 3 relies on computer help, and +likely cannot be verified manually by a human mathematician. (Our proofs of Theorems +1, 2, and 4 are all human-friendly, though.) +The author obtained Theorems 1–7 in 2018 after being introduced to the subject by +Knuth, in connection with the aforementioned exercise 172. Subsequently, Theorems 1, +3, and 5 were cited in a remark following the exercise’s solution. (Strictly speaking, at +that time the author derived Theorem 1 in a form referring to the number of essentially +distinct paths rather than the total number of paths. This is also how it was stated in [5].) +Then, in 2020, the author proposed Theorem 2, appropriately rephrased, as a mathe- +matical olympiad problem for the Cyberspace Mathematical Competition. It was featured +as problem 4 on day 1 of the contest. (The CMC was a one-off event intended to approx- +imate the International Mathematical Olympiad. Perhaps the problem’s difficulty was +not an ideal match for the contest’s format; out of 553 participants from 75 countries, +only two made substantial progress on it.) +3 + +Theorems 1–4 offer an interesting illustration of how the nature of Questions A–C +can change when we vary the underlying graph. In the setting of Theorem 1, Question +A is straightforward while Questions B and C are manageable. For Theorems 2 and +4, both Questions A and B become significantly more complicated, while a closed-form +answer to Question C is likely out of reach. Finally, in the setting of Theorem 3, already +with Question A it seems that an exact answer would be unfeasible, and even for our +asymptotic estimate we find ourselves in need of machine help. +2 +Preliminaries +Before we continue, let us briefly list some useful notations and observations. +Given two cells a′ = (x′, y′) and a′′ = (x′′, y′′), we write a′+a′′ for the cell (x′+x′′, y′+ +y′′). Given a cell a and a set of cells S, we write a + S for the set of cells {a + b | b ∈ S}. +A symmetry of a board A is the restriction to A of an isometry of the plane which +preserves A. Two objects defined with reference to A, such as two sets of cells on A or +two graphs on A, are essentially distinct (relative to A) when they are distinct under the +symmetries of A. +The grid graph on a set of cells S, denoted G(□, S), is defined similarly to the king and +knight graphs on S, except that the adjacency condition becomes {|x′ − x′′|, |y′ − y′′|} = +{0, 1} instead. Or, equivalently, cells a′ and a′′ must be at unit Euclidean distance from +one another. Of course, G(□, m × n) can also be viewed as the Cartesian product of two +paths with m and n vertices, respectively. +For the edge joining two vertices a and b in a graph G, we write either ab or a—b, +whichever one reads better in the situation at hand. +We introduce shorthand notation for certain paths within grid and king graphs. Given +two cells a and b of a grid or king graph G such that a and b are in the same row or +column, we write a∼b for the path in G connecting a and b whose remaining cells are the +ones between a and b in the corresponding row or column. For example, if a = (x′, y), +b = (x′′, y), and x′ ≤ x′′, then a∼b = (x′, y)—(x′ + 1, y)—(x′ + 2, y)—· · · —(x′′, y). +Let P be a path in some graph G. Given a vertex a of G, we write a ∈ P for “P visits +a”; given an edge e of G, we write e ∈ P for “P traverses e”; and, given a path Q in G, +we write Q ⊆ P for “Q is a subpath of P”. We will not use these abbreviations too often, +but the proofs of Lemmas 2 and 6 would be cumbersome to state without them. +Suppose, now, that P is a snake path in G. When ab′b′′ is a three-cycle in G, we have +that a ∈ P implies b′b′′ ̸∈ P. Similarly, when b, c′, and c′′ are three distinct neighbours +of a in G, we have that c′ac′′ ⊆ P implies b ̸∈ P. We will use these simple observations +repeatedly throughout the paper. +A subset S of the vertices of G is k-independent if, in the induced subgraph of G on +vertex set S, every vertex is of degree at most k. When k = 2, the induced subgraph +itself is a pseudosnake of G. Clearly, the number of vertices in the longest snake path or +cycle of G cannot exceed the number of vertices in its largest pseudosnake. When G is +finite, we define its pseudosnake density to be the ratio of the greatest number of vertices +in a pseudosnake of G to the total number of vertices in G. +4 + +3 +King Graphs on Even Boards +In this section, we prove Theorem 1. Let n be an even positive integer with n = 2k, +let A be the standard board of size n × n, and let G be the king graph on A. +For completeness, with n = 2 there are 6 longest snake paths, of which 2 are essentially +distinct; and with n = 4 there are 28 longest snake paths, of which 4 are essentially +distinct. From now on, let n ≥ 6. +The argument we give for the upper bound of Theorem 1 is not new; the special case +n = 8 is in [5], and the general case does not pose any additional difficulties. +Proof of the optimisation part of Theorem 1. For the upper bound, partition A into +k2 subboards of size 2×2 each. Since a snake path in G can visit at most two cells within +each subboard, its length cannot exceed 2k2 − 1. +Figure 1 +For the lower bound, let Pi be the path (i, i)∼(n−i−2, i)—(n−i−1, i+1)∼(n−i− +1, n−i−2)—(n−i−2, n−i−1)∼(i+1, n−i−1)—(i, n−i−2)∼(i, i+3)—(i+1, i+2)— +(i + 2, i + 2) for all even i with 0 ≤ i ≤ k − 3. When k is even and i = k − 2, we define +Pk−2 in the same way, except that we stop at cell (i, n − i − 2) = (k − 2, k); and, when +k is odd and i = k − 1, we also define Pk−1 to be the path (k − 1, k − 1)—(k, k). Then +the concatenation of these paths is a snake path in G of length n2/2 − 1. □ +For example, Figure 1 shows n = 8. +The enumeration part of Theorem 1 will be somewhat more complicated. +Let B be the standard board of size k × k and let H be the grid graph on B. +For each cell b of B, let Φ(b) denote the subboard 2b + [0; 1]2 of A, of size 2 × 2. (For +convenience, if b = (x, y), we also write simply Φ(x, y).) These subboards, taken over all +cells b of B, form a partitioning of A. +Let P be a longest snake path in G. The proof of the optimisation part of Theorem 1 +shows that P visits exactly two cells within each subboard of A of the form Φ(b). +Suppose that an edge of P joins one cell of Φ(b′) and one cell of Φ(b′′), with b′ ̸= b′′. +We claim that b′ and b′′ are then neighbours in H. +Indeed, if not, then b′ and b′′ must be diagonally adjacent in H, without loss of +generality with b′ + (1, 1) = b′′. So P contains a subpath of the form c′—(2b′ + (1, 1))— +2b′′—c′′, where c′ is in Φ(b′) and c′′ is in Φ(b′′). Since P is a snake path, it follows that +P cannot visit any cells in the set 2(b′ + (1, 0)) + {(0, 0), (0, 1), (1, 1)}. Consequently, P +visits at most one cell of Φ(b′ + (1, 0)), a contradiction. (Figure 2.) +For each edge of P joining one cell of Φ(b′) and one cell of Φ(b′′), with b′ ̸= b′′, take +the edge b′b′′ of H. Since P visits Φ(b) for all b, these edges form a Hamiltonian path +5 + +× +× × +Φ(b′) +Φ(b′′) +Figure 2 +in H. Denote this path by ̺. +From this point on, our plan for the proof will be as follows: First we obtain a +complete description of the structure of ̺. We do this by means of a series of mostly local +considerations, starting on the boundary of H and then working our way in. Once we +are done, we determine what paths P in the original graph G are associated with each +path ̺. +We continue with the details. +We define the i-th frame of B, denoted Fi, to be the subset +Fi = [i; k − i − 1]2 \ [i + 1; k − i − 2]2 +of B. Thus F0, F1, . . ., F⌈k/2⌉−1 form a partitioning of B. +We denote the four corner cells of Fi by ai = (i, i), bi = (k − i − 1, i), ci = (k − i − +1, k − i − 1), and di = (i, k − i − 1). +(a) +(b) +(c) +(d) +(e) +Figure 3 +We say that Fi is of type I, II, III, IV, or V (relative to ̺) when ̺ contains the +following subpaths: +For type I, ai∼bi∼ci∼di∼(ai + (0, 1))—(ai + (1, 1)). (Figure 3(a).) +For type II, ai∼bi∼ci∼di∼(ai + (0, 2))—(ai + (1, 2)) and (ai + (0, 1))—(ai + (1, 1)). +(Figure 3(b).) +For type III, ai∼bi∼ci∼(di +(1, 0))—(di +(1, −1)) and di∼(ai +(0, 1))—(ai +(1, 1)). +(Figure 3(c).) +For type IV, ai∼bi∼(ci+(0, −1))—(ci+(−1, −1)) and ci∼di∼(ai+(0, 1))—(ai+(1, 1)). +(Figure 3(d).) +For type V, ai∼(bi +(−1, 0))—(bi +(−1, 1)) and bi∼ci∼di∼(ai +(0, 1))—(ai +(1, 1)). +(Figure 3(e).) +We say that ̺ itself is of type I when all of F0, F1, . . ., F⌈k/2⌉−2 are of type I. +Furthermore, let T be one of the symbols II, III, IV, and V, and let s be a nonnegative +integer with 0 ≤ s ≤ ⌈k/2⌉ − 2. We say that ̺ is of type T(s) when all of F0, F1, . . ., +Fs−1 are of type I and all of Fs, Fs+1, . . ., F⌈k/2⌉−2 are of type T. When ̺ is of one of +the 4⌈k/2⌉ − 3 types we have just listed, we say that it is regular. +6 + +Figure 4 +For example, Figure 4 shows the unique ̺ of type III(1) when k = 7. +Observe that, when k is even, ̺ cannot be of type IV(s), for any s, as F⌈k/2⌉−2 being +of type IV prevents ̺ from visiting all cells of F⌈k/2⌉−1. Similarly, when k is odd, ̺ cannot +be of type II(s), for any s, as the frame F⌈k/2⌉−2 is too small to be of type II. +In all other cases, if k and the type of ̺ are fixed, there is a unique ̺ of that type, +with one exception: When k is even, there are two paths ̺ of type I, differing by just one +edge within the innermost frame F⌈k/2⌉−1. +Lemma 1. Suppose that ̺ makes a turn at cell b of B, for concreteness by means of +(b+(0, 1))—b—(b+(1, 0)). Then the two cells of Φ(b) in P are 2b+(0, 1) and 2b+(1, 0). +Of course, similar claims hold for the other three possible turns at b as well. +Proof. Suppose, for the sake of contradiction, that 2b+(1, 1) ∈ P. Since (b+(0, 1))— +b—(b + (1, 0)) ⊆ ̺, we get that there are two cells a′ and a′′ with a′ ∈ Φ(b + (0, 1)), +a′′ ∈ Φ(b + (1, 0)), and a′—(2b + (1, 1))—a′′ ⊆ P. However, then P cannot visit any cells +of Φ(b) other than 2b + (1, 1), a contradiction. +Φ(b) +Figure 5 +Thus 2b+(1, 1) ̸∈ P. Since (b+(0, 1))—b—(b+(1, 0)) ⊆ ̺, it follows that 2b+(0, 1) ∈ P +and 2b + (1, 0) ∈ P, as needed. (Figure 5.) □ +Lemma 2. A symmetry of B maps ̺ onto a regular Hamiltonian path in H. +Proof. First we consider the outermost frame F0 of B. Observe that all edges of H +within F0 form a cycle E. +Suppose, for the sake of contradiction, that (0, y)—(0, y + 1) ̸∈ ̺ for some y with +2 ≤ y ≤ k − 4. +Then P must miss both cells in at least one of the two pairs {(0, 2y + 1), (1, 2y + 1)} +and {(0, 2y + 2), (1, 2y + 2)}. Suppose, for concreteness, that this is true of the former +pair; the other case is similar. (Figure 6.) +Since P visits two cells in Φ(0, y), we get that (0, 2y) ∈ P and (1, 2y) ∈ P. So +(0, 2y − 1) ̸∈ P and (1, 2y − 1) ̸∈ P. Iterating this sequence of observations, we see that +all of (0, 2y − 2), (1, 2y − 2), (0, 2y − 4), and (1, 2y − 4) are in P while all of (0, 2y − 3), +7 + +× × +× × +× × +× × +Φ(0, y − 2) +Φ(0, y − 1) +Φ(0, y) +Φ(0, y + 1) +Figure 6 +(1, 2y − 3), (0, 2y − 5), and (1, 2y − 5) are not. (When y = 2, the lattermost couple of +cells will lie outside of A. Of course, then they will be outside of P, too.) +Thus all three of (0, 2y), (0, 2y − 2), and (0, 2y − 4) must be endpoints of P, a +contradiction. +Consequently, (0, y)—(0, y + 1) ∈ ̺ for all y with 2 ≤ y ≤ k − 4. By symmetry, it +follows that ̺ contains all edges of E except for, possibly, (0, 0)—(0, 1), (0, 1)—(0, 2), +and their images under the symmetries of B. +On the other hand, at least one edge of E must be outside of ̺. +If (0, 0)—(0, 1) ̸∈ ̺, then (0, 0) is an endpoint of ̺ and Φ(0, 0) contains an endpoint +of P. +If (0, 1)—(0, 2) ̸∈ ̺, then P must miss both cells in at least one of the two pairs +{(0, 3), (1, 3)} and {(0, 4), (1, 4)}. If the latter, then with k ≥ 5 we arrive at a contradic- +tion as before. If the former, then the same reasoning as before shows that Φ(0, 0) and +Φ(0, 1) must each contain an endpoint of P. When k = 4, the same conclusion holds up +to reflection with respect to the horizontal axis of symmetry of A. +Of course, these observations apply also to all images of the edges (0, 0)—(0, 1) and +(0, 1)—(0, 2) under the symmetries of B. +Since there are only two endpoints of P, it follows that ̺ cannot omit too many edges +of E. We are left to consider the following cases, up to the symmetries of B: +Case 1. The only edge of E outside of ̺ is (0, 0)—(0, 1). Then F0 is of type I. +Case 2. The only edge of E outside of ̺ is (0, 1)—(0, 2), and k ≥ 4. Then F0 is of +type II. +Case 3. The only edges of E outside of ̺ are (0, 0)—(0, 1) and (0, k − 1)—(1, k − 1). +Then F0 is of type III. +Case 4. The only edges of E outside of ̺ are (0, 0)—(0, 1) and (k − 1, k − 2)— +(k − 1, k − 1). Then F0 is of type IV. +Case 5. The only edges of E outside of ̺ are (0, 0)—(0, 1) and (k − 2, 0)—(k − 1, 0). +Then F0 is of type V. +Cases 6–9. The only edges of E outside of ̺ are (0, 0)—(0, 1) and one of (0, 0)—(1, 0), +(0, k −2)—(0, k −1), (k −2, k −1)—(k −1, k −1), and (k −1, 0)—(k −1, 1). The first one +of these cases cannot occur because then cell (0, 0) becomes isolated in ̺. The other three +8 + +cannot occur because, in each one of them, part of the edges of E in ̺ form a subpath +of ̺ which contains both endpoints of ̺ but does not coincide with ̺. +With this, we have established that F0 is of one of our types relative to ̺, up to the +symmetries of B. +We finish the proof by induction on k. Our base cases are k = 3 and k = 4, when +there is nothing left to prove. For the induction step, suppose that k ≥ 5 and that we +have already settled the question on all smaller boards. +Let A⋆ be the concentric subboard of A of size (n−4)×(n−4) given by A⋆ = [2; n−3]2 +and let G⋆ be the king graph on A⋆. +When F0 is of type I, the cells of P outside of A⋆ form a subpath of P containing one +endpoint of P. When F0 is of one of the remaining four types, the cells of P outside of +A⋆ form two subpaths of P containing the two endpoints of P. Either way, we see that +the restriction P ⋆ of P to A⋆ is a subpath of P. Since P is a snake path in G, also P ⋆ is +a snake path in G⋆. Furthermore, P ⋆ will be of the greatest possible length within G⋆. +Let B⋆ be the concentric subboard of B of size (k − 2) × (k − 2) given by B⋆ = +[1; k −2]2 = B \F0 and let H⋆ be the grid graph on B⋆. By the same reasoning as above, +the restriction ̺⋆ of ̺ to B⋆ is a subpath of ̺. Furthermore, ̺⋆ is a Hamiltonian path in +H⋆, and ̺⋆ and P ⋆ are related in the same way as ̺ and P, in the sense that ̺⋆ visits +the cells of B⋆ in the same order as P ⋆ visits their corresponding 2 × 2 subboards of A⋆. +By the induction hypothesis, either ̺⋆ or an image of it under a symmetry of B⋆ +must be regular relative to B⋆. +We consider five cases for the type of F0. +Case 1. F0 is of type I. +Then (0, 1)—(1, 1) ∈ ̺ and (1, 1) is an endpoint of ̺⋆. +Suppose, for the sake of contradiction, that (1, 1)—(1, 2) ∈ ̺ and ̺ makes a turn at +(1, 1). +If ̺ also makes a turn at (2, 1), then by Lemma 1 we get that (3, 3) ∈ P and (4, 3) ∈ P, +in contradiction with (1, 1)—(2, 1) ̸∈ ̺. +Otherwise, if ̺ does not make a turn at (2, 1), then (2, 1) is an endpoint of ̺⋆. By the +induction hypothesis, it follows that F1 satisfies the reflection, with respect to the line +x = y, of the conditions defining type II. Thus ̺ makes a turn at (3, 1), and by Lemma 1 +we get that (2, 2)—(3, 3)—a′ ⊆ P and (7, 2)—(6, 3)—a′′ ⊆ P for two cells a′ and a′′ with +a′ ∈ Φ(1, 2) and a′′ ∈ Φ(3, 2). Consequently, P cannot visit any cells in Φ(2, 1), another +contradiction. +We conclude that (1, 1)—(1, 2) ̸∈ ̺ and (1, 1)—(2, 1) ∈ ̺. By the induction hypoth- +esis, it follows that ̺⋆ is regular. (As in, it is ̺⋆ itself that is regular, rather than some +image of it under a symmetry of B⋆.) Then ̺ is regular as well, and of the same type +as ̺⋆. +Case 2. F0 is of type II. +Then (0, 1)—(1, 1) ∈ ̺, (0, 2)—(1, 2) ∈ ̺, and (1, 1) and (1, 2) are the two endpoints +of ̺⋆. By the induction hypothesis, it follows that ̺⋆ is of type II(0), and so is ̺ as well. +Case 3. F0 is of type III. +9 + +Then (0, 1)—(1, 1) ∈ ̺, (1, k − 1)—(1, k − 2) ∈ ̺, and (1, 1) and (1, k − 2) are the two +endpoints of ̺⋆. By the induction hypothesis, it follows that either ̺⋆ is of type III(0), +or its reflection with respect to the horizontal axis of symmetry of B⋆ is. (Here, we take +into account the fact that types III(0) and V(0) are related by quarter-turn rotation.) If +the latter, then ̺ makes a turn at both cells (1, 1) and (2, 1), and we get a contradiction +as in Case 1. Thus ̺⋆ is of type III(0), and so is ̺ as well. +Case 4. F0 is of type IV. +Then (0, 1)—(1, 1) ∈ ̺, (k − 1, k − 2)—(k − 2, k − 2) ∈ ̺, and (1, 1) and (k − 2, k − 2) +are the two endpoints of ̺⋆. By the induction hypothesis, it follows that either ̺⋆ is of +type IV(0), or its reflection with respect to the line x = y is. From here, the analysis +continues as in Case 3, and in the end we obtain that both of ̺⋆ and ̺ are of type IV(0). +Case 5. F0 is of type V. +This case is analogous to Case 3, and in it both of ̺⋆ and ̺ are of type V(0). +With this, the induction step is complete. □ +Lemma 3. Let ̺ be a regular Hamiltonian path in H. Then the number of longest +snake paths P in G associated with ̺ is as shown in Table 1. +type of ̺ +k even +k odd +I +8 + 1 +8 +II(s) +6 +— +III(s) +1 +1 +IV(s) +— +6 +V(s) +1 +1 +Table 1 +These paths are pairwise distinct under the symmetries of A, with the following ex- +ceptions: The unique P with ̺ of type III(0) and the unique P with ̺ of type V(0) are +related by quarter-turn rotation; and, when k is odd, two pairs of paths P with ̺ of type +IV(0) are related by central symmetry. +(The entry of Table 1 for k even and ̺ of type I includes one summand for each +regular path ̺ of that type. For the parametrised types, the total count does not depend +on the value of the parameter.) +Proof. First we strengthen the claim as follows: Let b be an arbitrary cell of F0. Then, +for all k with k ≥ 5: +(i) When ̺ goes straight through b, the two cells of Φ(b) in P are the ones on the +boundary of A, in the union of row 0, row n − 1, column 0, and column n − 1 of A; and +(ii) When b is an endpoint of ̺, by the definitions of our types we get that b is one +of (0, 0), (0, 1), (k − 1, 0), (k − 1, k − 1), and (0, k − 1). Then the two cells of Φ(b) in P +are, respectively, the ones in row 0, row 2, column n − 1, row n − 1, and column 0 of A. +For the proof, we work by induction on k. +10 + +(a) +(b) +(c) +(d) +(e) +Figure 7 +Our base cases are k = 3 and k = 4, when the strengthening is irrelevant and the +original claim follows by direct examination of all cases. These are shown in Figures 7 +and 8. For each situation, we depict the cells and edges which are forced to belong to +P in black, and the optional cells and edges of P in red. Figures 7(a)–(e) correspond to +the two paths ̺ of type I and the paths ̺ of types II(0), III(0), and V(0), respectively, +while Figures 8(a)–(d) correspond to the paths ̺ of types I, III(0), IV(0), and V(0), +respectively. +(a) +(b) +(c) +(d) +Figure 8 +For the induction step, suppose that k ≥ 5 and that we have already settled the +question on all smaller boards. Define A⋆, B⋆, G⋆, H⋆, P ⋆, and ̺⋆ as in the proof of +Lemma 2. +We begin with parts (i) and (ii) of the strengthening. +For (i), consider a cell b of F0 such that ̺ goes straight through b, using two edges +on H within F0. Let c be the unique neighbour of b in H which belongs to F1, let b′ and +b′′ be the two cells of Φ(b) which are adjacent in G to a cell in Φ(c), and, conversely, let +c′ and c′′ be the two cells of Φ(c) which are adjacent in G to a cell in Φ(b). +11 + +Then at least one of c′ and c′′ must be in P. Indeed, when ̺⋆ makes a turn at c, this +follows by Lemma 1. Otherwise, when ̺⋆ goes straight through c using two edges of H +within F1, or when c is an endpoint of ̺⋆, it follows by the catalogue in Figures 7 and 8 +when k = 5 or k = 6, and by the induction hypothesis when k ≥ 7. +Since bc ̸∈ ̺ and at least one of c′ and c′′ is in P, we conclude that b′ ̸∈ P and b′′ ̸∈ P. +So the other two cells of Φ(b) must be in P, confirming (i). +For (ii), consider a cell b of F0 such that b is an endpoint of ̺. When b is (0, 0), (0, 1), +(k − 1, 0), (k − 1, k − 1), or (0, k − 1), define c to be (0, 1), (0, 2), (k − 2, 0), (k − 1, k − 2), +or (1, k − 1), respectively, and also define cells b′, b′′, c′, and c′′ relative to b and c as in +our treatment of (i). +By the definitions of our types, bc ̸∈ ̺. Furthermore, with one exception, to be +considered shortly, ̺ makes a turn at c. By Lemma 1, it follows that exactly one of c′ +and c′′ is in P. From here, as in our treatment of (i), bc ̸∈ ̺ implies b′ ̸∈ P and b′′ ̸∈ P, +and so the other two cells of Φ(b) must be in P, as needed. The unique exception is when +F0 is of type II, b = (0, 0), and c = (0, 1). However, then c′ ∈ P and c′′ ∈ P by what we +just proved applied to the endpoint (0, 1) of ̺, and once again (ii) is confirmed. +With this, we have established both parts (i) and (ii) of the strengthening. +For the original claim, observe that the type of ̺ determines the type of F0 uniquely. +We are left to show that the type of F0, and the subpath P ⋆, determine P uniquely. We +only need to look at the cells of P in A \ A⋆, that is, in the union of the subboards Φ(b) +of A with b ∈ F0. +Let, then, b be an arbitrary cell of F0. When ̺ makes a turn at b, the cells of P in +Φ(b) are uniquely determined by Lemma 1. Otherwise, when ̺ goes straight through b, +or when b is an endpoint of ̺, the desired uniqueness follows by parts (i) and (ii) of the +strengthening, respectively. The induction step is complete. □ +Proof of Theorem 1. We get the number of essentially distinct paths by Lemmas 2 and +3, summing over all possible types of ̺ and then subtracting out the duplicates specified +in the statement of Lemma 3. +To convert this into the total number of paths, we analyse symmetries. +When k is even, all of the P’s are asymmetric. +Otherwise, when k is odd, two of the P’s which correspond to the unique ̺ of type +IV(0) are preserved under central symmetry, but not under any other symmetries of A; +and all other P’s corresponding to regular ̺’s are asymmetric. □ +Note that this argument also confirms our remark in the introduction regarding the +number of essentially distinct longest snake paths in G. +The proof of Theorem 1 shows that, roughly speaking, every longest snake path P in +G behaves as follows: Outside of some concentric square subboard of A, it is shaped as +a simple spiral; and then, within that subboard, it is shaped as a double spiral instead. +One endpoint of P lies on the boundary of A, and the other one lies on the boundary +between the two spirals. +12 + +4 +King Graphs on Odd Boards I +In this section, we prove Theorem 2. Let n be an odd positive integer with n = 2k−1, +let A be the standard square board of side n, and let G be the king graph on A. +Proof of the lower bound for Theorem 2. Consider the set of all cells of A of the form +(x, y) with 1 ≤ x ≤ n − 2 and y even. The king graph on it is the disjoint union of k +paths. To obtain the vertex set of a single snake path in G, we add in also the cells (0, 0); +(0, n − 1) if k is even and (n − 1, n − 1) if k is odd; (0, y) in A with y ≡ 3 (mod 4); and +(n − 1, y) in A with y ≡ 1 (mod 4). □ +For example, Figure 9 shows n = 9. +Figure 9 +We obtained the upper bound of Theorem 1 by summing over some subsets of A +such that, for every snake path P, the part of P within each subset must be small. +Our approach to the upper bound of Theorem 2 will follow a similar strategy, albeit +with significant complications. Instead of subsets of A, we sum over subgraphs of G. +Furthermore, our notion of smallness will be somewhat unusual: We consider the total +number of certain cells and edges of P within each subgraph. +We call a cell (x, y) of A even when both of x and y are even, and odd when both of +x and y are odd. We also call an edge of G regular when it is not incident with an odd +cell. +(a) +(b) +Figure 10 +Given an even cell a = (z, z) of A with z ≤ k − 2, we write Ѫ(a) for the subgraph +of G with vertices a + [0; 1]2 whose edges join a, a + (0, 1), and a + (1, 0) pairwise. Thus +Ѫ(a) contains four cells, one of which is odd, and three edges, all of which are regular. +(Figure 10(a).) +For every symmetry π of A, if b = π(a), then we also define Ѫ(b) = π(Ѫ(a)). We +call each subgraph of G of this form a little block. +Given an even cell a = (x, y) of A with x > y and x + y ≤ n − 3, we write Ѫ(a) +for the subgraph of G with vertices a + [−1; 1] × [0; 2] whose edges join a to a + (−1, 0) +and a + (1, 0) as well as a + (0, 1) to all elements of the set a + [−1; 1] × {0, 2}. Thus +13 + +Ѫ(a) contains nine cells, two of which are odd, and eight edges, all of which are regular. +(Figure 10(b).) +For every symmetry π of A, if b = π(a), then we also define Ѫ(b) = π(Ѫ(a)). We +call each subgraph of G of this form a large block. +(a) +(b) +Figure 11 +For example, Figure 11 shows n = 11 and n = 13. (The colouring is only for clarity.) +Observe that every odd cell belongs to at least two blocks and every regular edge +belongs to at least one block. +Consider an arbitrary snake path P in G. For every block Ѫ, let wCell(Ѫ) be the +number of odd cells of P in Ѫ, let wEdge(Ѫ) be the number of regular edges of P in Ѫ, +and let w(Ѫ) = wCell(Ѫ) + wEdge(Ѫ). +Lemma 4. When Ѫ is a little block, w(Ѫ) ≤ 1. Otherwise, when Ѫ is a large block, +w(Ѫ) ≤ 2. +Proof. By direct examination of all cases. □ +Proof of the upper bound for Theorem 2. Let wCell(P) be the number of odd cells in P +and let wEdge(P) be the number of regular edges in P. Then the length of P is bounded +from above by 2wCell(P)+wEdge(P). We sum the inequalities of Lemma 4 over all blocks +Ѫ, and we obtain that the latter expression cannot exceed (n2 − 1)/2. □ +5 +King Graphs on Odd Boards II +In this section, we give a complete description of the longest snake paths in king +graphs on odd square boards. We use the same notations as in Section 4. +We begin with brief overviews of two relevant topics. +Let s be a positive integer and let σ be a permutation of [0; s − 1]. For each i with +0 ≤ i ≤ s−2, draw a semicircle with endpoints (0, σ−1(i)) and (0, σ−1(i+1)) which lies on +the left of the coordinate axis Oy when i is even and on its right otherwise, when i is odd. +14 + +The union of all such semicircles is a curve κ in the plane with endpoints (0, σ−1(0)) and +(0, σ−1(s−1)). When this curve does not intersect itself, σ is a stamp-folding permutation. +For example, Figure 12 shows κ when σ is the permutation 1, 0, 2, 7, 4, 5, 6, 3. +Combinatorially, σ is a stamp-folding permutation if and only if there are no i and j +of the same parity with 0 ≤ i ≤ s − 2 and 0 ≤ j ≤ s − 2 such that exactly one of i and +i + 1 lies between j and j + 1 in σ. +Figure 12 +Figure 13 +The intuition is as follows: Imagine a paper strip of size 1 × s formed out of s stamps +of size 1 × 1 each. We fold this strip along the perforations between stamps so that all +stamps come to lie on top of one another. Then σ is a stamp-folding permutation if and +only if it can be obtained as the permutation of the stamps within the resulting stack. +The points of κ on the coordinate axis Oy correspond to the s stamps, and the semicircles +of κ correspond to the s − 1 creases between stamps. +Stamp-folding permutations have been studied extensively. +Consider a Hamiltonian path in the grid graph Γ of size s × s. The smallest number +of turns that such a path can make is 2s − 2. [3] We proceed to review some properties +of the paths which attain this minimum. [7] +(The author noticed the connection between stamp-folding permutations and fewest- +turn Hamiltonian paths when he obtained Theorems 2 and 4. Independently, [7] was +published before the present work was written.) +Let α be a fewest-turn Hamiltonian path in Γ. We partition α into 2s−1 subpaths at +the 2s − 2 cells where it makes a turn. (Each cell with a turn belongs to two subpaths.) +We call these subpaths the segments of α. Thus in each segment of α either all edges are +horizontal or all edges are vertical, and segments of these two types alternate. +Let us call α mostly-horizontal when it consists of s horizontal segments and s − 1 ver- +tical segments, and mostly-vertical otherwise, when it is the other way around. Suppose, +for concreteness, that α is mostly-horizontal. +Then every row of the board contains exactly one horizontal segment of α. Orient +α arbitrarily, and then number its segments from 0 to s − 1 in the order in which they +occur along α. When we assign to each row of the board the number of its horizontal +segment of α, we obtain a stamp-folding permutation. Conversely, every stamp-folding +permutation corresponds in this way to exactly two oriented mostly-horizontal fewest- +turn Hamiltonian paths. The two are reflections of one another with respect to the vertical +axis of symmetry of the board. +Explicitly, the correspondence is as follows: For each element i of [0; s−1], let ωLeft(i) +be the number of even nonnegative integers j such that i lies between j and j + 1 +in σ, and define ωRight(i) similarly, but with j odd. In one of the two oriented paths +15 + +(a) +(b) +(c) +(d) +Figure 14 +associated with σ, the path’s i-th horizontal segment goes from (s−ωRight(i)−1, σ−1(i)) +to (ωLeft(i), σ−1(i)) for all even i, and it goes in the opposite direction for all odd i. The +second oriented path associated with σ can be obtained by reflection. +For example, Figure 13 shows this for the stamp-folding permutation of Figure 12. +One corollary of this connection is that, for all s with s ≥ 2, the number of fewest-turn +Hamiltonian paths in the grid graph of size s × s is twice the number of stamp-folding +permutations of s elements. +This concludes the two overviews, and we return to our main topic. +Let H be the grid graph of size k × k and let ̺ be a Hamiltonian path in it. (Recall +from Section 4 that k = ⌈n/2⌉.) +We say that the cycle a′a′′b′b′′ in H is free (relative to ̺) when a′a′′ is an edge of ̺; +the other three edges of the cycle are outside of ̺; both of a′ and a′′ are cells that ̺ goes +straight through (so that, if a′ = (c′ + a′′)/2 and a′′ = (a′ + c′′)/2, then c′∼c′′ ⊆ ̺); and +both of b′ and b′′ are cells where ̺ makes a turn. +We proceed to associate ̺ with certain paths in G. Figure 14(a) shows one example +of a Hamiltonian path ̺ in H, and Figures 14(b)–(d) track the series of definitions given +below. +For each edge a′a′′ of ̺, we take the path 2a′—(a′ + a′′)—2a′′ in G. We denote the +concatenation of these paths by ϕ(̺). Intuitively, this operation scales ̺ up by a factor +of two. (Figure 14(b).) +Observe that the length of ϕ(̺) will be twice the length of ̺, namely 2(k2 − 1). +For each turn a′ba′′ in ̺, let us delete the subpath (a′ + b)—2b—(b + a′′) from ϕ(̺), +and let us replace it with the edge (a′ + b)—(b + a′′). We denote the resulting path in G +by ψ(̺). Intuitively, this operation smooths down the sharp turns in ϕ(̺). (Figure 14(c).) +16 + +Let t be the number of turns in ̺. Then the length of ψ(̺) will be 2(k2 − 1) − t. +Finally, for each edge a′a′′ of ̺ which is in a free cycle, either we do nothing or, +optionally, we choose one free cycle a′a′′b′b′′ which includes a′a′′, we set c = (a′ + a′′ + +b′ + b′′)/4, we delete the subpath 2a′—(a′ + a′′)—2a′′ from ψ(̺), and we replace it with +the subpath 2a′—2c—2a′′. We call a path in G which can be obtained in this way a lift +of ̺. Intuitively, this operation introduces some tiny aberrations in ψ(̺). (Figure 14(d).) +In particular, ψ(̺) is also a lift of ̺, namely the one in which we have selected the +do-nothing option everywhere. +Observe that every lift of ̺ is a snake path in G. Furthermore, all lifts of ̺ are of the +same length, namely 2(k2 − 1) − t. +We are ready to state and prove our structure theorem for the longest snake paths +in G. +Theorem 4. Let n be an odd positive integer with n = 2k − 1. Then every lift of a +fewest-turn Hamiltonian path in the grid graph of size k × k is a longest snake path in +the king graph of size n × n. Conversely, every longest snake path in the king graph of +size n × n can be obtained uniquely as a lift of some fewest-turn Hamiltonian path in the +grid graph of size k × k. +Theorem 4 yields the following recipe for the generation of all longest snake path in +G: First, we generate all stamp-folding permutations of k elements. Then we convert each +stamp-folding permutation into four oriented fewest-turn Hamiltonian paths in H, two +mostly-horizontal ones and two mostly-vertical ones. We forget about the orientations, +and discard the duplicates. Finally, for each fewest-turn Hamiltonian path in H, we +identify the corresponding free cycles, and we generate all of its lifts. +(We said in the introduction that each stamp-folding permutation yields two families +of longest snake paths. Strictly speaking, the reality is that each permutation yields four +families, and each quadruple of families is obtained in this way twice, out of two permu- +tations σ′ and σ′′ related by σ′(i) + σ′′(i) = k − 1 for all i. However, it is straightforward +to extract a one-to-two mapping from the two-to-four one.) +Before we go on to the proof of Theorem 4, let us briefly discuss the aberrations. +Proposition 1. Let ̺ be a fewest-turn Hamiltonian path in H and let f be the +number of its corresponding free cycles. Then ̺ yields exactly 2f lifts. Furthermore, f ≤ +max{0, k − 5}, and for all k this bound is attained by some ̺. +Proof. Suppose, for concreteness, that ̺ is mostly-horizontal. Let a′a′′ be an edge of +̺ and let a′a′′b′b′′ be a free cycle which includes a′a′′. +Suppose, for the sake of contradiction, that edge a′a′′ is horizontal. Then, since ̺ +makes a turn at both of b′ and b′′, the edges of ̺ in the row of b′ and b′′ cannot form one +contiguous subpath of ̺. This contradicts the fact that every row of the board contains +exactly one horizontal segment of ̺. +Thus edge a′a′′ must be vertical. +Suppose now, for the sake of contradiction, that edge a′a′′ is in a second free cycle +a′a′′c′c′′. Since ̺ must make a turn at both of c′ and c′′ as well, it follows that the edges +17 + +of ̺ in the row of b′ and c′ cannot form one contiguous subpath of ̺. From here, we get +a contradiction as before. +Thus no edge of ̺ can be in two distinct free cycles. Consequently, in the setting of +Theorem 4, when we construct a lift of ̺, we never have to choose between two free cycles +which include the same edge a′a′′. +We are left to show that f ≤ max{0, k − 5} and the bound is attained. We handle +the cases when k ≤ 6 directly, and from now on we assume that k ≥ 7. +Each free cycle contains two cells where ̺ makes a turn. Conversely, each such cell is +in at most one free cycle. +On the other hand, ̺ makes 2k − 2 turns altogether. However, a turn cell in an +outermost column of B cannot be in a free cycle. The lowermost turn cell and the +topmost turn cell in a non-outermost column of B cannot be in free cycles, either. +When at most one non-outermost column of B contains turns of ̺, since no turn cells +outside of that column can be in free cycles, it follows that 2f ≤ k−2. Then f ≤ k−5 by +virtue of k ≥ 7. Otherwise, when at least two non-outermost columns of B contain turns +of ̺, it follows that at least eight turn cells are not in free cycles, and so 2f ≤ 2k − 10. +The bound is attained, for example, when ̺ corresponds to the stamp-folding permu- +tation 0, 2, 3, 4, . . ., k − 1, 1. (The path in Figure 14(a) is of this form.) □ +In the setting of the proof of Proposition 1, our observation that edge a′a′′ must +be vertical allows us to characterise the free cycles corresponding to ̺ in terms of the +underlying stamp-folding permutation σ, as follows: +Consider the ordered pairs (ε, i) with ε ∈ {−1, 1} and 0 ≤ i ≤ k − 2 such that, in σ, +both of σ(i) and σ(i + 1) lie between σ(i) + ε(−1)σ(i) and σ(i + 1) + ε(−1)σ(i+1), and, +additionally, there is some j with 0 ≤ j ≤ k − 1 such that all four of these lie between j +and j + ε(−1)j. +Each such ordered pair yields a free cycle where edge a′a′′ joins rows i and i+1. When +ε = 1, cells a′ and a′′ are in column ωLeft(σ(i)) − 1 and cells b′ and b′′ are on the right of +them. Otherwise, when ε = −1, cells a′ and a′′ are in column k −ωRight(σ(i)) and cells b′ +and b′′ are on their left. Furthermore, this accounts for all free cycles corresponding to ̺. +Let NStamp(k) be the number of stamp-folding permutations of k elements and let +NKing(n) be the number of longest snake paths in the king graph of size n × n. Proposi- +tion 1 yields some loose bounds on NKing(n) in terms of NStamp(k). +Proposition 2. Let n be an odd positive integer with n = 2k−1. Then 2NStamp(k) < +NKing(n) < 2k−4NStamp(k) for all n with n ≥ 11, and NKing(n) = 2NStamp(k) when +3 ≤ n ≤ 9. Thus, in particular, log NKing(n) = Θ(n). +Proof. The first part follows by Theorem 4 and Proposition 1. (Strictly speaking, we +should also note that not all fewest-turn Hamiltonian paths in H attain the greatest +number of free cycles when n ≥ 11.) The second part is a corollary of the first part and +the well-known asymptotic estimate log NStamp(n) = Θ(n). [2] □ +We continue with the proof of Theorem 4. Let P be a longest snake path in G. +Here is a quick roadmap: Clearly, P must attain exact equality in all inequalities from +the proof of Theorem 2. We examine all blocks of G from this point of view, one by one, +18 + +in a certain order, and we see that P must satisfy certain purely local constraints. These +constraints allow us to conclude that P must be a lift. +The details, however, are somewhat technical. +We define an even cell a of A to be nice (relative to P) when either P visits a; or, +else, P does not visit a but it traverses exactly one edge of G between the four cells in the +set a + {(1, 0), (0, 1), (−1, 0), (0, −1)}. (Note that, for some a, some of these cells might +be outside of A.) +Lemma 5. Suppose that all even cells of A are nice and P does not visit any odd +cells. Then there is a Hamiltonian path ̺ in H such that P = ψ(̺). +Proof. Let us call an edge of G short when the Euclidean distance between its end- +points is unity, and long otherwise, when it is +√ +2. +For each long edge a′a′′ of P, we do the following: Since P does not visit any odd +cells, there is a unique even cell b such that both of a′b and ba′′ are edges of G. We delete +edge a′a′′ from P, and we replace it with these two edges. +Since all even cells of A are nice and P is a snake path, the result will be a path in +G which contains only short edges, which visits all even cells of A, and which does not +visit any odd cells. Denote this path by Q. +Let c be an endpoint of Q. Suppose, for the sake of contradiction, that c is not an +even cell. Since Q does not visit any odd cells, there are exactly two even cells d′ and d′′ +of A adjacent to c in G. Let cd′ be the unique edge of Q incident with c. Observe that c +is also an endpoint of P. But then d′′ cannot be nice because P is a snake path, and we +arrive at a contradiction. +So, to our previous observations about Q, we can add the fact that both of its end- +points are even cells. Consequently, Q is of the form Q = ϕ(ρ) for some Hamiltonian +path ̺ in H, and P = ψ(̺), as needed. □ +We define a rectifiable aberration in P to be a subpath of P of the form b′a′ba′′b′′ +such that a′ and a′′ are two even cells in the same row or column; the cell c = (a′ +a′′)/2 +is a common neighbour of a′ and a′′ in G; the cells b′ and b′′ satisfy a′ = (b′ + c)/2 and +a′′ = (c + b′′)/2; and b ̸= c. (Figure 15.) +b′ a′ +b +c a′′ b′′ +Figure 15 +To rectify a rectifiable aberration, we delete the subpath a′ba′′ from P, and we replace +it with the subpath a′ca′′. The result will be a new snake path in G of the same length +as P. This follows because P being a snake path implies that the only neighbours of c in +G which P visits are a′, b, and a′′. +Lemma 6. Suppose that P does not contain any rectifiable aberrations. Then all even +cells of A are nice and P does not visit any odd cells. +19 + +Proof. Since P is a longest snake path in G, it must attain exact equality in all +inequalities from the proof of Theorem 2. Thus: +(i) A cell of A of the form (z, z) with z odd cannot be in P because it is an odd +cell which is in more than two blocks. Same goes for the images of these cells under the +symmetries of A; +(ii) An edge of G of the form (z, z + 1)—(z + 1, z) with z odd and z ≤ k − 2 cannot +be in P because it is a regular edge which is in more than one block. Same goes for the +images of these edges under the symmetries of A; and +(iii) Every block Ѫ must attain exact equality in Lemma 4, so that w(Ѫ) = 1 when +Ѫ is a little block and w(Ѫ) = 2 otherwise, when Ѫ is a large block. +For all odd positive integers s with 1 ≤ s ≤ n, we write As for the concentric subboard +of A of size s × s given by As = [⌊n/2⌋ − ⌊s/2⌋; ⌊n/2⌋ + ⌊s/2⌋]2. +We will show by induction on s that all even cells of As are nice and all odd cells of +As are outside of P. +Our base case is s = 1. Then A1 consists of a single cell, namely (k −1, k −1). Denote +this cell by o. +When k is even, o is an odd cell, and o ̸∈ P by (i). +Figure 16 +When k is odd, o is an even cell. (Figure 16.) Suppose, for the sake of contradiction, +that k ≥ 3 and o is not nice. Then o ̸∈ P. By (i), o+(1, 1) ̸∈ P, and similarly for the images +of this cell under the symmetries of A. By (ii), (o+(1, 0))—(o+(0, 1)) ̸∈ P, and similarly +for the images of this edge under the symmetries of A. Thus (iii) for Ѫ(o + (0, −2)) +implies o + (−1, −2) ∈ P, o + (1, −2) ∈ P, and either o + (0, −1) ∈ P or o + (0, −2) ∈ P; +and similarly for the images of this block under the symmetries of P. However, the twelve +cells we just concluded must be in P are the vertices of a cycle in G, and we arrive at a +contradiction. +This settles the base case. +For the induction step, let s ≥ 3 and suppose that we have already established the +desired result for As−2. Let a = (x, y) be an arbitrary cell of As \ As−2. By symmetry, +we can assume without loss of generality that ⌊n/2⌋ − ⌊s/2⌋ ≤ x ≤ ⌊n/2⌋ and y = +⌊n/2⌋ − ⌊s/2⌋. +Figures 17 and 18 show some of the cells, edges, and blocks relevant to our reasoning. +Cell a is highlighted in all of them. Note that some blocks which are shown as large in +the figures might be little ones in reality. In all such cases, we emphasise this possibility +in the text. +Suppose, for the sake of contradiction, that a is an even cell but that it is not nice. +20 + +(a) +(b) +Figure 17 +Then a ̸∈ P. Furthermore, the odd cells of Ѫ(a) are not in P, either, by the induction +hypothesis. +Case 1. x = ⌊n/2⌋ − ⌊s/2⌋ and Ѫ(a) is a little block. (Figure 17(a).) +Then (iii) for Ѫ(a) implies (a + (1, 0))—(a + (0, 1)) ∈ P. Since a is not nice, at least +one more edge of G between the four cells in the set a + {(1, 0), (0, 1), (−1, 0), (0, −1)} +must be in P. It cannot be (a + (−1, 0))—(a + (0, −1)) because then P would contain +the vertices of a cycle in G. The other two subcases are symmetric with respect to the +line of unit slope through a, and we assume that (a + (0, −1))—(a + (1, 0)) ∈ P. +It follows that a + (2, 0) ̸∈ P and a + (2, 1) ̸∈ P. Thus no edges of Ѫ(a + (2, 0)) are +in P. Since also the odd cells of this block are outside of P by the induction hypothesis, +we arrive at a contradiction with (iii). (The conclusion holds regardless of whether the +block is a little one or a large one.) +Case 2. ⌊n/2⌋ − ⌊s/2⌋ < x ≤ ⌊n/2⌋ and Ѫ(a) is a large block. (Figure 17(b).) +By (iii) for Ѫ(a) and the induction hypothesis for a + (0, 2), exactly one of the +two edges (a + (1, 0))—(a + (0, 1)) and (a + (0, 1))—(a + (−1, 0)) is in P. The two +subcases are analogous, and we assume that the former edge is in P while the latter +one is not. As in Case 1, at least one more edge of G between the four cells in the set +a + {(1, 0), (0, 1), (−1, 0), (0, −1)} must be in P, and it cannot be (a + (−1, 0))—(a + +(0, −1)) because then P would contain the vertices of a cycle in G. Thus (a + (0, −1))— +(a + (1, 0)) ∈ P. +From here, we arrive at the exact same contradiction as in Case 1. (Once again, +regardless of the type of the block Ѫ(a + (2, 0)).) +We have established that if a is an even cell, then it is nice. +For the second half of the induction step, suppose, for the sake of contradiction, that +a is an odd cell with a ∈ P. By (i), it follows that ⌊n/2⌋ − ⌊s/2⌋ + 2 ≤ x ≤ ⌊n/2⌋. +Case 1. a + (−2, 0) ∈ P. (Figure 18(a).) +By (i), x ≥ ⌊n/2⌋ − ⌊s/2⌋ + 4. Thus Ѫ(a + (−1, 1)) is a large block. +Note that a ∈ P implies (a + (−1, 1))—(a + (0, 1)) ̸∈ P and a + (−2, 0) ∈ P implies +(a+(−1, 1))—(a+(−2, 1)) ̸∈ P. Furthermore, by the induction hypothesis, the odd cells +of Ѫ(a + (−1, 1)) are not in P and cell a + (−1, 3) is nice. Then (iii) for Ѫ(a + (−1, 1)) +implies that exactly one of the two edges (a + (0, 1))—(a + (−1, 2)) and (a + (−1, 2))— +(a+(−2, 1)) is in P. (Here, we take into account the fact that a ∈ P and a+(−2, 0) ∈ P +together imply (a + (−1, 1))—(a + (−1, 2)) ̸∈ P.) The two subcases are analogous, and +we assume that the former edge is in P while the latter one is not. +It follows that a—(a + (0, 1))—(a + (−1, 2)) ⊆ P, a + (1, 1) ̸∈ P, and a + (1, 2) ̸∈ P. +21 + +(a) +Ѫ′ +Ѫ′′ +(b) +Figure 18 +Thus no edges of Ѫ(a + (1, 1)) are in P. Since also the odd cells of this block are outside +of P by the induction hypothesis, we arrive at a contradiction with (iii). (The block +Ѫ(a + (1, 1)) will always be a large one because of our symmetry-breaking assumption +that x ≤ ⌊n/2⌋. However, in the analogous subcase when (a + (0, 1))—(a + (−1, 2)) ̸∈ P +and (a + (−1, 2))—(a + (−2, 1)) ∈ P, the contradiction occurs at block Ѫ(a + (−3, 1)) +which could happen to be a little one.) +Case 2. a + (2, 0) ∈ P. This case is analogous to Case 1. +Case 3. a + (−2, 0) ̸∈ P and a + (2, 0) ̸∈ P. (Figure 18(b).) +Observe that both of Ѫ′ = Ѫ(a + (−1, −1)) and Ѫ′′ = a + (1, −1) are large blocks. +Since a ∈ P, all edges of Ѫ′ other than (a+(−1, −1))—(a+(−2, −1)), (a+(−2, −1))— +(a + (−1, 0)), and (a + (−1, 0))—(a + (−2, 1)) are not in P. Then, in light of a ∈ P and +a + (−2, 0) ̸∈ P, (iii) for Ѫ′ implies that exactly one of these edges is in P. In particular, +exactly one cell b′ out of the pair a+{(−1, −1), (−1, 0)} is in P. Similar reasoning applies +to Ѫ′′, and we define b′′ analogously. +Suppose, for the sake of contradiction, that b′ = a + (−1, 0). Then ab′ ⊆ P implies +a + (0, 1) ̸∈ P and a + (−1, 1) ̸∈ P. Furthermore, the odd cells of Ѫ(a + (−1, 1)) are +outside of P by the induction hypothesis. When this block is a little one, we arrive at +a contradiction with (iii) immediately. Otherwise, when it is a large one, we arrive at a +contradiction with (iii) anyway once we take into account the fact that, by the induction +hypothesis, both of a + (−1, 1) and a + (−1, 3) are nice. +Consequently, b′ = a+(−1, −1). Similarly, b′′ = a+(1, −1). By (iii) for Ѫ′ and Ѫ′′, it +follows that also (a+(−1, −1))—(a+(−2, −1)) ∈ P and (a+(1, −1))—(a+(2, −1)) ∈ P. +However, then (a + (−2, −1))—(a + (−1, −1))—a—(a + (1, −1))—(a + (2, −1)) becomes +a rectifiable aberration in P, and we arrive at a contradiction. (In fact, this is the only +place in the proof where we use the constraint that P does not contain any rectifiable +aberrations.) +We have established that if a is an odd cell, then it cannot be in P. The induction +step is complete. □ +We are ready to tackle Theorem 4. +Proof of Theorem 4. For a start, let us rectify all rectifiable aberrations in P one by +one. The result will be a snake path Q in G of the same length as P and without any +rectifiable aberrations. +By Lemma 6, we get that all even cells are nice relative to Q and Q does not visit +any odd cells. +22 + +By Lemma 5, it follows that there is a Hamiltonian path ̺ in H such that Q = ψ(̺). +Let t be the number of turns in ̺. Then both of P and Q are of length 2(k2 − 1) − t. +Since the greatest length of a snake path in G is (n2−1)/2 by Theorem 2, we conclude +that t = 2k − 2, and so ̺ is in fact a fewest-turn Hamiltonian path in H. +Finally, in order to transform Q back into P, we must restore the rectifiable aberra- +tions which we removed in the beginning. However, it is straightforward to check that the +spots in Q where we can introduce a rectifiable aberration are exactly the ones associated +with the free cycles corresponding to ̺. Therefore, P is a lift of ̺, as needed. +The reasoning in the last few paragraphs shows also the converse: That every lift of +a fewest-turn Hamiltonian path in H is a longest snake path in G. □ +6 +Knight Graphs +In this section, we prove Theorem 3. Let m and n be positive integers, let A be the +standard board of size m × n, and let G be the knight graph on A. +We begin with the upper bound. +One natural approach would be as follows: First we find some finite knight graph H +with pseudosnake density 1/2. Then we sum over all translation copies of H contained +within our board. +The author was not able to implement this idea in its purest form. Below, we present +a slightly more complicated argument which relies on a weighted knight graph instead. +We define a weighted graph Γ to consist of a simple graph H and a weighting function +w which assigns a nonnegative real weight to each vertex of H. When H is finite, we +denote the total weight of all of its vertices by w(Γ), and we define the pseudosnake +density of Γ to be the ratio of the greatest total weight of the vertices in a pseudosnake +of H to w(Γ). +Lemma 7. Suppose that there is a weighted knight graph Γ with pseudosnake den- +sity τ. Then the number of vertices in a pseudosnake of G cannot exceed τmn+O(m+n). +Here and in the proof, the implicit constants in the O-terms depend on Γ. +Proof. Let P be a pseudosnake of G. +Consider all translation copies of Γ that fit within A. There are mn + O(m + n) of +them. For each such copy, the total weight of all cells of P within it cannot exceed τw(Γ). +Conversely, it is true of all but O(m + n) cells a of P that a is sufficiently far away +from the boundary of A for every translation copy of Γ which contains a to fit within +A. For each such cell of P, the sum of its weights over all translation copies of Γ which +contain it will be w(Γ). □ +Proof of the upper bound for Theorem 3. By Lemma 7, if suffices to exhibit one +concrete weighted knight graph with pseudosnake density 1/2. We claim that the one +in Figure 19 works. (The figure shows all cells of the graph together with the weights +assigned to them.) It has 68 cells of total weight 192. +Our claim would likely be extremely difficult to check by hand. However, it is straight- +forward to check with the help of a standard constraint satisfaction solver. The author +23 + +1 +1 +6 +1 +2 +1 +1 +3 +6 +1 4 +3 +4 +7 +7 +4 +1 +4 +1 +1 +4 +7 +4 +4 +6 +1 +1 +2 3 +1 +3 +6 +1 +4 +1 1 +1 +7 +6 +6 +1 +1 +1 +2 +1 +1 +1 +6 +2 +4 +2 +1 +3 +2 3 +2 +1 +1 +2 +1 +4 +1 +3 +4 +4 +1 +6 +3 +Figure 19 +has done this twice, using two different constraint satisfaction frameworks: the Copris +package for the Scala programming language and the OR-Tools package for the Python +programming language. □ +One might wonder how the weighted knight graph in Figure 19 was found. +Figure 20 +Figure 21 +For a start, let STess be the set of all 16 cells of the form ε1(2, 1)+ε2(1, 2)+ε3(−1, 2)+ +ε4(−2, 1), where εi ∈ {0, 1} for all i. Then the knight graph GTess on STess is isomorphic +to the tesseract graph. (Figure 20.) +It is not too difficult to check by hand that the pseudosnake density of the tesseract +graph is 9/16, and that it is attained by an essentially unique pseudosnake. Since 9/16 +is very close to 1/2 from above, we see that GTess works almost, but not quite. +We can attempt to fix this by taking the union of several overlapping copies of GTess. +Since GTess itself just barely manages to push through pseudosnake density 1/2, we can +hope that the interference between its copies will prevent too many of them from doing +the same. +We formalise this notion as follows: Let S′ and S′′ be two nonempty finite sets of +cells. We define their sum, denoted S′ + S′′, to be the multiset of cells which consists of +all cells of the form a′ + a′′ with a′ ∈ S′ and a′′ ∈ S′′, and where the multiplicity of each +cell is the number of ways that it can be expressed in this form. +Then, given a multiset of cells S, we define G(N, S) to be the weighted knight graph +on the cells of S where the weight of each cell is its multiplicity in S. +For each nonempty finite set of cells S, we can think of the weighted knight graph +G(N, S + STess) as constructed out of several overlapping copies of GTess. We experiment +24 + +with different S, and eventually we strike gold with SDia = [0; 3]2 \ {0, 3}2. This is the +Aztec diamond of order two. (Figure 21.) +We go on to the lower bound. +One natural approach would be as follows: First we find a doubly periodic pseudosnake +P∞ in G(N, Z2) with density 1/2, where furthermore every cell is of degree exactly two +and there are no finite cycles. (Here, “doubly periodic” means that there are two linearly +independent two-dimensional vectors u and v with a ∈ P∞ ⇔ a + u ∈ P∞ ⇔ a + v ∈ P∞ +for all cells a of Z2.) +Then, given a board A, we take the restriction P ⋆ of P∞ to A. Because of the struc- +ture of P∞, this restriction will be the disjoint union of several paths. We make some +modifications near the boundary of A, deleting some cells from P ⋆ and replacing them +with new ones, so as to stitch all of these paths together into a single snake path or cycle. +Since P∞ is doubly periodic with density 1/2, originally P ⋆ will contain mn/2+O(m+n) +cells. Finally, keeping our modifications close to the boundary of A ensures that they cost +us O(m + n) of these cells altogether. +Finding a suitable P∞ is straightforward enough. For example, the set of all cells +(x, y) with x mod 4 ∈ {0, 1} works. It consists of vertical strips of width two spaced two +units apart. +However, the second part of our plan runs into significant difficulties. Thus the con- +struction we present below is somewhat more complicated. We divide A into four large re- +gions; we fill up different regions using different pseudosnakes P∞; and we make stitching- +together modifications not only near the boundary of A, but also near the boundaries +between regions. +We continue with the details. +We define a twine to be a board of height two. When s ≥ 2, the knight graph on a +twine with width s is the disjoint union of four paths, two spanning ⌊s/2⌋ cells each and +two spanning ⌈s/2⌉ cells each. +Figure 22 +Figure 23 +Consider a twine E with lower left corner cell a. To tie off E on the left, we add to +it the four cells in the set a + {(−2, 1), (−1, 1), (−1, 3), (0, 3)}. (Figure 22.) Similarly, to +tie off on the right a twine E with lower right corner a, we add to it the reflections of +the four cells above with respect to the vertical line through a. +Consider, now, two twines E and F with lower left corners a and b satisfying a + +(1, 4) = b. To splice together E and F on the left, we add to them the ten cells in +the set a+{(−3, 4), (−3, 5), (−2, 2), (−2, 3), (−2, 6), (−1, 1), (−1, 2), (−1, 6), (0, 5), (0, 6)}. +25 + +(Figure 23.) Similarly, to splice together on the right two twines E and F whose lower +right corners a and b satisfy a + (−1, 4) = b, we add to them the reflections of the ten +cells above with respect to the vertical line through a. +Let k be a positive integer and let I = [x′; x′′] be an integer interval with |I| ≥ 8k −5. +For each i with 0 ≤ i ≤ k−1, if i is even then construct the twine Ei = [x′ +4i; x′′ −4i]× +[4i; 4i+1], and if i is odd then construct the twine Ei = [x′+4i+3; x′′−4i+3]×[4i; 4i+1]. +Tie off E0 on the left; for all i with 0 ≤ i ≤ k − 2, splice together Ei and Ei+1 on the +right if i is even, and on the left if i is odd; and, finally, if k is even then tie off Ek−1 on +the left, and if k is odd then tie it off on the right. +Figure 24 +We denote the resulting set of cells by U(k, I). For example, Figure 24 shows the +knight graph on U(4, [0; 32]). +Lemma 8. Suppose that k ≥ 2 and |I| is odd. Then the knight graph on U(k, I) is a +cycle. +Proof. Denote H = G(N, U(k, I)). It is straightforward to check that all cells of H +are of degree two. We are left to verify that H is connected. +Suppose, for the sake of clarity, that k is odd. The opposite case, when it is even, is +similar. +Let ai and bi be the lower left and lower right corner cells of Ei. Let also a′ +i and a′′ +i +be the two cells of Ei adjacent by side to ai, and define b′ +i and b′′ +i similarly for bi. +It is straightforward to verify that, since |I| is odd: (a) A path in H connects b′ +0 and +b′′ +0 and covers E0 together with the cells which tie it off; (b) For each i with 1 ≤ i ≤ k−2, +two paths in H connect the pairs {a′ +i, a′′ +i } and {b′ +i, b′′ +i } and cover Ei together with two +cells in the adjacent splices; and (c) A path in H connects a′ +k−1 and a′′ +k−1 and covers +Ek−1 together with the cells which tie it off. +Finally, the remaining cells of the splices of U(k, I) form additional knight paths in +H which connect the pairs {b′ +i, b′′ +i } and {b′ +i+1, b′′ +i+1} for all even i with 0 ≤ i ≤ k − 3 as +well as the pairs {a′ +i, a′′ +i } and {a′ +i+1, a′′ +i+1} for all odd i with 1 ≤ i ≤ k − 2. □ +Proof of the lower bound for Theorem 3. We construct a large snake cycle in G, and +for a large snake path we can simply delete one cell from that cycle. +Suppose, without loss of generality, that m ≤ n. Since we are already willing to accept +a tolerance of O(m+n), we can safely assume that m = 8k +14 for some positive integer +k with k ≥ 3, and that n is even. +26 + +Construct UI = U(k, [8; n−12]). Let also VI be the reflection of U(k, [8; m−12]) with +respect to the line x = y. Lastly, let UII and VII be symmetric to UI and VI with respect +to the center of A. +The knight graph on UI ∪VI ∪UII ∪VII is the disjoint union of four cycles. We proceed +to stitch these four cycles together into a single longer cycle. +Figure 25 +Define SDel = {(6, 9), (9, 6)} and SAdd = {(3, 6), (4, 4), (6, 3), (7, 10), (9, 9), (10, 7)}. +Delete the two cells of (4, 4) + SDel from UI and VI, and replace them with the six cells +of (4, 4) + SAdd. This stitches together the cycles of UI and VI. (Figure 25.) +We carry out two more such modifications. For one of them, we reflect the sets SDel +and SAdd with respect to vertical axis of symmetry of the board, we delete the two cells +in the image of SDel from UI and VII, and we replace them with the six cells in the image +of SAdd. This stitches together the cycles of UI and VII. For the other one, we proceed +similarly, except that the reflections are done with respect to horizontal axis of symmetry +of the board. This stitches together the cycles of VI and UII. +Let W be the final set of cells obtained in this way. For example, Figure 26 shows W +in the case when k = 5, m = 54, and n = 72. +Observe that the density of W within A is 1/2 everywhere except within five strips +of bounded width. (Four of these strips surround portions of the interior angle bisectors +at the four corners of A, and the fifth one surrounds a portion of the horizontal axis of +symmetry of A. The corresponding mostly hollow areas are clearly visible in Figure 26.) +Consequently, the number of cells in W is mn/2 + O(m + n). +On the other hand, W is the vertex set of a snake cycle in G. □ +27 + +Figure 26 +7 +Further Work on King Graphs +In this section, we collect some additional results and open problems on king graphs. +We saw that the behaviour of the longest snake paths in G(K, n × n) depends on the +parity of n. For cycles, it appears that there are four classes instead, depending on the +value of n mod 4. The techniques we developed for paths quickly resolve two of them. +Theorem 5. Let n be a positive integer with n ≡ 0 (mod 4) and n ≥ 8. Then the +greatest length of a snake cycle in the king graph of size n × n is n2/2 − 1. Furthermore, +for all such n, there are exactly 48 snake cycles which attain this greatest length. These +cycles are all asymmetric, and so six of them are essentially distinct. +For completeness, there is a unique snake cycle of the greatest length 8 when n = 4. +Each one of the cycles of Theorem 5 is shaped like a double spiral. +It is curious that the number of longest snake cycles freezes in this way, and the cycles +themselves crystallise into a single inflexible structure. A similar phenomenon occurs in +the setting of Theorem 7. +Proof. Let n = 2k and define A, B, G, H, and Φ as in Section 3. Let also C be a +snake cycle in G of length at least n2/2 − 1. +As in Section 3, for each cell b of B at most two cells of Φ(b) are in C. Let us call b +deficient when this bound is not attained. Thus there is at most one deficient cell. +28 + +Observe that, if an edge of C joins one cell of Φ(b′) and one cell of Φ(b′′), with +b′ ̸= b′′, then b′b′′ must still be an edge of H. Otherwise, assuming for concreteness that +b′ + (1, 1) = b′′, by an argument similar to the one in Section 3 we see that at least two +of the four cells in the set b′ + [0; 1]2 must be deficient, a contradiction. +This allows us to define the Hamiltonian cycle ̺ in H relative to C in the same +manner as in Section 3. +Define also the cycle E in H as in the proof of Lemma 2. Since E is not a Hamiltonian +cycle of H when k ≥ 4, at least one edge of E must be outside of ̺. +On the other hand, observe that Lemma 1 now admits a unique exception: When the +turn occurs at a deficient cell. Consequently, every edge of E outside of ̺ must possess +a deficient endpoint. +Define A⋆, B⋆, G⋆, and H⋆ as in Section 3. Let also C⋆ and ̺⋆ be the restrictions of C +and ̺ to A⋆ and B⋆, respectively. We obtain that: (a) There is exactly one deficient cell; +(b) There is exactly one edge β′β′′ of E outside of ̺; (c) ̺⋆ is a Hamiltonian path in H⋆ +whose endpoints are the two neighbours of β′ and β′′ in B⋆; and (d) C⋆ is a snake path +in G⋆ of the greatest possible length whose associated Hamiltonian path in H⋆ is ̺⋆. +However, the proof of Theorem 1 gives us a complete description of all Hamiltonian +paths in H⋆ associated with a longest snake path in G⋆. Since the two endpoints of ̺⋆ +are neighbours in H⋆, we conclude that when n ≥ 12 a symmetry of B⋆ must map ̺⋆ +onto the unique regular Hamiltonian path in H⋆ of type II(0), as defined in Section 3. +The rest is straightforward. □ +The other class we can tackle without too much extra effort is n ≡ 3 (mod 4). First, +though, we need to sort through some preliminaries. +Let s be an even positive integer. Given a permutation σ of [0; s − 1], consider a +closed curve in the plane defined in the same way as the curve κ in Section 5, except +that one additional arc on the right of the coordinate axis Oy joins points (0, σ−1(s−1)) +and (0, σ−1(0)). A curve of this form which does not intersect itself is known as a closed +meander. +Consider a Hamiltonian cycle in the grid graph Γ of size s × s. The smallest number +of turns that such a cycle can make is 2s. [3] Furthermore, the closed meanders with s +arcs and the fewest-turn Hamiltonian cycles in Γ are related in a way analogous to the +relation between stamp-folding permutations and fewest-turn Hamiltonian paths. +Observe, lastly, that our definition of a lift in Section 5 works just as well with cycles +instead of paths. +Theorem 6. Let n be a positive integer with n ≡ 3 (mod 4) and n = 2k − 1. Then +the greatest length of a snake cycle in the king graph of size n × n is (n2 − 1)/2. Every +lift of a fewest-turn Hamiltonian cycle in the grid graph of size k × k is a longest snake +cycle in the king graph of size n × n. Conversely, every longest snake cycle in the king +graph of size n × n can be obtained uniquely as a lift of some fewest-turn Hamiltonian +cycle in the grid graph of size k × k. +This time around, our analysis of paths carries over to cycles nearly verbatim. +29 + +Proof. The upper bound follows by the same argument as Theorem 2. The lower bound +is a corollary of the structure description. Finally, the structure description follows by +the same argument as Theorem 4. □ +With the remaining two classes, the main difficulty is this: In both Sections 3 and +5, we introduce the half-sized square board B of side ⌈n/2⌉ together with its grid graph +H, and our reasoning relies heavily on the properties of the Hamiltonian paths of H. +For Theorems 5 and 6, it is the Hamiltonian cycles of H that matter instead. When +n ≡ 1 (mod 4) or n ≡ 2 (mod 4), however, the side of B is odd and H does not admit +a Hamiltonian cycle. +This throws a substantial wrench in the works. While our upper bounds all go through +as before, the constructions that support the lower bounds do not, and the gap which +opens between the two appears to be difficult to close. +We continue with some tentative remarks. +Let Nr be the set of all positive integers n such that n ≡ r (mod 4). +Fix n⋆ in Nr with n⋆ ≥ 7. For each n in Nr with n ≥ n⋆, construct the subset Dn of +the standard board of size n×n as follows: Take all cells of the form (x, y) with x−y ≥ 1, +x + y ≤ n − 2, y even, and 0 ≤ y ≤ (n − n⋆)/2 together with their images under the +symmetries of the board. Delete the cells (0, 2), (0, 3), and (0, 4), and replace them with +the cells (1, 2) and (1, 4). Finally, for all even i with 2 ≤ i ≤ (n − n⋆)/2, delete the +three cells in the set (i, i) + {(0, 1), (0, 3), (0, 4)}, and replace them with the three cells +in the set (i, i) + {(0, 0), (1, 2), (1, 4)}. Note that Dn is the vertex set of a snake path in +G(K, n × n). +We say that Nr crystallises at n⋆ when, for all n in Nr with n ≥ n⋆ and every longest +snake cycle C in G(K, n×n), there is a symmetry π of the corresponding board such that +the set of all cells of π(C) outside of the concentric subboard of size (n⋆ − 4) × (n⋆ − 4) +coincides with Dn. +Thus, in particular, if Nr crystallises, then there are two constants ℓ and µ such +that the greatest length of a snake cycle in G(K, n × n) is n2/2 − ℓ and the number of +essentially distinct snake cycles which attain this greatest length is µ for all n in Nr with +n ≥ n⋆. Furthermore, each one of these cycles is asymmetric, and so for all such n the +total number of longest snake cycles in G(K, n × n) is 8µ. +The proof of Theorem 5 shows that the class N0 crystallises at n⋆ = 12 with ℓ = 1 +and µ = 6. +The author finds it reasonably plausible that each one of the classes N1 and N2 might +crystallise as well. Experimental data suggests that perhaps the class N1 crystallises at +n⋆ = 13 with ℓ = 5/2 and µ = 69 whereas the class N2 crystallises at n⋆ = 14 with ℓ = 3 +and µ = 72. +The next result might be helpful in the case of the class N1. The two cycles of +Theorem 7 appear to be related to the longest snake cycles of G(K, (2k − 1) × (2k − 1)) +in a way somewhat similar to how the fewest-turn Hamiltonian paths and cycles of grid +graphs are related to the paths and cycles of Theorems 4 and 6. +We define a near-Hamiltonian cycle of a graph G to be a cycle in G which visits all +but one vertices of G. +30 + +Theorem 7. Let k be an odd positive integer with k ≥ 5. Then the smallest number +of turns in a near-Hamiltonian cycle of the grid graph of size k × k is 2k. Furthermore, +for all such k, there are exactly 16 near-Hamiltonian cycles which attain this smallest +number. These cycles are all asymmetric, and so two of them are essentially distinct. +For completeness, there is a unique near-Hamiltonian cycle with the smallest number +of turns, namely four, when k = 3. +Just as in Theorem 5, the cycles of Theorem 7 are shaped like double spirals. +Proof. Let B be the standard board of size k × k, let H be the grid graph on B, and +let C be a near-Hamiltonian cycle in H. Denote the unique cell of B which C omits by o. +Suppose, for the sake of contradiction, that there are a row and a column of B +without an edge of C. Then the cell at their intersection must be o, and it cannot lie +on the boundary of B. Since C visits the neighbours of o in H but the row and column +of o do not contain edges of C, it follows that all edges of the cycle (o + (1, 1))∼(o + +(−1, 1))∼(o + (−1, −1))∼(o + (1, −1))∼(o + (1, 1)) must be in C. However, this cycle is +not near-Hamiltonian when k ≥ 5, a contradiction. +Suppose, for concreteness, that every row contains an edge of C. Define the segments +of C as in Section 5. Since every row contains a horizontal segment of C, and the end- +points of each such segment are turns, we get that C makes at least 2k turns altogether. +Suppose, from now on, that this bound is attained and that every row contains exactly +one horizontal segment of C. Thus, in particular, o cannot lie in the lowermost or topmost +row of B unless it is a corner cell of B. +Suppose, for the sake of contradiction, that the leftmost and rightmost columns of +B contain one vertical segment of C each. Then they cannot contain o unless it is a +corner cell of B. When, say, o = (0, 0), it follows that C must contain all edges of the +cycle (0, 1)—(1, 1)—(1, 0)∼(k − 1, 0)∼(k − 1, k − 1)∼(0, k − 1)∼(0, 1). Otherwise, when +o is not on the boundary of B, it follows that C must contain all edges of the cycle +(0, 0)∼(k − 1, 0)∼(k − 1, k − 1)∼(0, k − 1)∼(0, 0). However, in both cases the cycle in +question is not near-Hamiltonian when k ≥ 5, a contradiction. +Suppose, for concreteness, that the leftmost column of B contains at least two vertical +segments of C. Since the number of vertical segments in C is the same as its number of +horizontal segments, and we have already assumed that the latter number equals k, we +get that some column u of B does not contain any edges of C. +Consequently, C crosses over u every time when it visits this column. Since the total +number of crossings must be even, and u contains an odd number of cells, we obtain that +o must be in u. +It follows that there is exactly one column of B without vertical segments of C. (Since +each such column must contain o.) Thus the leftmost column of B must contain exactly +two vertical segments of C and all other columns except for u must contain exactly one +vertical segment of C each. (Since there are a total of k vertical segments in C.) +Let the two vertical segments of C in the leftmost column of B be (0, 0)∼(0, w) and +(0, w + 1)∼(0, k − 1). Suppose, for concreteness, that 1 ≤ w ≤ ⌊k/2⌋ − 1. +From this point on, we establish the identity w = 1 and the desired result together, by +induction on k. The base case k = 5 is straightforward. For the induction step, suppose +31 + +that k ≥ 7 and that we have already settled the question on all smaller boards. +Let us delete the subpath (1, w)—(0, w)∼(0, 0)∼(k − 1, 0)∼(k − 1, k − 1)∼(0, k − +1)∼(0, w + 1)—(1, w + 1) from C, and let us replace it with the edge (1, w)—(1, w + 1). +The result will be a near-Hamiltonian cycle C⋆ in the grid graph H⋆ on the concentric +subboard B⋆ of B of size (k − 2) × (k − 2) given by B⋆ = [1; k − 2]2. +Since we have deleted at least six turns from C and we have added at most two new +ones in their place, C⋆ can make at most 2k − 4 turns altogether. Thus our induction +hypothesis applies to it, and so in fact C⋆ makes exactly 2k − 4 turns, two of which are +at cells (1, w) and (1, w + 1) where they form the subpath (2, w)—(1, w)—(1, w + 1)— +(2, w + 1). Still by the induction hypothesis, C⋆ omits exactly one edge of the cycle +(1, 1)∼(k − 2, 1)∼(k − 2, k − 2)∼(1, k − 2)∼(1, 1), and that edge is an image of the edge +(1, 2)—(1, 3) under a symmetry of B⋆. +We conclude that w = 1 and exactly two of the fewest-turn near-Hamiltonian cycles +of H⋆ fit as a suitable C⋆. Therefore, there are exactly two essentially distinct fewest- +turn near-Hamiltonian cycles in H, both of them asymmetric, and the induction step is +complete. □ +8 +Further Work on Leaper Graphs +In this section, we collect some additional results and open problems on leaper graphs. +(To be defined shortly.) +For the knight, it would be interesting to see a human-friendly proof of the upper +bound in Theorem 3. Or, if not that, then at least it would be nice to know if there +is an unweighted knight graph with pseudosnake density 1/2 which we could have used +instead of the weighted one. +One natural direction of generalisation for our results in Section 6 is offered by leapers. +Let p and q be nonnegative integers with p ≤ q, not both zero. A (p, q)-leaper is a +fairy chess piece which moves as a generalised knight, leaping p units away along one +coordinate axis and q units away along the other. +Let L be a (p, q)-leaper. The leaper graph of L on a set of cells S, denoted G(L, S), is +defined similarly to the king and knight graphs on S, except that the adjacency condition +becomes {|x′ − x′′|, |y′ − y′′|} = {p, q} instead. +Let d = gcd(p, q). Then the leaper graph of L on the board of size m × n is the +disjoint union of several isomorphic copies of the leaper graphs of a (p/d, q/d)-leaper on +the boards of sizes ⌊m/d⌋ × ⌊n/d⌋, ⌊m/d⌋ × ⌈n/d⌉, ⌈m/d⌉ × ⌊n/d⌋, and ⌈m/d⌉ × ⌈n/d⌉, +each copy scaled up by a factor of d. Thus we can safely assume that d = 1. +For p and q relatively prime, L is known as free when p + q is odd and half-free when +it is even. Briefly, one reason for this distinction is that G(L, Z2) is connected when L is +free but consists of two connected components when L is half-free. +A skew leaper is one for which p and q are positive and distinct. The only non-skew +leapers with relatively prime p and q are the (0, 1)-leaper, known as the wazir, and the +(1, 1)-leaper, known as the fers. Of course, the wazir graph on a board coincides with the +32 + +grid graph on that board. Furthermore, G(Fers, m × n) can also be viewed as the direct +product of two paths with m and n vertices, respectively. +We take a look at the wazir and fers first, and after that we will focus on skew leapers. +Even though Question A for grid graphs on rectangular boards is a very natural thing +to ask, the only earlier reference for it known to the author as of the time of writing is [6], +a puzzle game website where players are invited to construct snake paths in grid graphs +on square boards, with longer paths scoring higher. +An asymptotic estimate is straightforward to obtain. The argument we give for the +upper bound is not new; it is essentially identical to the argument used in [4] to bound +from above the pseudosnake density of G(□, Z2). (We discuss one natural way to define +the pseudosnake density of certain infinite graphs below.) For the lower bound, the general +strategy we outlined in Section 6 goes through without a hitch. Once again, [4] contains +the same pseudosnake in G(□, Z2). +Proposition 3. Let m and n be positive integers. Then both the longest snake path +and the longest snake cycle in the grid graph of size m×n are of length 2mn/3+O(m+n). +Proof. Let A be the standard board of size m × n and let G be the grid graph on A. +For the upper bound, let P be a snake path in G; the argument for cycles is similar. +Let S be the vertex set of P and let T be the complement of S within A. Then nearly +every cell of S is adjacent to two cells of T; each exception is either an endpoint of P or +near the boundary of A, and so there are O(m + n) of them. On the other hand, every +cell of T is adjacent to at most four cells of S. Thus 2|S| ≤ 4|T| + O(m + n), and so +|S| ≤ 2|S ∪ T|/3 + O(m + n) as well. +We move on to the lower bound. Let S∞ be the set of all cells (x, y) in Z2 with x ̸≡ y +(mod 3). Then S∞ induces a pseudosnake P∞ in G(□, Z2). +Suppose without loss of generality that m ≥ 10 and n ≥ 10, let A⋆ be the subboard +of A given by A⋆ = [4; n − 5] × [4; m − 5], and also let P ⋆ be the restriction of P∞ to A⋆. +Then P ⋆ is the disjoint union of several paths, and it is straightforward to add several +cells out of A \ A⋆ to P ⋆ so as to stitch these paths together into a single snake path or +cycle. □ +The fers can be handled similarly. +Proposition 4. Let m and n be positive integers. Then both the longest snake path +and the longest snake cycle in the fers graph of size m×n are of length mn/3+O(m+n). +The proof is analogous to that of Proposition 3, and we omit it. +For wazir and fers graphs on rectangular boards, it might be possible to obtain exact +answers to Question A. By way of experimental data, [6] contains a table listing the +greatest length of a snake path in the grid graph of size n × n for all n with 2 ≤ n ≤ 15. +We continue on to skew leapers. Suppose, for concreteness, that p < q. +It seems highly likely that an exact answer to Question A would be out of reach for +skew leapers on arbitrary rectangular boards, or even on arbitrary square boards. For +this reason, we propose a weakened version of it. +33 + +Question 1. Let L be a skew leaper. What are some interesting lower and upper +bounds for the greatest length of a snake path or cycle of L on a given rectangular board? +Let us pick some of the low-hanging fruit. +When L is half-free, let Free(L) denote the free (p′, q′)-leaper with p′ = (q −p)/2 and +q′ = (p + q)/2. +Suppose, now, that L is a skew free leaper. We will consider this case first, and then +for Proposition 6 we will reduce the half-free case to the free case using the transformation +introduced above. +In all of the following asymptotic estimates, the implicit constants in the O-terms +depend on L. +One construction will be particularly useful to us, and so we introduce special notation +for it: +Let P be a pseudosnake in G(L, m × n) with vertex set S. Consider the union T of +all sets of cells of the form ((n + q)i, (m + q)j) + S, over all integers i and j. Then the +induced subgraph on vertex set T is a pseudosnake in G(L, Z2), as the translation copies +of P which this subgraph consists of are too far away from one another to interact in any +way. We denote this pseudosnake by Υ(m × n, P). +Now let τn be the pseudosnake density of G(L, n × n). +Observe that the sequence {τn}∞ +n=1 converges. Indeed, let n2 ≤ N. Since we can cover +the board of size N×N with ⌈N/n⌉2 subboards of size n×n, we get that τN ≤ τn+O(1/n). +On the other hand, fix a largest pseudosnake P in G(L, n × n). Then the restriction of +Υ(n × n, P) to the board of size N × N is a pseudosnake in G(L, N × N), and so +τN ≥ τn + O(1/n). +We denote τ(L) = limn→∞ τn, and we call this the pseudosnake density of G(L, Z2) +or, for short, the pseudosnake density of L. +Question 2. Let L be a skew free leaper. What is the pseudosnake density of L? Or, +alternatively, what are some interesting lower and upper bounds for it? +Consider the four-dimensional infinite grid graph G(□, Z4). We define its pseudosnake +density η similarly to how we defined τ(L). Observe that η is an absolute constant which +does not depend on L. The pseudosnake densities of infinite grid graphs with arbitrarily +many dimensions have been studied before. [4] +Proposition 5. For all skew free leapers L, the pseudosnake density of L satisfies +1/2 ≤ τ(L) ≤ η. +Proof. For the lower bound, it suffices to exhibit a doubly periodic pseudosnake in +G(L, Z2) with density 1/2. +Since L is free, exactly one of p and q is even. Denote that even value by r. When +r ≡ 2 (mod 4), let S be the set of all cells (x, y) such that ⌊x/2⌋ is even. Otherwise, +when r ≡ 0 (mod 4), let S be the set of all cells (x, y) such that ⌊x/2⌋ + y is even. Then +the induced subgraph of G(L, Z2) on vertex set S works. +For the upper bound, let n be a positive integer, fix a largest pseudosnake P in +G(L, n × n), and let Q = Υ(n × n, P). Then Q is a doubly periodic pseudosnake in +G(L, Z2) with density τn + O(1/n). +34 + +Consider, now, the induced subgraph R of G(□, Z4) whose vertex set consists of all +four-dimensional integer points (x1, x2, x3, x4) such that x1(p, q) + x2(q, p) + x3(−p, q) + +x4(−q, p) is a cell of Q. Then R is a quadruply periodic pseudosnake in G(□, Z4) with +the same density as Q. We let n grow without bound, and the conclusion follows. □ +Had it been the case that η = 1/2, Proposition 5 would have resolved Question 2 +immediately. However, it has been demonstrated that 649/1296 ≤ η ≤ 20/39, with +649/1296 ≈ 0.50077 and 20/39 ≈ 0.51282. [4] Still, Proposition 5 and this result together +imply, for all skew free leapers L, that 1/2 ≤ τ(L) ≤ 20/39. Since the gap between these +two bounds is rather narrow, and the graph G(L, Z2) is, in some intuitive sense, more +crowded than G(□, Z4), it seems plausible that in fact τ(L) = 1/2 for all skew free leapers +L. As we saw in Section 6, this is certainly true of the knight. +The natural connection between pseudosnake density and snake paths and cycles is +as follows: +Proposition 6. Let L be a skew leaper and let m and n be positive integers. When L +is free, the greatest length of a snake path or cycle of L on the board of size m × n does +not exceed τ(L) · mn + O(m + n). Furthermore, when L is half-free, it does not exceed +τ(L)/2 · mn + O(m + n). +Proof. Let P be a snake path or cycle in G(L, m × n) with s cells. +Suppose first that L is free. Since Υ(m × n, P) is a doubly periodic pseudosnake in +G(L, Z2), its density s/(m + q)(n + q) does not exceed τ(L). +Suppose, now, that L is half-free. Then ε = (x + y) mod 2 is constant over all cells +(x, y) of P. Instead of Υ(m × n, P), take its image under the transformation (x, y) → +((x − y + ε)/2, (x + y + ε)/2). This is a pseudosnake in G(Free(L), Z2), and from this +point on the argument continues as before. □ +The author finds it reasonably plausible that the upper bounds of Proposition 6 might +in fact be attained for all skew leapers. As we saw in Section 6, this is indeed the case +for the knight. +Note that our construction for the lower bound in the proof of Proposition 5 yields +a doubly periodic pseudosnake in G(L, Z2) where all cells are of degree exactly two and +there are no finite cycles. Thus for free leapers L with τ(L) = 1/2 and their corresponding +half-free leapers, this construction might play a role in a proof that the upper bounds +of Proposition 6 are attained which follows some variant of the strategy we outlined in +Section 6. +Acknowledgements +The author would like to thank Professor Donald Knuth for introducing him to the +subject of the longest snake paths and cycles in chess piece graphs. +35 + +References +[1] Thomas Dawson, Échecs Féeriques, L’Échiquier, volume 2, issue 2, 1930; issue 3, +1931. +[2] Neil Sloane, My Favorite Integer Sequences, Sequences and Their Applications: Pro- +ceedings of SETA ’98, 1999. Revised at http://neilsloane.com/doc/sg.pdf. +[3] George Jelliss, Knight’s Tour Notes, website, section Wazir Wanderings, 2001. Inter- +net Archive snapshot at https://web.archive.org/web/20020206053321/http:// +www.ktn.freeuk.com/9c.htm. Revised and collected in George Jelliss, Knight’s +Tour Notes, twelve-volume monograph, volume 2 (Walker Tours), 2019, https:// +www.mayhematics.com/p/p.htm. +[4] Martín Matamala, Erich Prisner, and Ivan Rapaport, k-Pseudosnakes in Large +Grids, LATIN 2002: Theoretical Informatics, 2002. +[5] Donald Knuth, The Art of Computer Programming, volume 4, pre-fascicle 5c (section +7.2.2.1, Dancing Links), 2019, https://cs.stanford.edu/~knuth/fasc5c.ps.gz. +Revised and collected in Donald Knuth, The Art of Computer Programming, volume +4B (Combinatorial Algorithms, Part 2), 2022. +[6] David Radcliffe, Build-a-Snake, website, 2020, https://snake.radcliffe.dev/. +Game idea by Christopher Danielson. Table of optimal path lengths by Alain Goupil +and Andrew Howroyd. +[7] Kendall Golder, Minimum Turn Hamiltonian Paths on Rectangular Grids, thesis +presented for the degree of Master of Science, Emporia State University, 2021. +36 + diff --git a/lNAzT4oBgHgl3EQfNvsP/content/tmp_files/load_file.txt b/lNAzT4oBgHgl3EQfNvsP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d56359a0a2aef536bc73e4cf1efdd3ed747a57d --- /dev/null +++ b/lNAzT4oBgHgl3EQfNvsP/content/tmp_files/load_file.txt @@ -0,0 +1,1082 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf,len=1081 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='01152v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='CO] 3 Jan 2023 Snake Paths in King and Knight Graphs Nikolai Beluhov Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' A snake path in a graph G is a path in G which is also an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For all n, we find the greatest length of a snake path in the n × n king graph and we give a complete description of the paths which attain this greatest length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The even and odd cases behave very differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We also estimate the greatest length of a snake path or cycle in the m × n knight graph, for all m and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 1 Introduction Let G be a simple graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' A snake path in G is a path in G which is also an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Equivalently, a path P in G is a snake path when, for all vertices u and v of P, we have that uv is an edge of G if and only if it is an edge of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Intuitively, a snake path never comes into contact with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Snake paths are also known as induced paths and chordless paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' A snake cycle is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Just like paths, snake cycles are alternatively called induced cycles and chordless cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Our focus will be mostly on snake paths, though we will touch upon snake cycles, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Given a graph G, some of the most natural questions we can ask about its snake paths are as follows: Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' What is the greatest length of a snake path in G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Note that we measure the length of a path by the number of edges that it traverses, rather than the number of vertices that it visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The answer of Question A coincides with the greatest diameter of an induced sub- graph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The greatest length of a snake path in G is also known as the induced detour number of G, and the greatest length of a snake cycle in G as the induced circumference of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In the special case when G is a hypercube graph, Question A is known as the snake- in-the-box problem, and its analogue for cycles as the coil-in-the-box problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Both of these problems have been studied extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Question B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' What is the structure of the longest snake paths in G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Question C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' How many longest snake paths are there in G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Note that we formalise paths as subgraphs, rather than as sequences of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Specifically, to us a “path” is a tree subgraph where all vertices are of degree at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The distinction between the two formalisations does not matter in most situations, but it does matter for enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For example, to us abc and cba are the same path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 1 We study Questions A–C for certain graphs G associated with chess pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Given a chess piece F and a board A, it is natural to consider the graph whose vertices are the cells of A and whose edges correspond to all possible moves of F on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We proceed to formalise this notion for the king and the knight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Other chess pieces can be handled similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' To us, a cell is an ordered pair of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Or, equivalently, an integer point in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let m and n be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' A board A of size m×n, with m rows and n columns, is a set of cells of the form I × J, where I and J are integer intervals with |I| = n and |J| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The standard board of size m × n has I = [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' n − 1] and J = [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' m − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since all boards of the same size are translation copies of one another, sometimes we refer to “the” board of a certain size, meaning the standard board of that size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Given a set of cells S, we define the king graph on S, denoted G(K, S), to be the graph on vertex set S where two distinct cells a′ = (x′, y′) and a′′ = (x′′, y′′) are joined by an edge if and only if |x′ − x′′| ≤ 1 and |y′ − y′′| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since all king graphs on boards of the same size are isomorphic, sometimes we refer to “the” king graph of a certain size, meaning the king graph on the standard board of that size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For convenience, we also use the notation G(K, m × n) for the king graph of size m × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Of course, G(K, m × n) can also be viewed as the strong product of two paths with m and n vertices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The knight graph on S, denoted G(N, S), is defined similarly, except that the adja- cency condition becomes {|x′ − x′′|, |y′ − y′′|} = {1, 2} instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Dawson, in problem 187 of [1], considers G(N, 8 × 8) and presents a snake path of length 31 as well as a snake cycle of length 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Knuth, in exercise 172 of [5], discusses the longest snake paths and cycles of various chess piece graphs in the context of algorithmic generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In particular, he determines that there are 16 essentially distinct snake paths of the greatest length 31 and 6 essentially distinct snake cycles of the greatest length 31 in G(K, 8 × 8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' as well as an essentially unique snake path of the greatest length 33 and 4 essentially distinct snake cycles of the greatest length 32 in G(N, 8 × 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Thus Dawson’s path was not optimal, but his cycle was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Our main results are as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let n be an even positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the greatest length of a snake path in the king graph of size n × n is n2/2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, when n ≥ 6, there are exactly 16n snake paths which attain this greatest length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This count includes rotations and reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In Section 3, we will see that with n ≥ 6 the number of essentially distinct longest snake paths is 2n when n/2 is even and 2n + 1 when n/2 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Over the course of the proof of Theorem 1, we will also give a complete description of these paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Roughly speaking, each one of them is shaped like a spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' That the total number of paths is given by such a nice formula is most likely only a happy coincidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Because of the overall structure of the proof, we should expect to see a linear function of n within each parity of n/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' however, there is no obvious reason a priori to expect these two functions to coincide, or their constant terms to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 2 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let n be an odd positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the greatest length of a snake path in the king graph of size n × n is (n2 − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Remarkably, the odd and even cases behave very differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Despite the surface simi- larity between the upper bounds of Theorems 1 and 2, the former bound is straightforward while the latter one poses considerable difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In Section 5, we will add to Theorem 2 a complete description of the paths which attain the greatest length, stated in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, the description is somewhat complicated, and relies on a long series of preceding definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus we do not reproduce it in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The gist is that each stamp-folding permutation of ⌈n/2⌉ elements yields two families of longest snake paths which share the same overall shape but differ from one another by some tiny aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Theorem 4 does not imply an exact answer to Question C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In fact, because of the connection to the stamp-folding problem, it seems unlikely that such an answer would be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Still, the theorem does yield some loose bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In particular, we will see (Proposition 2) that the logarithm of the number of longest snake paths grows as Θ(n), in stark contrast to the even case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Theorems 1, 2, and 4 together completely resolve the questions of the greatest length of a snake path and the structure of the longest snake paths in king graphs on square boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let m and n be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then both the longest snake path and the longest snake cycle in the knight graph of size m×n are of length mn/2+O(m+n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Note that we do not specify the sign of the error term: Since O-notation only bounds the absolute value of a function, the classes mn/2 + O(m + n) and mn/2 − O(m + n) consist of the same functions of m and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Same goes for the estimates in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Compared to the treatment of Question A in Theorems 1 and 2, with Theorem 3 we do not attempt to obtain an exact answer, and are content instead with an asymptotic estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' On the bright side, this asymptotic estimate applies to cycles as well as paths, and it is valid on all rectangular boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' It is worth noting that one step in our proof of Theorem 3 relies on computer help, and likely cannot be verified manually by a human mathematician.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Our proofs of Theorems 1, 2, and 4 are all human-friendly, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') The author obtained Theorems 1–7 in 2018 after being introduced to the subject by Knuth, in connection with the aforementioned exercise 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Subsequently, Theorems 1, 3, and 5 were cited in a remark following the exercise’s solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Strictly speaking, at that time the author derived Theorem 1 in a form referring to the number of essentially distinct paths rather than the total number of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This is also how it was stated in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Then, in 2020, the author proposed Theorem 2, appropriately rephrased, as a mathe- matical olympiad problem for the Cyberspace Mathematical Competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' It was featured as problem 4 on day 1 of the contest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (The CMC was a one-off event intended to approx- imate the International Mathematical Olympiad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Perhaps the problem’s difficulty was not an ideal match for the contest’s format;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' out of 553 participants from 75 countries, only two made substantial progress on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') 3 Theorems 1–4 offer an interesting illustration of how the nature of Questions A–C can change when we vary the underlying graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In the setting of Theorem 1, Question A is straightforward while Questions B and C are manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For Theorems 2 and 4, both Questions A and B become significantly more complicated, while a closed-form answer to Question C is likely out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Finally, in the setting of Theorem 3, already with Question A it seems that an exact answer would be unfeasible, and even for our asymptotic estimate we find ourselves in need of machine help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 2 Preliminaries Before we continue, let us briefly list some useful notations and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Given two cells a′ = (x′, y′) and a′′ = (x′′, y′′), we write a′+a′′ for the cell (x′+x′′, y′+ y′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Given a cell a and a set of cells S, we write a + S for the set of cells {a + b | b ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' A symmetry of a board A is the restriction to A of an isometry of the plane which preserves A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Two objects defined with reference to A, such as two sets of cells on A or two graphs on A, are essentially distinct (relative to A) when they are distinct under the symmetries of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The grid graph on a set of cells S, denoted G(□, S), is defined similarly to the king and knight graphs on S, except that the adjacency condition becomes {|x′ − x′′|, |y′ − y′′|} = {0, 1} instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Or, equivalently, cells a′ and a′′ must be at unit Euclidean distance from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Of course, G(□, m × n) can also be viewed as the Cartesian product of two paths with m and n vertices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the edge joining two vertices a and b in a graph G, we write either ab or a—b, whichever one reads better in the situation at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We introduce shorthand notation for certain paths within grid and king graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Given two cells a and b of a grid or king graph G such that a and b are in the same row or column, we write a∼b for the path in G connecting a and b whose remaining cells are the ones between a and b in the corresponding row or column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For example, if a = (x′, y), b = (x′′, y), and x′ ≤ x′′, then a∼b = (x′, y)—(x′ + 1, y)—(x′ + 2, y)—· · · —(x′′, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let P be a path in some graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Given a vertex a of G, we write a ∈ P for “P visits a”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' given an edge e of G, we write e ∈ P for “P traverses e”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and, given a path Q in G, we write Q ⊆ P for “Q is a subpath of P”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We will not use these abbreviations too often, but the proofs of Lemmas 2 and 6 would be cumbersome to state without them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, now, that P is a snake path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When ab′b′′ is a three-cycle in G, we have that a ∈ P implies b′b′′ ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Similarly, when b, c′, and c′′ are three distinct neighbours of a in G, we have that c′ac′′ ⊆ P implies b ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We will use these simple observations repeatedly throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' A subset S of the vertices of G is k-independent if, in the induced subgraph of G on vertex set S, every vertex is of degree at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When k = 2, the induced subgraph itself is a pseudosnake of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Clearly, the number of vertices in the longest snake path or cycle of G cannot exceed the number of vertices in its largest pseudosnake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When G is finite, we define its pseudosnake density to be the ratio of the greatest number of vertices in a pseudosnake of G to the total number of vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 4 3 King Graphs on Even Boards In this section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let n be an even positive integer with n = 2k, let A be the standard board of size n × n, and let G be the king graph on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For completeness, with n = 2 there are 6 longest snake paths, of which 2 are essentially distinct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and with n = 4 there are 28 longest snake paths, of which 4 are essentially distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' From now on, let n ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The argument we give for the upper bound of Theorem 1 is not new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' the special case n = 8 is in [5], and the general case does not pose any additional difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof of the optimisation part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the upper bound, partition A into k2 subboards of size 2×2 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since a snake path in G can visit at most two cells within each subboard, its length cannot exceed 2k2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Figure 1 For the lower bound, let Pi be the path (i, i)∼(n−i−2, i)—(n−i−1, i+1)∼(n−i− 1, n−i−2)—(n−i−2, n−i−1)∼(i+1, n−i−1)—(i, n−i−2)∼(i, i+3)—(i+1, i+2)— (i + 2, i + 2) for all even i with 0 ≤ i ≤ k − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When k is even and i = k − 2, we define Pk−2 in the same way, except that we stop at cell (i, n − i − 2) = (k − 2, k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and, when k is odd and i = k − 1, we also define Pk−1 to be the path (k − 1, k − 1)—(k, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the concatenation of these paths is a snake path in G of length n2/2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ For example, Figure 1 shows n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The enumeration part of Theorem 1 will be somewhat more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let B be the standard board of size k × k and let H be the grid graph on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For each cell b of B, let Φ(b) denote the subboard 2b + [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 1]2 of A, of size 2 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (For convenience, if b = (x, y), we also write simply Φ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') These subboards, taken over all cells b of B, form a partitioning of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let P be a longest snake path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The proof of the optimisation part of Theorem 1 shows that P visits exactly two cells within each subboard of A of the form Φ(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose that an edge of P joins one cell of Φ(b′) and one cell of Φ(b′′), with b′ ̸= b′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We claim that b′ and b′′ are then neighbours in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Indeed, if not, then b′ and b′′ must be diagonally adjacent in H, without loss of generality with b′ + (1, 1) = b′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' So P contains a subpath of the form c′—(2b′ + (1, 1))— 2b′′—c′′, where c′ is in Φ(b′) and c′′ is in Φ(b′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since P is a snake path, it follows that P cannot visit any cells in the set 2(b′ + (1, 0)) + {(0, 0), (0, 1), (1, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consequently, P visits at most one cell of Φ(b′ + (1, 0)), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') For each edge of P joining one cell of Φ(b′) and one cell of Φ(b′′), with b′ ̸= b′′, take the edge b′b′′ of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since P visits Φ(b) for all b, these edges form a Hamiltonian path 5 × × × Φ(b′) Φ(b′′) Figure 2 in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Denote this path by ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' From this point on, our plan for the proof will be as follows: First we obtain a complete description of the structure of ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We do this by means of a series of mostly local considerations, starting on the boundary of H and then working our way in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Once we are done, we determine what paths P in the original graph G are associated with each path ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We continue with the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We define the i-th frame of B, denoted Fi, to be the subset Fi = [i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' k − i − 1]2 \\ [i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' k − i − 2]2 of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus F0, F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=', F⌈k/2⌉−1 form a partitioning of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We denote the four corner cells of Fi by ai = (i, i), bi = (k − i − 1, i), ci = (k − i − 1, k − i − 1), and di = (i, k − i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (a) (b) (c) (d) (e) Figure 3 We say that Fi is of type I, II, III, IV, or V (relative to ̺) when ̺ contains the following subpaths: For type I, ai∼bi∼ci∼di∼(ai + (0, 1))—(ai + (1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') For type II, ai∼bi∼ci∼di∼(ai + (0, 2))—(ai + (1, 2)) and (ai + (0, 1))—(ai + (1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') For type III, ai∼bi∼ci∼(di +(1, 0))—(di +(1, −1)) and di∼(ai +(0, 1))—(ai +(1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') For type IV, ai∼bi∼(ci+(0, −1))—(ci+(−1, −1)) and ci∼di∼(ai+(0, 1))—(ai+(1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') For type V, ai∼(bi +(−1, 0))—(bi +(−1, 1)) and bi∼ci∼di∼(ai +(0, 1))—(ai +(1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 3(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') We say that ̺ itself is of type I when all of F0, F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=', F⌈k/2⌉−2 are of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, let T be one of the symbols II, III, IV, and V, and let s be a nonnegative integer with 0 ≤ s ≤ ⌈k/2⌉ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We say that ̺ is of type T(s) when all of F0, F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=', Fs−1 are of type I and all of Fs, Fs+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=', F⌈k/2⌉−2 are of type T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When ̺ is of one of the 4⌈k/2⌉ − 3 types we have just listed, we say that it is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 6 Figure 4 For example, Figure 4 shows the unique ̺ of type III(1) when k = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Observe that, when k is even, ̺ cannot be of type IV(s), for any s, as F⌈k/2⌉−2 being of type IV prevents ̺ from visiting all cells of F⌈k/2⌉−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Similarly, when k is odd, ̺ cannot be of type II(s), for any s, as the frame F⌈k/2⌉−2 is too small to be of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In all other cases, if k and the type of ̺ are fixed, there is a unique ̺ of that type, with one exception: When k is even, there are two paths ̺ of type I, differing by just one edge within the innermost frame F⌈k/2⌉−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose that ̺ makes a turn at cell b of B, for concreteness by means of (b+(0, 1))—b—(b+(1, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the two cells of Φ(b) in P are 2b+(0, 1) and 2b+(1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Of course, similar claims hold for the other three possible turns at b as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for the sake of contradiction, that 2b+(1, 1) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since (b+(0, 1))— b—(b + (1, 0)) ⊆ ̺, we get that there are two cells a′ and a′′ with a′ ∈ Φ(b + (0, 1)), a′′ ∈ Φ(b + (1, 0)), and a′—(2b + (1, 1))—a′′ ⊆ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, then P cannot visit any cells of Φ(b) other than 2b + (1, 1), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Φ(b) Figure 5 Thus 2b+(1, 1) ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since (b+(0, 1))—b—(b+(1, 0)) ⊆ ̺, it follows that 2b+(0, 1) ∈ P and 2b + (1, 0) ∈ P, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' A symmetry of B maps ̺ onto a regular Hamiltonian path in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' First we consider the outermost frame F0 of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Observe that all edges of H within F0 form a cycle E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for the sake of contradiction, that (0, y)—(0, y + 1) ̸∈ ̺ for some y with 2 ≤ y ≤ k − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then P must miss both cells in at least one of the two pairs {(0, 2y + 1), (1, 2y + 1)} and {(0, 2y + 2), (1, 2y + 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for concreteness, that this is true of the former pair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' the other case is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Since P visits two cells in Φ(0, y), we get that (0, 2y) ∈ P and (1, 2y) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' So (0, 2y − 1) ̸∈ P and (1, 2y − 1) ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Iterating this sequence of observations, we see that all of (0, 2y − 2), (1, 2y − 2), (0, 2y − 4), and (1, 2y − 4) are in P while all of (0, 2y − 3), 7 × × × × × × × × Φ(0, y − 2) Φ(0, y − 1) Φ(0, y) Φ(0, y + 1) Figure 6 (1, 2y − 3), (0, 2y − 5), and (1, 2y − 5) are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (When y = 2, the lattermost couple of cells will lie outside of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Of course, then they will be outside of P, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Thus all three of (0, 2y), (0, 2y − 2), and (0, 2y − 4) must be endpoints of P, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consequently, (0, y)—(0, y + 1) ∈ ̺ for all y with 2 ≤ y ≤ k − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By symmetry, it follows that ̺ contains all edges of E except for, possibly, (0, 0)—(0, 1), (0, 1)—(0, 2), and their images under the symmetries of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' On the other hand, at least one edge of E must be outside of ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' If (0, 0)—(0, 1) ̸∈ ̺, then (0, 0) is an endpoint of ̺ and Φ(0, 0) contains an endpoint of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' If (0, 1)—(0, 2) ̸∈ ̺, then P must miss both cells in at least one of the two pairs {(0, 3), (1, 3)} and {(0, 4), (1, 4)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' If the latter, then with k ≥ 5 we arrive at a contradic- tion as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' If the former, then the same reasoning as before shows that Φ(0, 0) and Φ(0, 1) must each contain an endpoint of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When k = 4, the same conclusion holds up to reflection with respect to the horizontal axis of symmetry of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Of course, these observations apply also to all images of the edges (0, 0)—(0, 1) and (0, 1)—(0, 2) under the symmetries of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since there are only two endpoints of P, it follows that ̺ cannot omit too many edges of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We are left to consider the following cases, up to the symmetries of B: Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The only edge of E outside of ̺ is (0, 0)—(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then F0 is of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The only edge of E outside of ̺ is (0, 1)—(0, 2), and k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then F0 is of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The only edges of E outside of ̺ are (0, 0)—(0, 1) and (0, k − 1)—(1, k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then F0 is of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The only edges of E outside of ̺ are (0, 0)—(0, 1) and (k − 1, k − 2)— (k − 1, k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then F0 is of type IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The only edges of E outside of ̺ are (0, 0)—(0, 1) and (k − 2, 0)—(k − 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then F0 is of type V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Cases 6–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The only edges of E outside of ̺ are (0, 0)—(0, 1) and one of (0, 0)—(1, 0), (0, k −2)—(0, k −1), (k −2, k −1)—(k −1, k −1), and (k −1, 0)—(k −1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The first one of these cases cannot occur because then cell (0, 0) becomes isolated in ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The other three 8 cannot occur because, in each one of them, part of the edges of E in ̺ form a subpath of ̺ which contains both endpoints of ̺ but does not coincide with ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' With this, we have established that F0 is of one of our types relative to ̺, up to the symmetries of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We finish the proof by induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Our base cases are k = 3 and k = 4, when there is nothing left to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the induction step, suppose that k ≥ 5 and that we have already settled the question on all smaller boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let A⋆ be the concentric subboard of A of size (n−4)×(n−4) given by A⋆ = [2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' n−3]2 and let G⋆ be the king graph on A⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When F0 is of type I, the cells of P outside of A⋆ form a subpath of P containing one endpoint of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When F0 is of one of the remaining four types, the cells of P outside of A⋆ form two subpaths of P containing the two endpoints of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Either way, we see that the restriction P ⋆ of P to A⋆ is a subpath of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since P is a snake path in G, also P ⋆ is a snake path in G⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, P ⋆ will be of the greatest possible length within G⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let B⋆ be the concentric subboard of B of size (k − 2) × (k − 2) given by B⋆ = [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' k −2]2 = B \\F0 and let H⋆ be the grid graph on B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By the same reasoning as above, the restriction ̺⋆ of ̺ to B⋆ is a subpath of ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, ̺⋆ is a Hamiltonian path in H⋆, and ̺⋆ and P ⋆ are related in the same way as ̺ and P, in the sense that ̺⋆ visits the cells of B⋆ in the same order as P ⋆ visits their corresponding 2 × 2 subboards of A⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By the induction hypothesis, either ̺⋆ or an image of it under a symmetry of B⋆ must be regular relative to B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We consider five cases for the type of F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' F0 is of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then (0, 1)—(1, 1) ∈ ̺ and (1, 1) is an endpoint of ̺⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for the sake of contradiction, that (1, 1)—(1, 2) ∈ ̺ and ̺ makes a turn at (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' If ̺ also makes a turn at (2, 1), then by Lemma 1 we get that (3, 3) ∈ P and (4, 3) ∈ P, in contradiction with (1, 1)—(2, 1) ̸∈ ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Otherwise, if ̺ does not make a turn at (2, 1), then (2, 1) is an endpoint of ̺⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By the induction hypothesis, it follows that F1 satisfies the reflection, with respect to the line x = y, of the conditions defining type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus ̺ makes a turn at (3, 1), and by Lemma 1 we get that (2, 2)—(3, 3)—a′ ⊆ P and (7, 2)—(6, 3)—a′′ ⊆ P for two cells a′ and a′′ with a′ ∈ Φ(1, 2) and a′′ ∈ Φ(3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consequently, P cannot visit any cells in Φ(2, 1), another contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We conclude that (1, 1)—(1, 2) ̸∈ ̺ and (1, 1)—(2, 1) ∈ ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By the induction hypoth- esis, it follows that ̺⋆ is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (As in, it is ̺⋆ itself that is regular, rather than some image of it under a symmetry of B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Then ̺ is regular as well, and of the same type as ̺⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' F0 is of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then (0, 1)—(1, 1) ∈ ̺, (0, 2)—(1, 2) ∈ ̺, and (1, 1) and (1, 2) are the two endpoints of ̺⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By the induction hypothesis, it follows that ̺⋆ is of type II(0), and so is ̺ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' F0 is of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 9 Then (0, 1)—(1, 1) ∈ ̺, (1, k − 1)—(1, k − 2) ∈ ̺, and (1, 1) and (1, k − 2) are the two endpoints of ̺⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By the induction hypothesis, it follows that either ̺⋆ is of type III(0), or its reflection with respect to the horizontal axis of symmetry of B⋆ is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Here, we take into account the fact that types III(0) and V(0) are related by quarter-turn rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') If the latter, then ̺ makes a turn at both cells (1, 1) and (2, 1), and we get a contradiction as in Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus ̺⋆ is of type III(0), and so is ̺ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' F0 is of type IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then (0, 1)—(1, 1) ∈ ̺, (k − 1, k − 2)—(k − 2, k − 2) ∈ ̺, and (1, 1) and (k − 2, k − 2) are the two endpoints of ̺⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By the induction hypothesis, it follows that either ̺⋆ is of type IV(0), or its reflection with respect to the line x = y is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' From here, the analysis continues as in Case 3, and in the end we obtain that both of ̺⋆ and ̺ are of type IV(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' F0 is of type V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This case is analogous to Case 3, and in it both of ̺⋆ and ̺ are of type V(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' With this, the induction step is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let ̺ be a regular Hamiltonian path in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the number of longest snake paths P in G associated with ̺ is as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' type of ̺ k even k odd I 8 + 1 8 II(s) 6 — III(s) 1 1 IV(s) — 6 V(s) 1 1 Table 1 These paths are pairwise distinct under the symmetries of A, with the following ex- ceptions: The unique P with ̺ of type III(0) and the unique P with ̺ of type V(0) are related by quarter-turn rotation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and, when k is odd, two pairs of paths P with ̺ of type IV(0) are related by central symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (The entry of Table 1 for k even and ̺ of type I includes one summand for each regular path ̺ of that type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the parametrised types, the total count does not depend on the value of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' First we strengthen the claim as follows: Let b be an arbitrary cell of F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then, for all k with k ≥ 5: (i) When ̺ goes straight through b, the two cells of Φ(b) in P are the ones on the boundary of A, in the union of row 0, row n − 1, column 0, and column n − 1 of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and (ii) When b is an endpoint of ̺, by the definitions of our types we get that b is one of (0, 0), (0, 1), (k − 1, 0), (k − 1, k − 1), and (0, k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the two cells of Φ(b) in P are, respectively, the ones in row 0, row 2, column n − 1, row n − 1, and column 0 of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the proof, we work by induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 10 (a) (b) (c) (d) (e) Figure 7 Our base cases are k = 3 and k = 4, when the strengthening is irrelevant and the original claim follows by direct examination of all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' These are shown in Figures 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For each situation, we depict the cells and edges which are forced to belong to P in black, and the optional cells and edges of P in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Figures 7(a)–(e) correspond to the two paths ̺ of type I and the paths ̺ of types II(0), III(0), and V(0), respectively, while Figures 8(a)–(d) correspond to the paths ̺ of types I, III(0), IV(0), and V(0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 8 For the induction step, suppose that k ≥ 5 and that we have already settled the question on all smaller boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Define A⋆, B⋆, G⋆, H⋆, P ⋆, and ̺⋆ as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We begin with parts (i) and (ii) of the strengthening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For (i), consider a cell b of F0 such that ̺ goes straight through b, using two edges on H within F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let c be the unique neighbour of b in H which belongs to F1, let b′ and b′′ be the two cells of Φ(b) which are adjacent in G to a cell in Φ(c), and, conversely, let c′ and c′′ be the two cells of Φ(c) which are adjacent in G to a cell in Φ(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 11 Then at least one of c′ and c′′ must be in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Indeed, when ̺⋆ makes a turn at c, this follows by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Otherwise, when ̺⋆ goes straight through c using two edges of H within F1, or when c is an endpoint of ̺⋆, it follows by the catalogue in Figures 7 and 8 when k = 5 or k = 6, and by the induction hypothesis when k ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since bc ̸∈ ̺ and at least one of c′ and c′′ is in P, we conclude that b′ ̸∈ P and b′′ ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' So the other two cells of Φ(b) must be in P, confirming (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For (ii), consider a cell b of F0 such that b is an endpoint of ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When b is (0, 0), (0, 1), (k − 1, 0), (k − 1, k − 1), or (0, k − 1), define c to be (0, 1), (0, 2), (k − 2, 0), (k − 1, k − 2), or (1, k − 1), respectively, and also define cells b′, b′′, c′, and c′′ relative to b and c as in our treatment of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By the definitions of our types, bc ̸∈ ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, with one exception, to be considered shortly, ̺ makes a turn at c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By Lemma 1, it follows that exactly one of c′ and c′′ is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' From here, as in our treatment of (i), bc ̸∈ ̺ implies b′ ̸∈ P and b′′ ̸∈ P, and so the other two cells of Φ(b) must be in P, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The unique exception is when F0 is of type II, b = (0, 0), and c = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, then c′ ∈ P and c′′ ∈ P by what we just proved applied to the endpoint (0, 1) of ̺, and once again (ii) is confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' With this, we have established both parts (i) and (ii) of the strengthening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the original claim, observe that the type of ̺ determines the type of F0 uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We are left to show that the type of F0, and the subpath P ⋆, determine P uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We only need to look at the cells of P in A \\ A⋆, that is, in the union of the subboards Φ(b) of A with b ∈ F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let, then, b be an arbitrary cell of F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When ̺ makes a turn at b, the cells of P in Φ(b) are uniquely determined by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Otherwise, when ̺ goes straight through b, or when b is an endpoint of ̺, the desired uniqueness follows by parts (i) and (ii) of the strengthening, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The induction step is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We get the number of essentially distinct paths by Lemmas 2 and 3, summing over all possible types of ̺ and then subtracting out the duplicates specified in the statement of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' To convert this into the total number of paths, we analyse symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When k is even, all of the P’s are asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Otherwise, when k is odd, two of the P’s which correspond to the unique ̺ of type IV(0) are preserved under central symmetry, but not under any other symmetries of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and all other P’s corresponding to regular ̺’s are asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ Note that this argument also confirms our remark in the introduction regarding the number of essentially distinct longest snake paths in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The proof of Theorem 1 shows that, roughly speaking, every longest snake path P in G behaves as follows: Outside of some concentric square subboard of A, it is shaped as a simple spiral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and then, within that subboard, it is shaped as a double spiral instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' One endpoint of P lies on the boundary of A, and the other one lies on the boundary between the two spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 12 4 King Graphs on Odd Boards I In this section, we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let n be an odd positive integer with n = 2k−1, let A be the standard square board of side n, and let G be the king graph on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof of the lower bound for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consider the set of all cells of A of the form (x, y) with 1 ≤ x ≤ n − 2 and y even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The king graph on it is the disjoint union of k paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' To obtain the vertex set of a single snake path in G, we add in also the cells (0, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (0, n − 1) if k is even and (n − 1, n − 1) if k is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (0, y) in A with y ≡ 3 (mod 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and (n − 1, y) in A with y ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ For example, Figure 9 shows n = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Figure 9 We obtained the upper bound of Theorem 1 by summing over some subsets of A such that, for every snake path P, the part of P within each subset must be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Our approach to the upper bound of Theorem 2 will follow a similar strategy, albeit with significant complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Instead of subsets of A, we sum over subgraphs of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, our notion of smallness will be somewhat unusual: We consider the total number of certain cells and edges of P within each subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We call a cell (x, y) of A even when both of x and y are even, and odd when both of x and y are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We also call an edge of G regular when it is not incident with an odd cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (a) (b) Figure 10 Given an even cell a = (z, z) of A with z ≤ k − 2, we write Ѫ(a) for the subgraph of G with vertices a + [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 1]2 whose edges join a, a + (0, 1), and a + (1, 0) pairwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus Ѫ(a) contains four cells, one of which is odd, and three edges, all of which are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 10(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') For every symmetry π of A, if b = π(a), then we also define Ѫ(b) = π(Ѫ(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We call each subgraph of G of this form a little block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Given an even cell a = (x, y) of A with x > y and x + y ≤ n − 3, we write Ѫ(a) for the subgraph of G with vertices a + [−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 1] × [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 2] whose edges join a to a + (−1, 0) and a + (1, 0) as well as a + (0, 1) to all elements of the set a + [−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 1] × {0, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus 13 Ѫ(a) contains nine cells, two of which are odd, and eight edges, all of which are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 10(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') For every symmetry π of A, if b = π(a), then we also define Ѫ(b) = π(Ѫ(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We call each subgraph of G of this form a large block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (a) (b) Figure 11 For example, Figure 11 shows n = 11 and n = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (The colouring is only for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Observe that every odd cell belongs to at least two blocks and every regular edge belongs to at least one block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consider an arbitrary snake path P in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For every block Ѫ, let wCell(Ѫ) be the number of odd cells of P in Ѫ, let wEdge(Ѫ) be the number of regular edges of P in Ѫ, and let w(Ѫ) = wCell(Ѫ) + wEdge(Ѫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When Ѫ is a little block, w(Ѫ) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Otherwise, when Ѫ is a large block, w(Ѫ) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By direct examination of all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ Proof of the upper bound for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let wCell(P) be the number of odd cells in P and let wEdge(P) be the number of regular edges in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the length of P is bounded from above by 2wCell(P)+wEdge(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We sum the inequalities of Lemma 4 over all blocks Ѫ, and we obtain that the latter expression cannot exceed (n2 − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ 5 King Graphs on Odd Boards II In this section, we give a complete description of the longest snake paths in king graphs on odd square boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We use the same notations as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We begin with brief overviews of two relevant topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let s be a positive integer and let σ be a permutation of [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' s − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For each i with 0 ≤ i ≤ s−2, draw a semicircle with endpoints (0, σ−1(i)) and (0, σ−1(i+1)) which lies on the left of the coordinate axis Oy when i is even and on its right otherwise, when i is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 14 The union of all such semicircles is a curve κ in the plane with endpoints (0, σ−1(0)) and (0, σ−1(s−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When this curve does not intersect itself, σ is a stamp-folding permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For example, Figure 12 shows κ when σ is the permutation 1, 0, 2, 7, 4, 5, 6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Combinatorially, σ is a stamp-folding permutation if and only if there are no i and j of the same parity with 0 ≤ i ≤ s − 2 and 0 ≤ j ≤ s − 2 such that exactly one of i and i + 1 lies between j and j + 1 in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Figure 12 Figure 13 The intuition is as follows: Imagine a paper strip of size 1 × s formed out of s stamps of size 1 × 1 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We fold this strip along the perforations between stamps so that all stamps come to lie on top of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then σ is a stamp-folding permutation if and only if it can be obtained as the permutation of the stamps within the resulting stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The points of κ on the coordinate axis Oy correspond to the s stamps, and the semicircles of κ correspond to the s − 1 creases between stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Stamp-folding permutations have been studied extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consider a Hamiltonian path in the grid graph Γ of size s × s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The smallest number of turns that such a path can make is 2s − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [3] We proceed to review some properties of the paths which attain this minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [7] (The author noticed the connection between stamp-folding permutations and fewest- turn Hamiltonian paths when he obtained Theorems 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Independently, [7] was published before the present work was written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Let α be a fewest-turn Hamiltonian path in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We partition α into 2s−1 subpaths at the 2s − 2 cells where it makes a turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Each cell with a turn belongs to two subpaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') We call these subpaths the segments of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus in each segment of α either all edges are horizontal or all edges are vertical, and segments of these two types alternate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let us call α mostly-horizontal when it consists of s horizontal segments and s − 1 ver- tical segments, and mostly-vertical otherwise, when it is the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for concreteness, that α is mostly-horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then every row of the board contains exactly one horizontal segment of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Orient α arbitrarily, and then number its segments from 0 to s − 1 in the order in which they occur along α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When we assign to each row of the board the number of its horizontal segment of α, we obtain a stamp-folding permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Conversely, every stamp-folding permutation corresponds in this way to exactly two oriented mostly-horizontal fewest- turn Hamiltonian paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The two are reflections of one another with respect to the vertical axis of symmetry of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Explicitly, the correspondence is as follows: For each element i of [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' s−1], let ωLeft(i) be the number of even nonnegative integers j such that i lies between j and j + 1 in σ, and define ωRight(i) similarly, but with j odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In one of the two oriented paths 15 (a) (b) (c) (d) Figure 14 associated with σ, the path’s i-th horizontal segment goes from (s−ωRight(i)−1, σ−1(i)) to (ωLeft(i), σ−1(i)) for all even i, and it goes in the opposite direction for all odd i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The second oriented path associated with σ can be obtained by reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For example, Figure 13 shows this for the stamp-folding permutation of Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' One corollary of this connection is that, for all s with s ≥ 2, the number of fewest-turn Hamiltonian paths in the grid graph of size s × s is twice the number of stamp-folding permutations of s elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This concludes the two overviews, and we return to our main topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let H be the grid graph of size k × k and let ̺ be a Hamiltonian path in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Recall from Section 4 that k = ⌈n/2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') We say that the cycle a′a′′b′b′′ in H is free (relative to ̺) when a′a′′ is an edge of ̺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' the other three edges of the cycle are outside of ̺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' both of a′ and a′′ are cells that ̺ goes straight through (so that, if a′ = (c′ + a′′)/2 and a′′ = (a′ + c′′)/2, then c′∼c′′ ⊆ ̺);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and both of b′ and b′′ are cells where ̺ makes a turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We proceed to associate ̺ with certain paths in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Figure 14(a) shows one example of a Hamiltonian path ̺ in H, and Figures 14(b)–(d) track the series of definitions given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For each edge a′a′′ of ̺, we take the path 2a′—(a′ + a′′)—2a′′ in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We denote the concatenation of these paths by ϕ(̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Intuitively, this operation scales ̺ up by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 14(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Observe that the length of ϕ(̺) will be twice the length of ̺, namely 2(k2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For each turn a′ba′′ in ̺, let us delete the subpath (a′ + b)—2b—(b + a′′) from ϕ(̺), and let us replace it with the edge (a′ + b)—(b + a′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We denote the resulting path in G by ψ(̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Intuitively, this operation smooths down the sharp turns in ϕ(̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 14(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') 16 Let t be the number of turns in ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the length of ψ(̺) will be 2(k2 − 1) − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Finally, for each edge a′a′′ of ̺ which is in a free cycle, either we do nothing or, optionally, we choose one free cycle a′a′′b′b′′ which includes a′a′′, we set c = (a′ + a′′ + b′ + b′′)/4, we delete the subpath 2a′—(a′ + a′′)—2a′′ from ψ(̺), and we replace it with the subpath 2a′—2c—2a′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We call a path in G which can be obtained in this way a lift of ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Intuitively, this operation introduces some tiny aberrations in ψ(̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 14(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') In particular, ψ(̺) is also a lift of ̺, namely the one in which we have selected the do-nothing option everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Observe that every lift of ̺ is a snake path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, all lifts of ̺ are of the same length, namely 2(k2 − 1) − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We are ready to state and prove our structure theorem for the longest snake paths in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let n be an odd positive integer with n = 2k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then every lift of a fewest-turn Hamiltonian path in the grid graph of size k × k is a longest snake path in the king graph of size n × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Conversely, every longest snake path in the king graph of size n × n can be obtained uniquely as a lift of some fewest-turn Hamiltonian path in the grid graph of size k × k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Theorem 4 yields the following recipe for the generation of all longest snake path in G: First, we generate all stamp-folding permutations of k elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then we convert each stamp-folding permutation into four oriented fewest-turn Hamiltonian paths in H, two mostly-horizontal ones and two mostly-vertical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We forget about the orientations, and discard the duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Finally, for each fewest-turn Hamiltonian path in H, we identify the corresponding free cycles, and we generate all of its lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (We said in the introduction that each stamp-folding permutation yields two families of longest snake paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Strictly speaking, the reality is that each permutation yields four families, and each quadruple of families is obtained in this way twice, out of two permu- tations σ′ and σ′′ related by σ′(i) + σ′′(i) = k − 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, it is straightforward to extract a one-to-two mapping from the two-to-four one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Before we go on to the proof of Theorem 4, let us briefly discuss the aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let ̺ be a fewest-turn Hamiltonian path in H and let f be the number of its corresponding free cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then ̺ yields exactly 2f lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, f ≤ max{0, k − 5}, and for all k this bound is attained by some ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for concreteness, that ̺ is mostly-horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let a′a′′ be an edge of ̺ and let a′a′′b′b′′ be a free cycle which includes a′a′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for the sake of contradiction, that edge a′a′′ is horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then, since ̺ makes a turn at both of b′ and b′′, the edges of ̺ in the row of b′ and b′′ cannot form one contiguous subpath of ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This contradicts the fact that every row of the board contains exactly one horizontal segment of ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus edge a′a′′ must be vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose now, for the sake of contradiction, that edge a′a′′ is in a second free cycle a′a′′c′c′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since ̺ must make a turn at both of c′ and c′′ as well, it follows that the edges 17 of ̺ in the row of b′ and c′ cannot form one contiguous subpath of ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' From here, we get a contradiction as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus no edge of ̺ can be in two distinct free cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consequently, in the setting of Theorem 4, when we construct a lift of ̺, we never have to choose between two free cycles which include the same edge a′a′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We are left to show that f ≤ max{0, k − 5} and the bound is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We handle the cases when k ≤ 6 directly, and from now on we assume that k ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Each free cycle contains two cells where ̺ makes a turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Conversely, each such cell is in at most one free cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' On the other hand, ̺ makes 2k − 2 turns altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, a turn cell in an outermost column of B cannot be in a free cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The lowermost turn cell and the topmost turn cell in a non-outermost column of B cannot be in free cycles, either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When at most one non-outermost column of B contains turns of ̺, since no turn cells outside of that column can be in free cycles, it follows that 2f ≤ k−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then f ≤ k−5 by virtue of k ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Otherwise, when at least two non-outermost columns of B contain turns of ̺, it follows that at least eight turn cells are not in free cycles, and so 2f ≤ 2k − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The bound is attained, for example, when ̺ corresponds to the stamp-folding permu- tation 0, 2, 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=', k − 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (The path in Figure 14(a) is of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') □ In the setting of the proof of Proposition 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' our observation that edge a′a′′ must be vertical allows us to characterise the free cycles corresponding to ̺ in terms of the underlying stamp-folding permutation σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' as follows: Consider the ordered pairs (ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' i) with ε ∈ {−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 1} and 0 ≤ i ≤ k − 2 such that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' in σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' both of σ(i) and σ(i + 1) lie between σ(i) + ε(−1)σ(i) and σ(i + 1) + ε(−1)σ(i+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' additionally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' there is some j with 0 ≤ j ≤ k − 1 such that all four of these lie between j and j + ε(−1)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Each such ordered pair yields a free cycle where edge a′a′′ joins rows i and i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When ε = 1, cells a′ and a′′ are in column ωLeft(σ(i)) − 1 and cells b′ and b′′ are on the right of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Otherwise, when ε = −1, cells a′ and a′′ are in column k −ωRight(σ(i)) and cells b′ and b′′ are on their left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, this accounts for all free cycles corresponding to ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let NStamp(k) be the number of stamp-folding permutations of k elements and let NKing(n) be the number of longest snake paths in the king graph of size n × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proposi- tion 1 yields some loose bounds on NKing(n) in terms of NStamp(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let n be an odd positive integer with n = 2k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then 2NStamp(k) < NKing(n) < 2k−4NStamp(k) for all n with n ≥ 11, and NKing(n) = 2NStamp(k) when 3 ≤ n ≤ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus, in particular, log NKing(n) = Θ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The first part follows by Theorem 4 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Strictly speaking, we should also note that not all fewest-turn Hamiltonian paths in H attain the greatest number of free cycles when n ≥ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') The second part is a corollary of the first part and the well-known asymptotic estimate log NStamp(n) = Θ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [2] □ We continue with the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let P be a longest snake path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Here is a quick roadmap: Clearly, P must attain exact equality in all inequalities from the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We examine all blocks of G from this point of view, one by one, 18 in a certain order, and we see that P must satisfy certain purely local constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' These constraints allow us to conclude that P must be a lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The details, however, are somewhat technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We define an even cell a of A to be nice (relative to P) when either P visits a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' or, else, P does not visit a but it traverses exactly one edge of G between the four cells in the set a + {(1, 0), (0, 1), (−1, 0), (0, −1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Note that, for some a, some of these cells might be outside of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose that all even cells of A are nice and P does not visit any odd cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then there is a Hamiltonian path ̺ in H such that P = ψ(̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let us call an edge of G short when the Euclidean distance between its end- points is unity, and long otherwise, when it is √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For each long edge a′a′′ of P, we do the following: Since P does not visit any odd cells, there is a unique even cell b such that both of a′b and ba′′ are edges of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We delete edge a′a′′ from P, and we replace it with these two edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since all even cells of A are nice and P is a snake path, the result will be a path in G which contains only short edges, which visits all even cells of A, and which does not visit any odd cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Denote this path by Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let c be an endpoint of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for the sake of contradiction, that c is not an even cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since Q does not visit any odd cells, there are exactly two even cells d′ and d′′ of A adjacent to c in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let cd′ be the unique edge of Q incident with c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Observe that c is also an endpoint of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' But then d′′ cannot be nice because P is a snake path, and we arrive at a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' So, to our previous observations about Q, we can add the fact that both of its end- points are even cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consequently, Q is of the form Q = ϕ(ρ) for some Hamiltonian path ̺ in H, and P = ψ(̺), as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ We define a rectifiable aberration in P to be a subpath of P of the form b′a′ba′′b′′ such that a′ and a′′ are two even cells in the same row or column;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' the cell c = (a′ +a′′)/2 is a common neighbour of a′ and a′′ in G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' the cells b′ and b′′ satisfy a′ = (b′ + c)/2 and a′′ = (c + b′′)/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and b ̸= c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') b′ a′ b c a′′ b′′ Figure 15 To rectify a rectifiable aberration, we delete the subpath a′ba′′ from P, and we replace it with the subpath a′ca′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The result will be a new snake path in G of the same length as P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This follows because P being a snake path implies that the only neighbours of c in G which P visits are a′, b, and a′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose that P does not contain any rectifiable aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then all even cells of A are nice and P does not visit any odd cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since P is a longest snake path in G, it must attain exact equality in all inequalities from the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus: (i) A cell of A of the form (z, z) with z odd cannot be in P because it is an odd cell which is in more than two blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Same goes for the images of these cells under the symmetries of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (ii) An edge of G of the form (z, z + 1)—(z + 1, z) with z odd and z ≤ k − 2 cannot be in P because it is a regular edge which is in more than one block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Same goes for the images of these edges under the symmetries of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and (iii) Every block Ѫ must attain exact equality in Lemma 4, so that w(Ѫ) = 1 when Ѫ is a little block and w(Ѫ) = 2 otherwise, when Ѫ is a large block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For all odd positive integers s with 1 ≤ s ≤ n, we write As for the concentric subboard of A of size s × s given by As = [⌊n/2⌋ − ⌊s/2⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' ⌊n/2⌋ + ⌊s/2⌋]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We will show by induction on s that all even cells of As are nice and all odd cells of As are outside of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Our base case is s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then A1 consists of a single cell, namely (k −1, k −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Denote this cell by o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When k is even, o is an odd cell, and o ̸∈ P by (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Figure 16 When k is odd, o is an even cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Suppose, for the sake of contradiction, that k ≥ 3 and o is not nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then o ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By (i), o+(1, 1) ̸∈ P, and similarly for the images of this cell under the symmetries of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By (ii), (o+(1, 0))—(o+(0, 1)) ̸∈ P, and similarly for the images of this edge under the symmetries of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus (iii) for Ѫ(o + (0, −2)) implies o + (−1, −2) ∈ P, o + (1, −2) ∈ P, and either o + (0, −1) ∈ P or o + (0, −2) ∈ P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and similarly for the images of this block under the symmetries of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, the twelve cells we just concluded must be in P are the vertices of a cycle in G, and we arrive at a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This settles the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the induction step, let s ≥ 3 and suppose that we have already established the desired result for As−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let a = (x, y) be an arbitrary cell of As \\ As−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By symmetry, we can assume without loss of generality that ⌊n/2⌋ − ⌊s/2⌋ ≤ x ≤ ⌊n/2⌋ and y = ⌊n/2⌋ − ⌊s/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Figures 17 and 18 show some of the cells, edges, and blocks relevant to our reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Cell a is highlighted in all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Note that some blocks which are shown as large in the figures might be little ones in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In all such cases, we emphasise this possibility in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for the sake of contradiction, that a is an even cell but that it is not nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 20 (a) (b) Figure 17 Then a ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, the odd cells of Ѫ(a) are not in P, either, by the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' x = ⌊n/2⌋ − ⌊s/2⌋ and Ѫ(a) is a little block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 17(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Then (iii) for Ѫ(a) implies (a + (1, 0))—(a + (0, 1)) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since a is not nice, at least one more edge of G between the four cells in the set a + {(1, 0), (0, 1), (−1, 0), (0, −1)} must be in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' It cannot be (a + (−1, 0))—(a + (0, −1)) because then P would contain the vertices of a cycle in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The other two subcases are symmetric with respect to the line of unit slope through a, and we assume that (a + (0, −1))—(a + (1, 0)) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' It follows that a + (2, 0) ̸∈ P and a + (2, 1) ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus no edges of Ѫ(a + (2, 0)) are in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since also the odd cells of this block are outside of P by the induction hypothesis, we arrive at a contradiction with (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (The conclusion holds regardless of whether the block is a little one or a large one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' ⌊n/2⌋ − ⌊s/2⌋ < x ≤ ⌊n/2⌋ and Ѫ(a) is a large block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 17(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') By (iii) for Ѫ(a) and the induction hypothesis for a + (0, 2), exactly one of the two edges (a + (1, 0))—(a + (0, 1)) and (a + (0, 1))—(a + (−1, 0)) is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The two subcases are analogous, and we assume that the former edge is in P while the latter one is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' As in Case 1, at least one more edge of G between the four cells in the set a + {(1, 0), (0, 1), (−1, 0), (0, −1)} must be in P, and it cannot be (a + (−1, 0))—(a + (0, −1)) because then P would contain the vertices of a cycle in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus (a + (0, −1))— (a + (1, 0)) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' From here, we arrive at the exact same contradiction as in Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Once again, regardless of the type of the block Ѫ(a + (2, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') We have established that if a is an even cell, then it is nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the second half of the induction step, suppose, for the sake of contradiction, that a is an odd cell with a ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By (i), it follows that ⌊n/2⌋ − ⌊s/2⌋ + 2 ≤ x ≤ ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' a + (−2, 0) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 18(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') By (i), x ≥ ⌊n/2⌋ − ⌊s/2⌋ + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus Ѫ(a + (−1, 1)) is a large block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Note that a ∈ P implies (a + (−1, 1))—(a + (0, 1)) ̸∈ P and a + (−2, 0) ∈ P implies (a+(−1, 1))—(a+(−2, 1)) ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, by the induction hypothesis, the odd cells of Ѫ(a + (−1, 1)) are not in P and cell a + (−1, 3) is nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then (iii) for Ѫ(a + (−1, 1)) implies that exactly one of the two edges (a + (0, 1))—(a + (−1, 2)) and (a + (−1, 2))— (a+(−2, 1)) is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Here, we take into account the fact that a ∈ P and a+(−2, 0) ∈ P together imply (a + (−1, 1))—(a + (−1, 2)) ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') The two subcases are analogous, and we assume that the former edge is in P while the latter one is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' It follows that a—(a + (0, 1))—(a + (−1, 2)) ⊆ P, a + (1, 1) ̸∈ P, and a + (1, 2) ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 21 (a) Ѫ′ Ѫ′′ (b) Figure 18 Thus no edges of Ѫ(a + (1, 1)) are in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since also the odd cells of this block are outside of P by the induction hypothesis, we arrive at a contradiction with (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (The block Ѫ(a + (1, 1)) will always be a large one because of our symmetry-breaking assumption that x ≤ ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, in the analogous subcase when (a + (0, 1))—(a + (−1, 2)) ̸∈ P and (a + (−1, 2))—(a + (−2, 1)) ∈ P, the contradiction occurs at block Ѫ(a + (−3, 1)) which could happen to be a little one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' a + (2, 0) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This case is analogous to Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' a + (−2, 0) ̸∈ P and a + (2, 0) ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 18(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Observe that both of Ѫ′ = Ѫ(a + (−1, −1)) and Ѫ′′ = a + (1, −1) are large blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since a ∈ P, all edges of Ѫ′ other than (a+(−1, −1))—(a+(−2, −1)), (a+(−2, −1))— (a + (−1, 0)), and (a + (−1, 0))—(a + (−2, 1)) are not in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then, in light of a ∈ P and a + (−2, 0) ̸∈ P, (iii) for Ѫ′ implies that exactly one of these edges is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In particular, exactly one cell b′ out of the pair a+{(−1, −1), (−1, 0)} is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Similar reasoning applies to Ѫ′′, and we define b′′ analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for the sake of contradiction, that b′ = a + (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then ab′ ⊆ P implies a + (0, 1) ̸∈ P and a + (−1, 1) ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, the odd cells of Ѫ(a + (−1, 1)) are outside of P by the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When this block is a little one, we arrive at a contradiction with (iii) immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Otherwise, when it is a large one, we arrive at a contradiction with (iii) anyway once we take into account the fact that, by the induction hypothesis, both of a + (−1, 1) and a + (−1, 3) are nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consequently, b′ = a+(−1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Similarly, b′′ = a+(1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By (iii) for Ѫ′ and Ѫ′′, it follows that also (a+(−1, −1))—(a+(−2, −1)) ∈ P and (a+(1, −1))—(a+(2, −1)) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, then (a + (−2, −1))—(a + (−1, −1))—a—(a + (1, −1))—(a + (2, −1)) becomes a rectifiable aberration in P, and we arrive at a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (In fact, this is the only place in the proof where we use the constraint that P does not contain any rectifiable aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') We have established that if a is an odd cell, then it cannot be in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The induction step is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ We are ready to tackle Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For a start, let us rectify all rectifiable aberrations in P one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The result will be a snake path Q in G of the same length as P and without any rectifiable aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By Lemma 6, we get that all even cells are nice relative to Q and Q does not visit any odd cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 22 By Lemma 5, it follows that there is a Hamiltonian path ̺ in H such that Q = ψ(̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let t be the number of turns in ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then both of P and Q are of length 2(k2 − 1) − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since the greatest length of a snake path in G is (n2−1)/2 by Theorem 2, we conclude that t = 2k − 2, and so ̺ is in fact a fewest-turn Hamiltonian path in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Finally, in order to transform Q back into P, we must restore the rectifiable aberra- tions which we removed in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, it is straightforward to check that the spots in Q where we can introduce a rectifiable aberration are exactly the ones associated with the free cycles corresponding to ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Therefore, P is a lift of ̺, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The reasoning in the last few paragraphs shows also the converse: That every lift of a fewest-turn Hamiltonian path in H is a longest snake path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ 6 Knight Graphs In this section, we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let m and n be positive integers, let A be the standard board of size m × n, and let G be the knight graph on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We begin with the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' One natural approach would be as follows: First we find some finite knight graph H with pseudosnake density 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then we sum over all translation copies of H contained within our board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The author was not able to implement this idea in its purest form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Below, we present a slightly more complicated argument which relies on a weighted knight graph instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We define a weighted graph Γ to consist of a simple graph H and a weighting function w which assigns a nonnegative real weight to each vertex of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When H is finite, we denote the total weight of all of its vertices by w(Γ), and we define the pseudosnake density of Γ to be the ratio of the greatest total weight of the vertices in a pseudosnake of H to w(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose that there is a weighted knight graph Γ with pseudosnake den- sity τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the number of vertices in a pseudosnake of G cannot exceed τmn+O(m+n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Here and in the proof, the implicit constants in the O-terms depend on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let P be a pseudosnake of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consider all translation copies of Γ that fit within A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' There are mn + O(m + n) of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For each such copy, the total weight of all cells of P within it cannot exceed τw(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Conversely, it is true of all but O(m + n) cells a of P that a is sufficiently far away from the boundary of A for every translation copy of Γ which contains a to fit within A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For each such cell of P, the sum of its weights over all translation copies of Γ which contain it will be w(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ Proof of the upper bound for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By Lemma 7, if suffices to exhibit one concrete weighted knight graph with pseudosnake density 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We claim that the one in Figure 19 works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (The figure shows all cells of the graph together with the weights assigned to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') It has 68 cells of total weight 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Our claim would likely be extremely difficult to check by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, it is straight- forward to check with the help of a standard constraint satisfaction solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The author 23 1 1 6 1 2 1 1 3 6 1 4 3 4 7 7 4 1 4 1 1 4 7 4 4 6 1 1 2 3 1 3 6 1 4 1 1 1 7 6 6 1 1 1 2 1 1 1 6 2 4 2 1 3 2 3 2 1 1 2 1 4 1 3 4 4 1 6 3 Figure 19 has done this twice, using two different constraint satisfaction frameworks: the Copris package for the Scala programming language and the OR-Tools package for the Python programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ One might wonder how the weighted knight graph in Figure 19 was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Figure 20 Figure 21 For a start, let STess be the set of all 16 cells of the form ε1(2, 1)+ε2(1, 2)+ε3(−1, 2)+ ε4(−2, 1), where εi ∈ {0, 1} for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the knight graph GTess on STess is isomorphic to the tesseract graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') It is not too difficult to check by hand that the pseudosnake density of the tesseract graph is 9/16, and that it is attained by an essentially unique pseudosnake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since 9/16 is very close to 1/2 from above, we see that GTess works almost, but not quite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We can attempt to fix this by taking the union of several overlapping copies of GTess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since GTess itself just barely manages to push through pseudosnake density 1/2, we can hope that the interference between its copies will prevent too many of them from doing the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We formalise this notion as follows: Let S′ and S′′ be two nonempty finite sets of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We define their sum, denoted S′ + S′′, to be the multiset of cells which consists of all cells of the form a′ + a′′ with a′ ∈ S′ and a′′ ∈ S′′, and where the multiplicity of each cell is the number of ways that it can be expressed in this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then, given a multiset of cells S, we define G(N, S) to be the weighted knight graph on the cells of S where the weight of each cell is its multiplicity in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For each nonempty finite set of cells S, we can think of the weighted knight graph G(N, S + STess) as constructed out of several overlapping copies of GTess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We experiment 24 with different S, and eventually we strike gold with SDia = [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 3]2 \\ {0, 3}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This is the Aztec diamond of order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') We go on to the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' One natural approach would be as follows: First we find a doubly periodic pseudosnake P∞ in G(N, Z2) with density 1/2, where furthermore every cell is of degree exactly two and there are no finite cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Here, “doubly periodic” means that there are two linearly independent two-dimensional vectors u and v with a ∈ P∞ ⇔ a + u ∈ P∞ ⇔ a + v ∈ P∞ for all cells a of Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Then, given a board A, we take the restriction P ⋆ of P∞ to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Because of the struc- ture of P∞, this restriction will be the disjoint union of several paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We make some modifications near the boundary of A, deleting some cells from P ⋆ and replacing them with new ones, so as to stitch all of these paths together into a single snake path or cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since P∞ is doubly periodic with density 1/2, originally P ⋆ will contain mn/2+O(m+n) cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Finally, keeping our modifications close to the boundary of A ensures that they cost us O(m + n) of these cells altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Finding a suitable P∞ is straightforward enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For example, the set of all cells (x, y) with x mod 4 ∈ {0, 1} works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' It consists of vertical strips of width two spaced two units apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, the second part of our plan runs into significant difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus the con- struction we present below is somewhat more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We divide A into four large re- gions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' we fill up different regions using different pseudosnakes P∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and we make stitching- together modifications not only near the boundary of A, but also near the boundaries between regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We continue with the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We define a twine to be a board of height two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When s ≥ 2, the knight graph on a twine with width s is the disjoint union of four paths, two spanning ⌊s/2⌋ cells each and two spanning ⌈s/2⌉ cells each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Figure 22 Figure 23 Consider a twine E with lower left corner cell a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' To tie off E on the left, we add to it the four cells in the set a + {(−2, 1), (−1, 1), (−1, 3), (0, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Similarly, to tie off on the right a twine E with lower right corner a, we add to it the reflections of the four cells above with respect to the vertical line through a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consider, now, two twines E and F with lower left corners a and b satisfying a + (1, 4) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' To splice together E and F on the left, we add to them the ten cells in the set a+{(−3, 4), (−3, 5), (−2, 2), (−2, 3), (−2, 6), (−1, 1), (−1, 2), (−1, 6), (0, 5), (0, 6)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 25 (Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Similarly, to splice together on the right two twines E and F whose lower right corners a and b satisfy a + (−1, 4) = b, we add to them the reflections of the ten cells above with respect to the vertical line through a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let k be a positive integer and let I = [x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' x′′] be an integer interval with |I| ≥ 8k −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For each i with 0 ≤ i ≤ k−1, if i is even then construct the twine Ei = [x′ +4i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' x′′ −4i]× [4i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 4i+1], and if i is odd then construct the twine Ei = [x′+4i+3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' x′′−4i+3]×[4i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 4i+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Tie off E0 on the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' for all i with 0 ≤ i ≤ k − 2, splice together Ei and Ei+1 on the right if i is even, and on the left if i is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and, finally, if k is even then tie off Ek−1 on the left, and if k is odd then tie it off on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Figure 24 We denote the resulting set of cells by U(k, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For example, Figure 24 shows the knight graph on U(4, [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose that k ≥ 2 and |I| is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the knight graph on U(k, I) is a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Denote H = G(N, U(k, I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' It is straightforward to check that all cells of H are of degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We are left to verify that H is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for the sake of clarity, that k is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The opposite case, when it is even, is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let ai and bi be the lower left and lower right corner cells of Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let also a′ i and a′′ i be the two cells of Ei adjacent by side to ai, and define b′ i and b′′ i similarly for bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' It is straightforward to verify that, since |I| is odd: (a) A path in H connects b′ 0 and b′′ 0 and covers E0 together with the cells which tie it off;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (b) For each i with 1 ≤ i ≤ k−2, two paths in H connect the pairs {a′ i, a′′ i } and {b′ i, b′′ i } and cover Ei together with two cells in the adjacent splices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and (c) A path in H connects a′ k−1 and a′′ k−1 and covers Ek−1 together with the cells which tie it off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Finally, the remaining cells of the splices of U(k, I) form additional knight paths in H which connect the pairs {b′ i, b′′ i } and {b′ i+1, b′′ i+1} for all even i with 0 ≤ i ≤ k − 3 as well as the pairs {a′ i, a′′ i } and {a′ i+1, a′′ i+1} for all odd i with 1 ≤ i ≤ k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ Proof of the lower bound for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We construct a large snake cycle in G, and for a large snake path we can simply delete one cell from that cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, without loss of generality, that m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since we are already willing to accept a tolerance of O(m+n), we can safely assume that m = 8k +14 for some positive integer k with k ≥ 3, and that n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 26 Construct UI = U(k, [8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' n−12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let also VI be the reflection of U(k, [8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' m−12]) with respect to the line x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Lastly, let UII and VII be symmetric to UI and VI with respect to the center of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The knight graph on UI ∪VI ∪UII ∪VII is the disjoint union of four cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We proceed to stitch these four cycles together into a single longer cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Figure 25 Define SDel = {(6, 9), (9, 6)} and SAdd = {(3, 6), (4, 4), (6, 3), (7, 10), (9, 9), (10, 7)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Delete the two cells of (4, 4) + SDel from UI and VI, and replace them with the six cells of (4, 4) + SAdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This stitches together the cycles of UI and VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') We carry out two more such modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For one of them, we reflect the sets SDel and SAdd with respect to vertical axis of symmetry of the board, we delete the two cells in the image of SDel from UI and VII, and we replace them with the six cells in the image of SAdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This stitches together the cycles of UI and VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the other one, we proceed similarly, except that the reflections are done with respect to horizontal axis of symmetry of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This stitches together the cycles of VI and UII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let W be the final set of cells obtained in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For example, Figure 26 shows W in the case when k = 5, m = 54, and n = 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Observe that the density of W within A is 1/2 everywhere except within five strips of bounded width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Four of these strips surround portions of the interior angle bisectors at the four corners of A, and the fifth one surrounds a portion of the horizontal axis of symmetry of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The corresponding mostly hollow areas are clearly visible in Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Consequently, the number of cells in W is mn/2 + O(m + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' On the other hand, W is the vertex set of a snake cycle in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ 27 Figure 26 7 Further Work on King Graphs In this section, we collect some additional results and open problems on king graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We saw that the behaviour of the longest snake paths in G(K, n × n) depends on the parity of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For cycles, it appears that there are four classes instead, depending on the value of n mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The techniques we developed for paths quickly resolve two of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let n be a positive integer with n ≡ 0 (mod 4) and n ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the greatest length of a snake cycle in the king graph of size n × n is n2/2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, for all such n, there are exactly 48 snake cycles which attain this greatest length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' These cycles are all asymmetric, and so six of them are essentially distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For completeness, there is a unique snake cycle of the greatest length 8 when n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Each one of the cycles of Theorem 5 is shaped like a double spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' It is curious that the number of longest snake cycles freezes in this way, and the cycles themselves crystallise into a single inflexible structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' A similar phenomenon occurs in the setting of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let n = 2k and define A, B, G, H, and Φ as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let also C be a snake cycle in G of length at least n2/2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' As in Section 3, for each cell b of B at most two cells of Φ(b) are in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let us call b deficient when this bound is not attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus there is at most one deficient cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 28 Observe that, if an edge of C joins one cell of Φ(b′) and one cell of Φ(b′′), with b′ ̸= b′′, then b′b′′ must still be an edge of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Otherwise, assuming for concreteness that b′ + (1, 1) = b′′, by an argument similar to the one in Section 3 we see that at least two of the four cells in the set b′ + [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 1]2 must be deficient, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This allows us to define the Hamiltonian cycle ̺ in H relative to C in the same manner as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Define also the cycle E in H as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since E is not a Hamiltonian cycle of H when k ≥ 4, at least one edge of E must be outside of ̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' On the other hand, observe that Lemma 1 now admits a unique exception: When the turn occurs at a deficient cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consequently, every edge of E outside of ̺ must possess a deficient endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Define A⋆, B⋆, G⋆, and H⋆ as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let also C⋆ and ̺⋆ be the restrictions of C and ̺ to A⋆ and B⋆, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We obtain that: (a) There is exactly one deficient cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (b) There is exactly one edge β′β′′ of E outside of ̺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (c) ̺⋆ is a Hamiltonian path in H⋆ whose endpoints are the two neighbours of β′ and β′′ in B⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' and (d) C⋆ is a snake path in G⋆ of the greatest possible length whose associated Hamiltonian path in H⋆ is ̺⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, the proof of Theorem 1 gives us a complete description of all Hamiltonian paths in H⋆ associated with a longest snake path in G⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since the two endpoints of ̺⋆ are neighbours in H⋆, we conclude that when n ≥ 12 a symmetry of B⋆ must map ̺⋆ onto the unique regular Hamiltonian path in H⋆ of type II(0), as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The rest is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ The other class we can tackle without too much extra effort is n ≡ 3 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' First, though, we need to sort through some preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let s be an even positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Given a permutation σ of [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' s − 1], consider a closed curve in the plane defined in the same way as the curve κ in Section 5, except that one additional arc on the right of the coordinate axis Oy joins points (0, σ−1(s−1)) and (0, σ−1(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' A curve of this form which does not intersect itself is known as a closed meander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consider a Hamiltonian cycle in the grid graph Γ of size s × s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The smallest number of turns that such a cycle can make is 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [3] Furthermore, the closed meanders with s arcs and the fewest-turn Hamiltonian cycles in Γ are related in a way analogous to the relation between stamp-folding permutations and fewest-turn Hamiltonian paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Observe, lastly, that our definition of a lift in Section 5 works just as well with cycles instead of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let n be a positive integer with n ≡ 3 (mod 4) and n = 2k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the greatest length of a snake cycle in the king graph of size n × n is (n2 − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Every lift of a fewest-turn Hamiltonian cycle in the grid graph of size k × k is a longest snake cycle in the king graph of size n × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Conversely, every longest snake cycle in the king graph of size n × n can be obtained uniquely as a lift of some fewest-turn Hamiltonian cycle in the grid graph of size k × k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This time around, our analysis of paths carries over to cycles nearly verbatim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The upper bound follows by the same argument as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The lower bound is a corollary of the structure description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Finally, the structure description follows by the same argument as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ With the remaining two classes, the main difficulty is this: In both Sections 3 and 5, we introduce the half-sized square board B of side ⌈n/2⌉ together with its grid graph H, and our reasoning relies heavily on the properties of the Hamiltonian paths of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For Theorems 5 and 6, it is the Hamiltonian cycles of H that matter instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When n ≡ 1 (mod 4) or n ≡ 2 (mod 4), however, the side of B is odd and H does not admit a Hamiltonian cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This throws a substantial wrench in the works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' While our upper bounds all go through as before, the constructions that support the lower bounds do not, and the gap which opens between the two appears to be difficult to close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We continue with some tentative remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let Nr be the set of all positive integers n such that n ≡ r (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Fix n⋆ in Nr with n⋆ ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For each n in Nr with n ≥ n⋆, construct the subset Dn of the standard board of size n×n as follows: Take all cells of the form (x, y) with x−y ≥ 1, x + y ≤ n − 2, y even, and 0 ≤ y ≤ (n − n⋆)/2 together with their images under the symmetries of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Delete the cells (0, 2), (0, 3), and (0, 4), and replace them with the cells (1, 2) and (1, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Finally, for all even i with 2 ≤ i ≤ (n − n⋆)/2, delete the three cells in the set (i, i) + {(0, 1), (0, 3), (0, 4)}, and replace them with the three cells in the set (i, i) + {(0, 0), (1, 2), (1, 4)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Note that Dn is the vertex set of a snake path in G(K, n × n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We say that Nr crystallises at n⋆ when, for all n in Nr with n ≥ n⋆ and every longest snake cycle C in G(K, n×n), there is a symmetry π of the corresponding board such that the set of all cells of π(C) outside of the concentric subboard of size (n⋆ − 4) × (n⋆ − 4) coincides with Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus, in particular, if Nr crystallises, then there are two constants ℓ and µ such that the greatest length of a snake cycle in G(K, n × n) is n2/2 − ℓ and the number of essentially distinct snake cycles which attain this greatest length is µ for all n in Nr with n ≥ n⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, each one of these cycles is asymmetric, and so for all such n the total number of longest snake cycles in G(K, n × n) is 8µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The proof of Theorem 5 shows that the class N0 crystallises at n⋆ = 12 with ℓ = 1 and µ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The author finds it reasonably plausible that each one of the classes N1 and N2 might crystallise as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Experimental data suggests that perhaps the class N1 crystallises at n⋆ = 13 with ℓ = 5/2 and µ = 69 whereas the class N2 crystallises at n⋆ = 14 with ℓ = 3 and µ = 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The next result might be helpful in the case of the class N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The two cycles of Theorem 7 appear to be related to the longest snake cycles of G(K, (2k − 1) × (2k − 1)) in a way somewhat similar to how the fewest-turn Hamiltonian paths and cycles of grid graphs are related to the paths and cycles of Theorems 4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We define a near-Hamiltonian cycle of a graph G to be a cycle in G which visits all but one vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 30 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let k be an odd positive integer with k ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the smallest number of turns in a near-Hamiltonian cycle of the grid graph of size k × k is 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, for all such k, there are exactly 16 near-Hamiltonian cycles which attain this smallest number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' These cycles are all asymmetric, and so two of them are essentially distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For completeness, there is a unique near-Hamiltonian cycle with the smallest number of turns, namely four, when k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Just as in Theorem 5, the cycles of Theorem 7 are shaped like double spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let B be the standard board of size k × k, let H be the grid graph on B, and let C be a near-Hamiltonian cycle in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Denote the unique cell of B which C omits by o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for the sake of contradiction, that there are a row and a column of B without an edge of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the cell at their intersection must be o, and it cannot lie on the boundary of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since C visits the neighbours of o in H but the row and column of o do not contain edges of C, it follows that all edges of the cycle (o + (1, 1))∼(o + (−1, 1))∼(o + (−1, −1))∼(o + (1, −1))∼(o + (1, 1)) must be in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, this cycle is not near-Hamiltonian when k ≥ 5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for concreteness, that every row contains an edge of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Define the segments of C as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since every row contains a horizontal segment of C, and the end- points of each such segment are turns, we get that C makes at least 2k turns altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, from now on, that this bound is attained and that every row contains exactly one horizontal segment of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus, in particular, o cannot lie in the lowermost or topmost row of B unless it is a corner cell of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for the sake of contradiction, that the leftmost and rightmost columns of B contain one vertical segment of C each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then they cannot contain o unless it is a corner cell of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When, say, o = (0, 0), it follows that C must contain all edges of the cycle (0, 1)—(1, 1)—(1, 0)∼(k − 1, 0)∼(k − 1, k − 1)∼(0, k − 1)∼(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Otherwise, when o is not on the boundary of B, it follows that C must contain all edges of the cycle (0, 0)∼(k − 1, 0)∼(k − 1, k − 1)∼(0, k − 1)∼(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, in both cases the cycle in question is not near-Hamiltonian when k ≥ 5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for concreteness, that the leftmost column of B contains at least two vertical segments of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since the number of vertical segments in C is the same as its number of horizontal segments, and we have already assumed that the latter number equals k, we get that some column u of B does not contain any edges of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consequently, C crosses over u every time when it visits this column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since the total number of crossings must be even, and u contains an odd number of cells, we obtain that o must be in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' It follows that there is exactly one column of B without vertical segments of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Since each such column must contain o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Thus the leftmost column of B must contain exactly two vertical segments of C and all other columns except for u must contain exactly one vertical segment of C each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (Since there are a total of k vertical segments in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') Let the two vertical segments of C in the leftmost column of B be (0, 0)∼(0, w) and (0, w + 1)∼(0, k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for concreteness, that 1 ≤ w ≤ ⌊k/2⌋ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' From this point on, we establish the identity w = 1 and the desired result together, by induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The base case k = 5 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the induction step, suppose 31 that k ≥ 7 and that we have already settled the question on all smaller boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let us delete the subpath (1, w)—(0, w)∼(0, 0)∼(k − 1, 0)∼(k − 1, k − 1)∼(0, k − 1)∼(0, w + 1)—(1, w + 1) from C, and let us replace it with the edge (1, w)—(1, w + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The result will be a near-Hamiltonian cycle C⋆ in the grid graph H⋆ on the concentric subboard B⋆ of B of size (k − 2) × (k − 2) given by B⋆ = [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' k − 2]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since we have deleted at least six turns from C and we have added at most two new ones in their place, C⋆ can make at most 2k − 4 turns altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus our induction hypothesis applies to it, and so in fact C⋆ makes exactly 2k − 4 turns, two of which are at cells (1, w) and (1, w + 1) where they form the subpath (2, w)—(1, w)—(1, w + 1)— (2, w + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Still by the induction hypothesis, C⋆ omits exactly one edge of the cycle (1, 1)∼(k − 2, 1)∼(k − 2, k − 2)∼(1, k − 2)∼(1, 1), and that edge is an image of the edge (1, 2)—(1, 3) under a symmetry of B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We conclude that w = 1 and exactly two of the fewest-turn near-Hamiltonian cycles of H⋆ fit as a suitable C⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Therefore, there are exactly two essentially distinct fewest- turn near-Hamiltonian cycles in H, both of them asymmetric, and the induction step is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ 8 Further Work on Leaper Graphs In this section, we collect some additional results and open problems on leaper graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (To be defined shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') For the knight, it would be interesting to see a human-friendly proof of the upper bound in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Or, if not that, then at least it would be nice to know if there is an unweighted knight graph with pseudosnake density 1/2 which we could have used instead of the weighted one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' One natural direction of generalisation for our results in Section 6 is offered by leapers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let p and q be nonnegative integers with p ≤ q, not both zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' A (p, q)-leaper is a fairy chess piece which moves as a generalised knight, leaping p units away along one coordinate axis and q units away along the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let L be a (p, q)-leaper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The leaper graph of L on a set of cells S, denoted G(L, S), is defined similarly to the king and knight graphs on S, except that the adjacency condition becomes {|x′ − x′′|, |y′ − y′′|} = {p, q} instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let d = gcd(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the leaper graph of L on the board of size m × n is the disjoint union of several isomorphic copies of the leaper graphs of a (p/d, q/d)-leaper on the boards of sizes ⌊m/d⌋ × ⌊n/d⌋, ⌊m/d⌋ × ⌈n/d⌉, ⌈m/d⌉ × ⌊n/d⌋, and ⌈m/d⌉ × ⌈n/d⌉, each copy scaled up by a factor of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus we can safely assume that d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For p and q relatively prime, L is known as free when p + q is odd and half-free when it is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Briefly, one reason for this distinction is that G(L, Z2) is connected when L is free but consists of two connected components when L is half-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' A skew leaper is one for which p and q are positive and distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The only non-skew leapers with relatively prime p and q are the (0, 1)-leaper, known as the wazir, and the (1, 1)-leaper, known as the fers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Of course, the wazir graph on a board coincides with the 32 grid graph on that board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, G(Fers, m × n) can also be viewed as the direct product of two paths with m and n vertices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We take a look at the wazir and fers first, and after that we will focus on skew leapers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Even though Question A for grid graphs on rectangular boards is a very natural thing to ask, the only earlier reference for it known to the author as of the time of writing is [6], a puzzle game website where players are invited to construct snake paths in grid graphs on square boards, with longer paths scoring higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' An asymptotic estimate is straightforward to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The argument we give for the upper bound is not new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' it is essentially identical to the argument used in [4] to bound from above the pseudosnake density of G(□, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' (We discuss one natural way to define the pseudosnake density of certain infinite graphs below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=') For the lower bound, the general strategy we outlined in Section 6 goes through without a hitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Once again, [4] contains the same pseudosnake in G(□, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let m and n be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then both the longest snake path and the longest snake cycle in the grid graph of size m×n are of length 2mn/3+O(m+n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let A be the standard board of size m × n and let G be the grid graph on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the upper bound, let P be a snake path in G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' the argument for cycles is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let S be the vertex set of P and let T be the complement of S within A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then nearly every cell of S is adjacent to two cells of T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' each exception is either an endpoint of P or near the boundary of A, and so there are O(m + n) of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' On the other hand, every cell of T is adjacent to at most four cells of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus 2|S| ≤ 4|T| + O(m + n), and so |S| ≤ 2|S ∪ T|/3 + O(m + n) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We move on to the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let S∞ be the set of all cells (x, y) in Z2 with x ̸≡ y (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then S∞ induces a pseudosnake P∞ in G(□, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose without loss of generality that m ≥ 10 and n ≥ 10, let A⋆ be the subboard of A given by A⋆ = [4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' n − 5] × [4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' m − 5], and also let P ⋆ be the restriction of P∞ to A⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then P ⋆ is the disjoint union of several paths, and it is straightforward to add several cells out of A \\ A⋆ to P ⋆ so as to stitch these paths together into a single snake path or cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ The fers can be handled similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let m and n be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then both the longest snake path and the longest snake cycle in the fers graph of size m×n are of length mn/3+O(m+n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The proof is analogous to that of Proposition 3, and we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For wazir and fers graphs on rectangular boards, it might be possible to obtain exact answers to Question A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' By way of experimental data, [6] contains a table listing the greatest length of a snake path in the grid graph of size n × n for all n with 2 ≤ n ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We continue on to skew leapers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, for concreteness, that p < q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' It seems highly likely that an exact answer to Question A would be out of reach for skew leapers on arbitrary rectangular boards, or even on arbitrary square boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For this reason, we propose a weakened version of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 33 Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let L be a skew leaper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' What are some interesting lower and upper bounds for the greatest length of a snake path or cycle of L on a given rectangular board?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let us pick some of the low-hanging fruit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When L is half-free, let Free(L) denote the free (p′, q′)-leaper with p′ = (q −p)/2 and q′ = (p + q)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, now, that L is a skew free leaper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We will consider this case first, and then for Proposition 6 we will reduce the half-free case to the free case using the transformation introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' In all of the following asymptotic estimates, the implicit constants in the O-terms depend on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' One construction will be particularly useful to us, and so we introduce special notation for it: Let P be a pseudosnake in G(L, m × n) with vertex set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consider the union T of all sets of cells of the form ((n + q)i, (m + q)j) + S, over all integers i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the induced subgraph on vertex set T is a pseudosnake in G(L, Z2), as the translation copies of P which this subgraph consists of are too far away from one another to interact in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We denote this pseudosnake by Υ(m × n, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Now let τn be the pseudosnake density of G(L, n × n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Observe that the sequence {τn}∞ n=1 converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Indeed, let n2 ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since we can cover the board of size N×N with ⌈N/n⌉2 subboards of size n×n, we get that τN ≤ τn+O(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' On the other hand, fix a largest pseudosnake P in G(L, n × n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the restriction of Υ(n × n, P) to the board of size N × N is a pseudosnake in G(L, N × N), and so τN ≥ τn + O(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We denote τ(L) = limn→∞ τn, and we call this the pseudosnake density of G(L, Z2) or, for short, the pseudosnake density of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let L be a skew free leaper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' What is the pseudosnake density of L?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Or, alternatively, what are some interesting lower and upper bounds for it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Consider the four-dimensional infinite grid graph G(□, Z4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We define its pseudosnake density η similarly to how we defined τ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Observe that η is an absolute constant which does not depend on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The pseudosnake densities of infinite grid graphs with arbitrarily many dimensions have been studied before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [4] Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For all skew free leapers L, the pseudosnake density of L satisfies 1/2 ≤ τ(L) ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the lower bound, it suffices to exhibit a doubly periodic pseudosnake in G(L, Z2) with density 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since L is free, exactly one of p and q is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Denote that even value by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When r ≡ 2 (mod 4), let S be the set of all cells (x, y) such that ⌊x/2⌋ is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Otherwise, when r ≡ 0 (mod 4), let S be the set of all cells (x, y) such that ⌊x/2⌋ + y is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then the induced subgraph of G(L, Z2) on vertex set S works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' For the upper bound, let n be a positive integer, fix a largest pseudosnake P in G(L, n × n), and let Q = Υ(n × n, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then Q is a doubly periodic pseudosnake in G(L, Z2) with density τn + O(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 34 Consider, now, the induced subgraph R of G(□, Z4) whose vertex set consists of all four-dimensional integer points (x1, x2, x3, x4) such that x1(p, q) + x2(q, p) + x3(−p, q) + x4(−q, p) is a cell of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then R is a quadruply periodic pseudosnake in G(□, Z4) with the same density as Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' We let n grow without bound, and the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ Had it been the case that η = 1/2, Proposition 5 would have resolved Question 2 immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' However, it has been demonstrated that 649/1296 ≤ η ≤ 20/39, with 649/1296 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='50077 and 20/39 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='51282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [4] Still, Proposition 5 and this result together imply, for all skew free leapers L, that 1/2 ≤ τ(L) ≤ 20/39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since the gap between these two bounds is rather narrow, and the graph G(L, Z2) is, in some intuitive sense, more crowded than G(□, Z4), it seems plausible that in fact τ(L) = 1/2 for all skew free leapers L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' As we saw in Section 6, this is certainly true of the knight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' The natural connection between pseudosnake density and snake paths and cycles is as follows: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let L be a skew leaper and let m and n be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' When L is free, the greatest length of a snake path or cycle of L on the board of size m × n does not exceed τ(L) · mn + O(m + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Furthermore, when L is half-free, it does not exceed τ(L)/2 · mn + O(m + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Let P be a snake path or cycle in G(L, m × n) with s cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose first that L is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Since Υ(m × n, P) is a doubly periodic pseudosnake in G(L, Z2), its density s/(m + q)(n + q) does not exceed τ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Suppose, now, that L is half-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Then ε = (x + y) mod 2 is constant over all cells (x, y) of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Instead of Υ(m × n, P), take its image under the transformation (x, y) → ((x − y + ε)/2, (x + y + ε)/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' This is a pseudosnake in G(Free(L), Z2), and from this point on the argument continues as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' □ The author finds it reasonably plausible that the upper bounds of Proposition 6 might in fact be attained for all skew leapers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' As we saw in Section 6, this is indeed the case for the knight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Note that our construction for the lower bound in the proof of Proposition 5 yields a doubly periodic pseudosnake in G(L, Z2) where all cells are of degree exactly two and there are no finite cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Thus for free leapers L with τ(L) = 1/2 and their corresponding half-free leapers, this construction might play a role in a proof that the upper bounds of Proposition 6 are attained which follows some variant of the strategy we outlined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Acknowledgements The author would like to thank Professor Donald Knuth for introducing him to the subject of the longest snake paths and cycles in chess piece graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 35 References [1] Thomas Dawson, Échecs Féeriques, L’Échiquier, volume 2, issue 2, 1930;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' issue 3, 1931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [2] Neil Sloane, My Favorite Integer Sequences, Sequences and Their Applications: Pro- ceedings of SETA ’98, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Revised at http://neilsloane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='com/doc/sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [3] George Jelliss, Knight’s Tour Notes, website, section Wazir Wanderings, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Inter- net Archive snapshot at https://web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='org/web/20020206053321/http:// www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='ktn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='freeuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='com/9c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='htm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Revised and collected in George Jelliss, Knight’s Tour Notes, twelve-volume monograph, volume 2 (Walker Tours), 2019, https:// www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='mayhematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='com/p/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='htm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [4] Martín Matamala, Erich Prisner, and Ivan Rapaport, k-Pseudosnakes in Large Grids, LATIN 2002: Theoretical Informatics, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [5] Donald Knuth, The Art of Computer Programming, volume 4, pre-fascicle 5c (section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='1, Dancing Links), 2019, https://cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='edu/~knuth/fasc5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='gz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Revised and collected in Donald Knuth, The Art of Computer Programming, volume 4B (Combinatorial Algorithms, Part 2), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [6] David Radcliffe, Build-a-Snake, website, 2020, https://snake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='radcliffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content='dev/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Game idea by Christopher Danielson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' Table of optimal path lengths by Alain Goupil and Andrew Howroyd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' [7] Kendall Golder, Minimum Turn Hamiltonian Paths on Rectangular Grids, thesis presented for the degree of Master of Science, Emporia State University, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} +page_content=' 36' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAzT4oBgHgl3EQfNvsP/content/2301.01152v1.pdf'} diff --git a/lNE4T4oBgHgl3EQftQ2k/content/tmp_files/2301.05223v1.pdf.txt b/lNE4T4oBgHgl3EQftQ2k/content/tmp_files/2301.05223v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..098e91d1fac607b0a4b33c02cd9d70495a88aea6 --- /dev/null +++ b/lNE4T4oBgHgl3EQftQ2k/content/tmp_files/2301.05223v1.pdf.txt @@ -0,0 +1,1089 @@ +NOPA: Neurally-guided Online Probabilistic Assistance +for Building Socially Intelligent Home Assistants +Xavier Puig∗, Tianmin Shu∗, Joshua B. Tenenbaum, Antonio Torralba +Abstract— In this work, we study how to build socially +intelligent robots to assist people in their homes. In particular, +we focus on assistance with online goal inference, where robots +must simultaneously infer humans’ goals and how to help +them achieve those goals. Prior assistance methods either lack +the adaptivity to adjust helping strategies (i.e., when and +how to help) in response to uncertainty about goals or the +scalability to conduct fast inference in a large goal space. +Our NOPA (Neurally-guided Online Probabilistic Assistance) +method addresses both of these challenges. NOPA consists of +(1) an online goal inference module combining neural goal +proposals with inverse planning and particle filtering for robust +inference under uncertainty, and (2) a helping planner that +discovers valuable subgoals to help with and is aware of +the uncertainty in goal inference. We compare NOPA against +multiple baselines in a new embodied AI assistance challenge: +Online Watch-And-Help, in which a helper agent needs to +simultaneously watch a main agent’s action, infer its goal, and +help perform a common household task faster in realistic virtual +home environments. Experiments show that our helper agent +robustly updates its goal inference and adapts its helping plans +to the changing level of uncertainty.1 +I. INTRODUCTION +There has been growing interest in engineering socially +intelligent robots that can safely and productively work with +humans in the real world. Prior work on robot assistance has +achieved some success in scenarios where robots are given +the true human goals a priori or only need to help humans +in simple environments with a small state space. However, it +remains very challenging to build robot assistants that can +help humans perform all the activities of daily life in more +natural settings, such as in our homes, where the space of +human goals is vast and a person’s goal at any point in time +will not generally be known with certainty. +Our goal here is to build robot assistants that are able to +help people perform a wide range of tasks in complex home +environments. Our robot assistants must have the ability to +infer the true goals of humans based on past observations in +an online fashion, plan how to help humans without disrupting +them, and adapt to their behaviors by simultaneously updating +goal inference and helping strategies as the task progresses +(as illustrated in Fig. 1). Such ability has proven difficult for +robots to date due to two main technical obstacles. On the +one hand, online goal inference in realistic environments is +extremely difficult due to large state, action, and goal spaces; +on the other hand, inaccurate or ambiguous goal inferences +∗ Equal contribution. All authors are with MIT. {xpuig, tshu, jbt, +torralba}@mit.edu +1Code +is +available +at +https://github.com/xavierpuigf/ +online_watch_and_help. Project website: https://www.tshu. +io/online_watch_and_help. +Cabinet +Dining Table +Cabinet +Action +Walk to cabinet +Fridge +Kitchen +Counter +Kitchen +Dining Room +Living Room +Sofa +1 +2 +1 +Action +Grab 2 forks +2 +Action +Put 2 forks on dining table +3 +2 +Coffee Table +TV +3 +Cabinet +1 +3 +Inferred Goals +Apple on dining table +Apple in Fridge +Apple inside fridge +Plate(s) + Fork(s) on dining table +Plate(s) + Fork(s) on coffee table +Glass inside cabinet +Glass on dining table +Help +No action +Inferred Goals +Apple on dining table +Apple in Fridge +Apple inside fridge +2 Plates + 2 Forks on dining table +2 Plates + 2 Forks on coffee table +Glass inside cabinet +Glass on dining table +Help +Hand over 2 plates to human +Inferred Goals +Apple on dining table +Apple in Fridge +Apple inside fridge +2 Plates + 2 Forks on dining table +2 Plates + 2 Forks on coffee table +Glass inside cabinet +Glass on dining table +Help +Put 2 plates on dining table +Human +Robot +Fig. 1: Illustration of successful online assistance. The robot +initially has no knowledge about the human’s goal and thus +would opt to observe. As it observes more human actions, it +becomes more and more confident in its goal inference, so it +would dynamically adjust its helping subgoal. For instance, in +this figure, the robot first sees the human walking towards a +cabinet and consequently infers that the goal involves objects +inside the cabinet. After the human grabs 2 forks, the robot +infers that the goal is to put 2 sets of dining pieces (plates and +forks) on the dining table or the coffee table but is uncertain +about the goal location. Thus, it hands over 2 plates to the +human instead of randomly guessing a location. +often lead to ineffective or even counterproductive attempts +to help in systems that are not aware of their own uncertainty. +To address these challenges, we propose a novel online as- +sistance method, NOPA (Neurally-guided Online Probabilistic +Assistance). As illustrated in Fig. 2a, NOPA consists of two +main components: (1) a neurally-guided online goal inference +module and (2) an uncertainty-aware helping planner. The +neurally-guided online goal inference module first produces +bottom-up goal proposals from a neural network and then +maintains a set of predictions of goals and future trajectories +consistent with the observed actions via particle filtering and +inverse planning. This ensures that inferences are both fast +and robust. Given the latest predictions and their certainty, the +helping planner first identifies a subgoal that is most valuable +to help with and then plans the corresponding helping actions +using a symbolic planner. The resulting helping plan can adapt +to all levels of uncertainty in the predictions. For instance, +when there are multiple possible target locations for a goal +object, the robot assistant will deliver the object to the human +agent instead of risking misplacing the object. +For evaluation, we present a new embodied AI assis- +arXiv:2301.05223v1 [cs.RO] 12 Jan 2023 + +Goal Proposal Net +apple chips +main +in +fridge kitchen +table +kitchen +cabinet +in +on + +AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GPRi8eK9gPaWDbTbt0swm7E6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvSKQw6Lrf +TmFldW19o7hZ2tre2d0r7x80TZxqxhslrFuB9RwKRvoEDJ24nmNAokbwWjm6nfeuLaiFg94DjhfkQHSoSCUbTSvXnEXrniVt0ZyDLx +clKBHPVe+avbj1kacYVMUmM6npugn1GNgk+KXVTwxPKRnTAO5YqGnHjZ7NTJ+TEKn0SxtqWQjJTf09kNDJmHAW2M6I4NIveVPzP6QY +XvmZUEmKXLH5ojCVBGMy/Zv0heYM5dgSyrSwtxI2pJoytOmUbAje4svLpHlW9S6q7t15pXadx1GEIziGU/DgEmpwC3VoAIMBPMrvDnS +eXHenY95a8HJZw7hD5zPH2m/jeI= +st +hold +cupcake +… +… +… +… +… +… +… +… +AB9XicbVBNS8NAEJ34WetX1aOXYBE 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We represent +states and goals using scene graphs. st is the state at time t; at +M is the main agent’s action at time t; ˆgk is the k-th goal +proposal; and ˆΓk is the prediction of the main agent’s future trajectory corresponding to ˆgk. (b) Goal proposal network. +∆st is a matrix encoding the difference between the predicate counts in the states st and s0, where each row is a one-hot +vector, indicating the change in the count of a specific predicate. p = 136 is the number of all predicate types, c = 9 is the +maximum number of counts, and d = 100 and h = 128 are the dimensions of the intermediate layers. +tance challenge, Online Watch-And-Help (O-WAH), which +builds off a recent multi-agent virtual home environment, +VirtualHome-Social. Unlike most existing challenges (e.g., +[1], [2]), in O-WAH, a helper agent needs to infer the goal +of a main agent in an online fashion and simultaneously help +achieve the inferred goal as efficiently as possible. We evaluate +agents built with NOPA and several baselines in a range of +household tasks, helping the main agent controlled by either +a human player or a planner-based agent. The experimental +results show that NOPA significantly outperforms all baselines. +We are also able to observe intelligent helping strategies +emerging from NOPA adapting to new observations. +In sum, our main contribution includes (1) a new online +assistance method for building socially intelligent home +assistants in complex settings and (2) an embodied AI +assistance challenge, Online Watch-And-Help, as a testbed for +training and testing embodied agents to perform online goal +inference and helping in realistic virtual home environments. +II. RELATED WORK +Online goal inference. Online goal inference approaches +generally fall into two categories – (1) feedforward prediction +that directly maps observed past trajectories to possible goals, +typically enabled by goal prediction networks [3]–[11], or +(2) generative approaches such as inverse planning [12]–[19] +which conduct inference by comparing generated plans of +given goal hypotheses with observed actions. Feedforward +methods are fast and can perform well in simple tasks (such +as destination prediction for pedestrians [5]) when trained +with a large amount of data. However, in unfamiliar scenar- +ios, inverse planning methods often outperform feedfoward +prediction due to their ability to imagine rational behaviors +under various conditions. One of the limitations of inverse +planning methods is that they can be slow if the goal space +is large and rely on manually designed heuristics to speed +up the inference [17], [18]. Our work integrates these two +types of approaches to achieve both speed and robustness. +Embodied AI assistance with unknown goals. There has +been a rich history of research on embodied AI assistance. +Many of the existing works assume a known common goal +shared among human and AI/robot partners [1], [20]–[27]. +However, in the real world, robots often need to infer humans’ +goals on the fly. There has been work on helping with +inferred goals [2], [28]–[32] which shows that an accurate +goal inference can improve the objective and perceived +performance of embodied AI assistants. However, when the +goal inference is uncertain, helping with inferred goals often +leads to counterproductive behaviors such as undoing finished +goals [2]. For this, some prior work devised planners under +uncertain goal inference in simple environments [33]. A +recent study proposed a goal-agnostic assistance framework +via empowerment [34], which aims at changing the states +to maximize an agent’s ability to reach as many goals as +possible regardless of its true intent. Despite its success +in certain domains, assisting humans in real-world settings +without the knowledge of their goals would often result in +counterproductive behaviors. Our work investigates how to +design an uncertainty-aware planner that intelligently adjusts +the helping behavior ranging from goal-agnostic strategies to +goal-specific plans in a complex environment. +Embodied AI assistance benchmarks. There have been +benchmarks designed to evaluate AI agents’ ability to +collaborate with human teammates [1], [25]. However, most +of them focus on simple game environments and assume +a common goal given to both the AI and human agents a +priori. Based on a realistic virtual home environment, [2] +proposed a challenge, Watch-And-Help, in which a helper +agent must infer a main agent’s goal from a pre-recorded +demonstration, and then help the main agent to achieve the +same goal. We extend this challenge to an online assistance +setting, where the demonstration is no longer available. As a +result, helper agents have to pay attention to the main agent’s +actions and constantly update the inference and its uncertainty +while working towards inferred goals, which is closer to what +robots are expected to do in real homes. Such benchmark +complements conventional robot assistance studies conducted +in lab environments [20], [21], providing a reproducible and + +Algorithm 1 NOPA +1: Input: Γ0 +M = {(s0, a0 +M)}, st, K, Tmax, Tprop, q, wr, wc, wm, Lmax +2: t ← 1, l ← 0 +3: Q ← ∅ +4: repeat +5: +Q, l ← GoalInf(t, Q, Γt−1 +M +, st, q, K, l, Tprop) +6: +Γt +H ← HelpPlanner(Q, s0, st, wr, wc, wm, Lmax) +7: +Execute the first action from the helping plan at +H +8: +Observe at +M, st+1 from the environment +9: +t ← t + 1 +10: until t = Tmax or the true goal has not been reached +scalable way to compare different methods. +III. METHOD +A. Problem Setup +We define the online assistance problem as a mixed- +observability Markov decision process (MOMDP) [35], where +a helper agent needs to infer a main agent’ goal and help +the main agent achieve its goal faster. This can be formalized +by ⟨S, G, AH, O, TS, TG, Z, RH, γ⟩. The overall state has two +components: the world state, s ∈ S, which is fully observable +to both agents, and the main agent’s goal, g ∈ G, which is +partially observable to the helper agent. AH is the action space +of the helper agent. The helper agent’s observation consists of +the world state and the main agent’s action, i.e., O = S ×AM. +TS(s, g, aH, s′) = p(s′|s, g, aH) is the transition function for +the world state, and TG(s, g, aH, s′, g′) = p(g′|s, g, aH, s′) is +the transition function for the goal. Z(s′, g′, aH, o) = p(o = +(s, aM)|s′, g′, aH) is the conditional probability function for +the observation result. At step t + 1, the helper infers the +main’s goal given main’s past trajectory upon t, i.e., Γt = +{(sτ, aτ +M)}t +τ=1. The expected reward function for the helper +agent is defined as RH(s, a|Γt) = Ep(g|Γt)[1(s = g)]−cH(a), +where cH(a) is the cost for action a, and 1(·) checks if the +goal is satisfied in the current world state s. γ is the discount +factor. Note that the assumption that the world states are +fully observable for both agents is common in prior work on +assistance with unknown goals [29], [31], [32]. +B. Method Overview +To solve the online assistance problem formalized above, +we propose NOPA (Neurally-guided Online Probabilistic +assistance). Fig. 2a provides an overview of NOPA, showing +the two main components: i) Neurally-guided online goal +inference, and ii) an uncertainty-aware helping planner. As +sketched in Algorithm 1, NOPA updates a set of particles +conditioned on observed states and the main agent’s actions. +Each particle corresponds to a possible final goal. Common +assistance frameworks [2], [29], [31], [32] typically only +consider the final goal for helping. However, when there +is uncertainty in the goal inference, an intelligent assistant +should seek intermediate subgoals that can be helpful with +high certainty. For that, we also predict the main agent’s future +trajectory for each particle. We represent both intermediate +states and final goals as a set of edges in a scene graph [36], +[37], ⟨O, E⟩, as shown in Fig. 2a. Each node, o ∈ O, repre- +sents an entity (agent/object); each edge, e ∈ E, corresponds +Algorithm 2 GoalInf +1: Input: t, Q, Γt−1 +M += {(sτ, aτ +M)}t−1 +τ=0, st, q, K, l, Tprop +2: Output: Updated proposals Q′ and steps since last proposal l′ +3: Q′ ← ∅ +4: if Q ̸= ∅ and l < Tprop then +5: +for k = 1, · · · , |Q| do +6: +if at−1 +M +is part of the plan ˆΓk then +7: +Q′ ← Q′ ∪ {(ˆgk, ˆΓk)} +8: +end if +9: +end for +10: end if +11: if Q′ = ∅ then +12: +for k = 1, · · · , K do +13: +ˆgk ∼ q(g|s0, st) // Sample a goal proposal +14: +ˆΓk ← MCTS(st, ˆgk, Tprop) // Sample a plan +15: +Q′ ← Q′ ∪ {(ˆgk, ˆΓk)} +16: +end for +17: +l′ ← 0 +18: else +19: +l′ ← l + 1 +20: end if +21: return Q′, l′ +to a predicate (e.g., IN(apple, kitchencabinet)), +indicating the spatial relationship between two entities. Such +representations have been widely adopted in robotics and +embodied AI [38]–[41]. Given the particles, the helping +planner assesses the value of the edges in the intermediate +states and the final goals and selects the most valuable edge +as the helping subgoal. We introduce the details below. +C. Neurally-guided Online Goal Inference +Unlike prior work on online goal inference, the objective +of the online goal inference in this work is to help the +downstream task, i.e., assistance. This poses additional +challenges: (1) the helper agent has to estimate the uncertainty +in the inference instead of only predicting the most probable +goal; (2) it has to ensure that the inference is resilient in +a dynamic environment; and (3) the inference has to be +efficient so that the helper can have a prompt reaction to offer +assistance. For this, we propose a neurally-guided online goal +inference algorithm as summarized in Algorithm 2, which +combines inverse planning and a neural network. +We use a goal proposal network (GPN), as depicted in +Fig. 2b, to learn a proposal distribution q(g|s0, st), from +which we can sample K goal proposals {ˆgk}K +k=1 given the +initial state s0 and the current state st. Each goal proposal is +a set of goal predicates. Here, we only consider the first state +and the current state instead of a sequence of past states for the +input to GPN so that the GPN trained on episodes with only +the main agent performing the tasks can be robustly applied +to the helping condition where the sequence of state changes +could become very different from the training sequences. +We use inverse planning to evaluate the goals proposed by +the GPN and reject the ones that are inconsistent with the +observed main agent’s actions. To model an agent’s behavior +given each goal ˆgk with bounded rationality, we use the built- +in planner to predict the future trajectory of the main agent in +the next Tprop steps ˆΓk = {(ˆst+τk, ˆat+τk)}Tprop +τ=1. Specifically, +the level of rationality can be adjusted by the number of + +Algorithm 3 HelpPlanner +1: Input: Q, s0, st, wr, wc, wm, Lmax +2: Output: helping plan Γt +H +3: for e ∈ E do +4: +LM(e) ← ∞ +5: +p(e) ← 0 +6: +V (e) ← −∞ +7: +ΓH(e) ← ∅ +8: +if e appears in the initial state then +9: +LM(e) ← 0 +10: +p(e) ← 1 +11: +else +12: +for k = 1, · · · , |Q| do +13: +if e appears in the future traj. ˆΓk then +14: +Let τk(e) be the first step when e appears in ˆΓk +15: +LM(e) ← min(LM(e), τk(e)) +16: +p(e) ← p(e) + 1/|Q| +17: +else if e appears in the predicted goal ˆgk then +18: +LM(e) ← min(LM(e), Lmax) +19: +p(e) ← p(e) + 1/|Q| +20: +end if +21: +end for +22: +end if +23: +if p(e) > 0 then +24: +ΓH(e) ← MCTS(st, {e}, ∞) // Helper’s plan for subgoal e +25: +LH(e) ← |ΓH(e)| +26: +Compute V (st, e) as Eq (1) +27: +end if +28: end for +29: e∗ ← arg max V (st, e) +30: return ΓH(e∗) +simulations and the length of rollouts. We then create K +particles Q = {ˆgk, ˆΓk}. Whenever the main agent takes +a new action, at +M, we check if it is part of the predicted +plan for each particle. If for a particle k, the action is not +included in the predicted plan, then it suggests that the rational +behavior under the corresponding goal is not consistent with +the observed action. Thus the goal is likely to be wrong and +the particle needs to be rejected. When there is no particle left +or we have reached the prediction horizon Tprop, we resample +another K goals from the GPN based on the latest state +and create new particles. Since some particles may share the +same goal but have different predicted plans, our approach +can consider different ways to reach a goal. +D. Uncertainty-aware Helping Planner +Finding the optimal helping plan at each step is expensive. +To balance the speed and the optimality, our helping planner +(Algorithm 3) focuses on finding valuable subgoals instead of +updating the whole plan at each step. It considers all edges +that appear in the final goal and the intermediate states in the +predicted main agent’s plans as candidate helping subgoals. +Additionally, the helper agent may find that the objects it +grabs are no longer needed when it updates goal inference or +after the main agent achieves the corresponding subgoals. To +allow the helper agent to return those objects to their initial +locations, the helping subgoal space also includes edges in +the initial state. For each edge e, we estimate how long it +would take the main agent to reach that subgoal, LM(e). If +this edge appears in one of the predicted trajectories in the +particles, we can conveniently estimate LM(e) based on when +it appears in the trajectories. If it only appears in the final +goals, we then use a fixed length, Lmax, to anticipate how +many steps it would take the main agent to reach that subgoal. +We can also use MCTS to search for a plan for the helper +agent, ΓH(e), to reach the same subgoal. Let LH(e) be the +length of the helper’s plan, we then define the benefit of +helping with subgoal e as the speed up the helper agent can +offer by reaching the subgoal e, i.e., max(LM(e)−LH(e), 0). +To account for the uncertainty in inference, we estimate how +likely e is going to be necessary, p(e), by counting how many +particles include e in either the intermediate states or the +final goal. Finally, we define a value function for selecting +the best subgoal for the helper agent: +V (st, e) = +wrp(e)|LM(e) − LH(e)|+ − wcLH(e) +−wm(D(s0, ˆs(e)) − D(s0, st)), +(1) +where wr, wc, and wm are constant weights; D measures +the difference between two states; and ˆs(e) is the state after +reaching the subgoal e from the current state st. The three +terms in Eq. (1) evaluate i) the expected benefit of helping +reach the subgoal, ii) the cost of the helper agent, and iii) +the additional state change (compared with the initial state) +introduced by the subgoal. These three terms make sure that +the helper agent selects a subgoal that i) is likely to speed up +the task with high certainty, ii) is not too costly for the helper +agent, and iii) could restore the initial states of objects that +are not needed for the task respectively. Given the subgoal +e∗, we execute the first action of the helping plan ΓH(e∗). +IV. ONLINE WATCH-AND-HELP +To evaluate different assistance methods, we propose On- +line Watch-And-Help (O-WAH), an embodied AI assistance +challenge, in which a helper agent has to infer a main agent’s +goal and help reach the goal as fast as possible. This extends +an existing challenge, Watch-And-Help (WAH), to an online +assistance problem. O-WAH is built in a realistic multi-agent +virtual platform, VirtualHome-Social [2], simulating daily +household tasks (as shown in Fig. 3). The goal for each task +is defined by a set of predicates and their counts, representing +the target locations of different objects in the environment. +We sample each goal in the challenge from five general +types of household tasks: set table, put dishwasher, stock +fridge, prepare meal, and get snacks. Note that we define +task types only to ensure that the goals are emulating real-life +household tasks, but that this information is not provided to +the helper agents. As summarized in Table I, different kinds +of uncertainty may arise from these tasks: i) uncertainty in +the number of objects, ii) uncertainty in which objects are +needed, and iii) uncertainty in the target locations. Compared +to prior work, the goal space in O-WAH is 1 or 2 orders of +magnitude larger. We adopt the same action space in [2]. +To create a training episode, we first sample a goal and an +initial environment using one of the five training apartments +and then use a built-in planner to control the main agent +to perform the task alone. The built-in planner is the same +hierarchical planner as in [2]. We create a large training +set with 6,000 episodes and a small training set with 300 + +Helper agent: +infer Main’s goal +and help reach +the goal faster +Main agent: +set up a dinner table +Fig. 3: An example setup of O-WAH +in one of the simulated apartments. +TABLE I: The goal definition and the number of unique goals for each task type. +Task Name +Goal definition +#Goals +Set table +Put N plate, N fork, N OBJ on LOC, where N ∼ U(1,3), +12 +OBJ ∼ choice([waterglass, wineglass]), +LOC ∼ choice([kitchentable, coffeetable]) +Put dishwasher +Put N objects from OBJ_POOL in dishwasher, where N ∼ U(3,7), +315 +OBJ_POOL = [fork, plates, waterglass, wineglass] +Stock fridge +Put N objects from OBJ_POOL in fridge, where N ∼ U(3,7), +315 +OBJ_POOL = [salmon, apple, cupcake, pudding] +Prepare meal +Put N salmon, N apple, N OBJ on LOC, where N ∼ U(1,3), +18 +OBJ ∼ choice([cupcake, pudding]), +LOC ∼ choice([kitchentable, coffeetable, stove]) +Get snacks +Put 1 remote, 1 condiment, 1 chips on coffeetable +1 +a. +b. +c. +d. +Main controlled by a planner +Main controlled by humans +Fig. 4: (a) Speedup of different methods (striped bars indicate using the small training set). Errors are standard errors. (b) +F1-scores of the predicted goal over the course of a task. The x axis is normalized in proportion to the number of steps +needed for the main agent to perform each task alone. The curves show the means and the shaded regions show the standard +errors. c) F1-scores over time for different approaches in a single test episode, a dot indicates the number of steps a given +baseline took to complete the task. The dashed lines in (b) and (c) indicate using the small training set. (d) Results of the +human experiment. Here we show the speedup of different methods when the main agent is controlled by the built-in planner +or by human players (note that the results under the two conditions are based on the same 10 testing episodes). +episodes. The testing set has 100 episodes in the two testing +apartments unseen during training. +We use F1-score over the goal predicates to measure the +goal inference accuracy. To evaluate the helping performance, +we use speedup, where we compare the episode length when +the helper agent works with the main agent (LH) against the +episode length when the main agent works alone (LM), i.e., +LM/LH − 1. For each episode, set a time limit of 250 steps +and report the average performance across 3 runs. +V. EXPERIMENTS +A. Baselines +We compare NOPA against several baselines. +HPGPN: We adopt the best performing approach in the original +Watch-And-Help challenge [2] for this baseline, which is a +hierarchical planner (HP) based on the most probable goal +according to the GPN. In particular, at each step, HPGPN +uses the goal ˆg = arg maxg q(g|s0, st). +AFGPN: We extend HPGPN by using NOPA’s online goal +inference. We generate a plan for each predicated goal using +HP and execute the most frequent first action among all plans. +Empowerment: By adopting the idea of empowerment [34], +this baseline uniformly samples K goals at each step, predicts +plans and intermediate states for the goals, and selects the +most frequent edge in the intermediate state as the helping +subgoal (i.e., the most common subgoal for any goal). +HPRG: A hierarchical planner based on a randomly sampled +goal at the beginning of the episode. +We consider the following ablated methods to evaluate the +effect of different components of NOPA. +OursRG: We replace the proposal distribution q in Algo- +rithm 2 with a uniform distribution. +Ours-InvPlan: Ours without inverse planning. +Ours-Return: Ours without returning irrelevant objects to their +initial locations (wm = 0 in Eq.(1)). +By default, the GPN is trained on the large training set. To +evaluate the sample efficiency of NOPA, we also report the +performance of Ours and HPGPN, when the GPN is trained +on a small training set, indicated by the subscript GPN-S. +To measure the upper bound on the helping performance, +we also implement an oracle helper HPGT, which knows the +ground-truth goal and is controlled by an HP. +We set Tmax = 250, Tprop = 15, wr = 1, wc = 1, wd = 5, +and Lmax = 100 for NOPA. For all approaches that propose +multiple goals, we use K = 20 proposals. We train the GPN +using Adam [42] with a learning rate of .0009 and a batch +size of 256. +B. Results +1) Main Controlled by a Planner: We evaluate all methods +with a main agent controlled by the built-in planner and +report the helping speedup (average and standard error across +episodes) in Fig. 4a. For methods that have different goal +inference modules, we also report the F1-score of their goal +inference results in Fig. 4b. The speedup of the oracle agent, +operating with true knowledge about the goal, HPGT is 1.29. + +Grab fork +Grab wineglass +Walk +to dishwasher +Walk to plate +Grab wineglass +Grab fork +Main +Helper +Walk to kitchentable +Walk to wineglass +Put fork to kitchentable +Probability +Steps +Grab fork +Fig. 5: Goal inference and plans by NOPA for the same +task shown in Fig. 4c, which is setting up a kitchen table +for 3 persons. We show the posterior probabilities of the +top predicates and their counts based on the particles at +each step, key actions of the main agent (indicated by red +dots), and key helping actions (indicated by blue dots). At +step 2, after watching the main agent walking towards the +dishwasher, NOPA rejects proposals involving nearby objects +(e.g., apples, salmons) that are not inside of the dishwasher. +After the main agent grabs a fork at step 8, NOPA infers with +high confidence that the goal is setting up a table for at least +one person. So at the following step, the helper agent takes +its very first action – walking to grab a plate. Upon seeing +Main walking to the kitchen table at step 10, the goal location +becomes certain. After observing more actions, the inference +converges to setting up the kitchen table for 3 persons. +Helper grabs an apple. +Helper hands the apple over to +Main. +Main puts the apple in the +fridge, and Helper goes to grab +other objects. +Helper gets a fork, as Main +goes to get a fork too. +Main puts the fork on the table. +Helper infers that no more forks +are needed. +To avoid messing up the +environment, Helper puts back +the fork it grabbed. +a. Helper hands over an object to Main +b. Helper returning an extra object to its original location +Fig. 6: Examples of helping plans that are beyond directly +achieving final goals. Main is in red, and Helper is in blue. +NOPA (Ours) outperforms all baselines, offering the highest +speedup. It also achieves the best goal inference accuracy at +the early stage of the tasks, which serves as the foundation +of its successful assistance. This benefit can be more clearly +seen from Fig. 4c (the improvement margin appears to be +smaller since the temporal normalization for each episode is +different). The low speedup by Empowerment suggests that +online goal inference is necessary for effective assistance, +despite its success in certain domains shown in prior work +[34]. Given that the predicted goal may be uncertain, using +multiple goal proposals leads to a better helping performance, +as seen by comparing OursGPN with HPGPN. The effect is +more pronounced when the GPN is trained with fewer data +and is consequently less accurate (GPN-S). We also find +that the neurally-guided goal proposals can greatly improve +the goal inference over uniform goal proposals (OursRG). +Moreover, the results demonstrate that inverse planning is +important for filtering spurious goal proposals from GPN, +significantly improving the speedup over Ours-InvPlan since it +allows the goal inference to reach a relatively high accuracy +much earlier than Ours-InvPlan and other baselines do (see +Fig. 4c). Finally, by comparing NOPA with Ours-Return, we +can see a marginal improvement in speedup by avoiding +unnecessarily distorting the environment; Ours-Return also +causes 11.2% more unnecessary state changes. +Fig. 5 shows a typical successful example by NOPA, where +the task is the same as the one in Fig. 4c. It demonstrates +that NOPA can i) achieve accurate goal inference early on +by filtering out goal proposals that are inconsistent with the +main agent’s actions, ii) correctly update the goal inference +and its uncertainty estimation based on more observation, and +iii) plan for effective helping actions based on the filtered +goal proposals and the uncertainty in the inference. Note that +the helper remains idle but takes a useful helping action as +soon as the goal inference becomes confident and is able to +avoid grabbing extra objects thanks to its gradual update of +the number of objects needed. +We also observe diverse helping behaviors enabled by +NOPA that are not just about directly achieving the final +goals as shown in Fig. 6. First, the helper agent sometimes +selects a subgoal of handing over objects to the main agent. +For example, as illustrated in Fig. 6a, the helper agent hands +over the apple to the main agent who is right next to the +fridge so that the task execution can be faster. Second, the +helper agent can return extra objects to their initial locations +once it realizes that they are not needed for reaching the goal +(Fig. 6b). The supplementary video2 shows more examples. +2) Main Controlled by Humans: To evaluate how effective +helper agents are at assisting real humans, we conducted a +human experiment where the main agent is controlled by +human players. We used 10 testing episodes to run 40 trials. +In each trial, a human participant was asked to either perform +the task alone (to estimate the number of steps needed for +completing each task alone) or work with a helper agent +controlled by one of the three approaches, NOPA, HPGPN, +and HPRG. Participants did not know which helper agent they +were working with. We recruited 10 participants (mean age += 32.3; 4 female) who had no prior exposure to our system. +As shown in Fig. 4(d), the ranking of the methods remains +consistent when the main agent is controlled by human players. +There is no significant difference in NOPA’s performance +under the two conditions (t(9) = 0.87, ρ = 0.40). +2The supplementary video is available at https://youtu.be/Oawo9pynPL0. + +VI. CONCLUSION +In this work, we propose a novel method for building +socially intelligent home assistants, Neurally-guided Online +Probabilistic Assistance (NOPA), which integrates (1) a hybrid +online goal inference algorithm combining a goal proposal +network and inverse planning and (2) an uncertainty-aware +helping planner that identifies valuable helping subgoals from +both the final goals and intermediate states. For a systematic +and scalable evaluation, we introduce a new embodied AI +assistance challenge, Online Watch-And-Help, based on a +realistic virtual home platform. We evaluate NOPA with +several baselines in our challenge with a main agent controlled +either by a built-in planner or by humans. Our experiments +show that NOPA significantly outperforms baselines and +achieves great sample efficiency for training. In the future, +we plan to extend NOPA to a partial observability setting +and apply it to robots in real homes. +ACKNOWLEDGMENT +This work was supported by the DARPA Machine Common +Sense program, ONR MURI N00014-13-1-0333, and a grant +from Lockheed Martin. +REFERENCES +[1] M. Carroll, R. Shah, M. K. Ho, T. Griffiths, S. Seshia, P. Abbeel, +and A. Dragan, “On the utility of learning about humans for human- +AI coordination,” Advances in neural information processing systems, +vol. 32, 2019. +[2] X. Puig, T. Shu, S. Li, Z. Wang, Y.-H. Liao, J. B. Tenenbaum, S. Fidler, +and A. 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Ba, “Adam: A method for stochastic optimization,” +arXiv preprint arXiv:1412.6980, 2014. + diff --git a/lNE4T4oBgHgl3EQftQ2k/content/tmp_files/load_file.txt b/lNE4T4oBgHgl3EQftQ2k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e365254ffa5b40a52da3d1bdaea3148b324af37 --- /dev/null +++ b/lNE4T4oBgHgl3EQftQ2k/content/tmp_files/load_file.txt @@ -0,0 +1,1002 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf,len=1001 +page_content='NOPA: Neurally-guided Online Probabilistic Assistance for Building Socially Intelligent Home Assistants Xavier Puig∗, Tianmin Shu∗, Joshua B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' Tenenbaum, Antonio Torralba Abstract— In this work, we study how to build socially intelligent robots to assist people in their homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' In particular, we focus on assistance with online goal inference, where robots must simultaneously infer humans’ goals and how to help them achieve those goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' Prior assistance methods either lack the adaptivity to adjust helping strategies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=', when and how to help) in response to uncertainty about goals or the scalability to conduct fast inference in a large goal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' Our NOPA (Neurally-guided Online Probabilistic Assistance) method addresses both of these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' NOPA consists of (1) an online goal inference module combining neural goal proposals with inverse planning and particle filtering for robust inference under uncertainty, and (2) a helping planner that discovers valuable subgoals to help with and is aware of the uncertainty in goal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' We compare NOPA against multiple baselines in a new embodied AI assistance challenge: Online Watch-And-Help, in which a helper agent needs to simultaneously watch a main agent’s action, infer its goal, and help perform a common household task faster in realistic virtual home environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' Experiments show that our helper agent robustly updates its goal inference and adapts its helping plans to the changing level of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' INTRODUCTION There has been growing interest in engineering socially intelligent robots that can safely and productively work with humans in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' Prior work on robot assistance has achieved some success in scenarios where robots are given the true human goals a priori or only need to help humans in simple environments with a small state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' However, it remains very challenging to build robot assistants that can help humans perform all the activities of daily life in more natural settings, such as in our homes, where the space of human goals is vast and a person’s goal at any point in time will not generally be known with certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' Our goal here is to build robot assistants that are able to help people perform a wide range of tasks in complex home environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' Our robot assistants must have the ability to infer the true goals of humans based on past observations in an online fashion, plan how to help humans without disrupting them, and adapt to their behaviors by simultaneously updating goal inference and helping strategies as the task progresses (as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' Such ability has proven difficult for robots to date due to two main technical obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' On the one hand, online goal inference in realistic environments is extremely difficult due to large state, action, and goal spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' on the other hand, inaccurate or ambiguous goal inferences ∗ Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' All authors are with MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' {xpuig, tshu, jbt, torralba}@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='edu 1Code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='com/xavierpuigf/ online_watch_and_help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' Project website: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='tshu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' io/online_watch_and_help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Cabinet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Dining Table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Cabinet ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Cabinet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Inferred Goals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Apple on dining table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Apple in Fridge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Apple inside fridge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Plate(s) + Fork(s) on dining table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Plate(s) + Fork(s) on coffee table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Glass inside cabinet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Glass on dining table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Help ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='No action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Inferred Goals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Apple on dining table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Apple in Fridge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Apple inside fridge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='2 Plates + 2 Forks on dining table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='2 Plates + 2 Forks on coffee table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Glass inside cabinet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Glass on dining table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Help ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Hand over 2 plates to human ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Inferred Goals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Apple on dining table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Apple in Fridge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Apple inside fridge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='2 Plates + 2 Forks on dining table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='2 Plates + 2 Forks on coffee table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Glass inside cabinet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Glass on dining table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Help ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Put 2 plates on dining table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Human ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Robot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' 1: Illustration of successful online assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' The robot initially has no knowledge about the human’s goal and thus would opt to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' As it observes more human actions, it becomes more and more confident in its goal inference, so it would dynamically adjust its helping subgoal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' For instance, in this figure, the robot first sees the human walking towards a cabinet and consequently infers that the goal involves objects inside the cabinet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' After the human grabs 2 forks, the robot infers that the goal is to put 2 sets of dining pieces (plates and forks) on the dining table or the coffee table but is uncertain about the goal location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' Thus, it hands over 2 plates to the human instead of randomly guessing a location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' often lead to ineffective or even counterproductive attempts to help in systems that are not aware of their own uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' To address these challenges, we propose a novel online as- sistance method, NOPA (Neurally-guided Online Probabilistic Assistance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' 2a, NOPA consists of two main components: (1) a neurally-guided online goal inference module and (2) an uncertainty-aware helping planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' The neurally-guided online goal inference module first produces bottom-up goal proposals from a neural network and then maintains a set of predictions of goals and future trajectories consistent with the observed actions via particle filtering and inverse planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' This ensures that inferences are both fast and robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' Given the latest predictions and their certainty, the helping planner first identifies a subgoal that is most valuable to help with and then plans the corresponding helping actions using a symbolic planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' The resulting helping plan can adapt to all levels of uncertainty in the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' For instance, when there are multiple possible target locations for a goal object, the robot assistant will deliver the object to the human agent instead of risking misplacing the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content=' For evaluation, we present a new embodied AI assis- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='05223v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='RO] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='12 Jan 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='Goal Proposal Net ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='apple chips ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE4T4oBgHgl3EQftQ2k/content/2301.05223v1.pdf'} +page_content='main ' metadata={'source': 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r(k−1) +j +for k ∈ [K] with r(0) +j +:= 0 and r(K) +j +:= 0 for all j, we have +∆r(k) +j += +� +� +� +� +� +1 +K − qj +where k = gj, +1 +K − (1 − qj) +where k = hj, +1 +K +otherwise. +(2) +Note that ∆r(k) +j +has its minimum at k = gj and the second smallest value at k = hj for qj ∈ (1/2, 1]. If one +can specify gj, the task difficulty qj can also be revealed from +1 +K − ∆r(gj) +j +. In the next section, we use this +structure of r(k) for k ∈ [K] to infer the top two answers and the confusion probability.2 +3 +Proposed Algorithm +Our algorithm consists of two stages. In Stage 1, we compute an initial estimate on top-two answers and +the confusion probability q. In Stage 2, we estimate the worker reliability vector p by using the result of the +first stage, and use the estimated p and q to refine our estimates for the top two answers. Assume that we +randomly split the original response matrix A into A1 and A2 with probability s1 and 1 − s1, respectively, +and use only A1 for stage 1 and (A1, A2) for stage 2. +3.1 +Stage 1: Initial estimates using SVD +The first stage begins with randomly splitting A1 again into two independent matrices B and C with +equal probabilities. We then convert B and C into (K − 1)-binary matrices B(k) and C(k) as explained +in Sec. 2. Define X(k) and Y (k) as X(k) := B(k) − s′(K−2k) +K +1n×m and Y (k) := C(k) − s′(K−2k) +K +1n×m for +s′ = s · s1/2. We have E[X(k)] = E[Y (k)] = s′p(r(k))⊤ from Prop. 1. +We use X(k) and Y (k) to estimate p∗ := p/∥p∥2 and ∥p∥2r(k), respectively. The estimators are denoted +by u(k) and v(k), respectively. We define u(k) as the left singular vector of X(k) with the largest singular +value. Sign ambiguity of the singular vector is resolved by defining u(k) as the one between {u(k), −u(k)} +in which at least half of the entries are positive. After trimming abnormally large components of u(k) and +defining the trimmed vector as ˜u(k), we calculate v(k) := 1 +s′ (Y (k))⊤ ˜u(k), which is an estimate for ∥p∥2r(k). +By using v(k) for 1 ≤ k < K, we get estimates for top-two answers (ˆgj, ˆhj) based on the observation in +equation 2. Lastly, we estimate ∥p∥2 and use v(k)/∥p∥2 ≈ r(k) to estimate the confusion probability vector +q. See Algorithm 1 for details. +3.2 +Stage 2: Plug-in Maximum Likelihood Estimator (MLE) +The second stage uses the result of Stage 1 to estimate the worker reliability vector p. We first propose +an estimate for the worker reliability vector p by using the estimated top-two answers {(gj, hj)}m +j=1 from +Algorithm 1. We randomly split the original response matrix A into A1 and A2 with probability s1 and +2We assume that η√n ≤ ∥p∥2 ≤ √n for some η > 0, i.e., there are only o(n) spammers (pi = 0), and ∥r(k)∥2 = Θ(√m) for +every k ∈ [K], which can be easily satisfied except exceptional cases from equation 2. +4 + +Algorithm 1 Spectral Method for Initial Estimation (TopTwo1 Algorithm) +1: Input: data matrix A1 ∈ {0, 1, . . . , K}n×m and parameter η > 0 where η√n ≤ ∥p∥2 ≤ √n. +2: Randomly split (with equal probabilities) and convert A1 into binary matrices X(k) ∈ {−1, 0, 1}n×m +and Y (k) ∈ {−1, 0, 1}n×m for 1 ≤ k < K as described in Sec. 3.1. +3: Let u(k) be the leading normalized left singular vector of X(k). Trim the abnormally large components +of u(k) by letting it be zero if u(k) +i +> +2 +η√n and denote the resulting vector as ˜u(k). +4: Calculate the estimate of ∥p∥r(k) by v(k) := 1 +s′ (Y (k))⊤ ˜u(k). Assume v(0) := 0 and v(K) := 0. +5: For k ∈ [K], calculate ∆v(k) +j +:= v(k) +j +− v(k−1) +j +. Estimate the top-two answers for j ∈ [m] by +ˆgj := arg min +k∈[K] +∆v(k) +j +; +ˆhj := arg min +k̸=ˆgj,k∈[K] +∆v(k) +j +. +(3) +6: Estimate ∥p∥2 by defining lj := +K +K−2 +� +k̸=ˆgj,k̸=ˆhj ∆v(k) +j +and l := 1 +m +�m +j=1 lj. +7: Estimate qj for j ∈ [m] by defining +ˆqj := 1/K − ∆v(ˆgj) +j +/l. +(4) +8: Output: estimated top-two answers {(ˆgj, ˆhj)}m +j=1 and confusion probability vector ˆq. +Algorithm 2 Plug-in MLE (TopTwo2 Algorithm) +1: Input: data matrix A ∈ {0, 1, . . . , K}n×m and the sample splitting rate s1 > 0. +2: Randomly split A into A1 and A2 by defining A1 := A ◦ S and A2 = A ◦ (1n×m − S) where S is an +n × m matrix whose entries are i.i.d. with Bern(s1) and ◦ is an entrywise product. +3: Apply Algorithm 1 to A1 to yield estimates for top-two answers {(ˆgj, ˆhj)}m +j=1 and confusion probability +vector ˆq. +4: By using {(ˆgj, ˆhj)}m +j=1 and A2, calculate the estimate ˆp as in equation 5. +5: By using the whole A and (ˆp, ˆq), find the plug-in MLE estimates (ˆgMLE +j +, ˆhMLE +j +) by +arg max +a,b∈[K]2,a̸=b +n +� +i=1 +log +�K ˆpiˆqj +1 − ˆpi ++ 1 +� +1(Aij = a) + log +�K ˆpi(1 − ˆqj) +1 − ˆpi ++ 1 +� +1(Aij = b). +(6) +6: Output: estimated top-two answers {(ˆgMLE +j +, ˆhMLE +j +)}m +j=1. +1 − s1, respectively, and use A1 only for Algorithm 1 and A2 only for calculating the estimator ˆp. Our +estimate for the worker reliability pi is defined as +ˆpi = +K +(K − 2) +� +� +1 +s(1 − s1) +� +� 1 +m +m +� +j=1 +1(A2 +ij = ˆgj or ˆhj) +� +� − 2 +K +� +� . +(5) +Our plug-in MLE uses the estimated (ˆp, ˆq) in the place of (p, q) at the oracle MLE, which finds (ˆgj, ˆhj) ∈ +[K]2\{(1, 1), (1, 2), . . . , (K, K)} such that (ˆgj, ˆhj) := arg max(a,b)∈[K]2,a̸=b +�n +i=1 log P(Aij|p, qj, (a, b)) as in +equation 6. Details can be found in Alg.2. +The time complexity of Alg. 2 is O(m2 log m + nmK2), since the SVD in Alg. 1 can be computed +via power iterations within O(m2 log m) steps (Boutsidis et al., 2015), and the step for finding the pair of +answers maximizing equation 6 requires O(nmK2) steps. +5 + +4 +Performance Analysis +To state our main theoretical results, we first need to introduce some notation and assumptions. Let µ(i,j) +(a,b),k +denote the probability that a worker i ∈ [n] gives label k ∈ [K] for the assigned task j ∈ [m] of which +the top-two answers are (gj, hj) = (a, b). Note that µ(i,j) +(a,b),k can be written in terms of (pi, qj) from the +distribution in equation 1 for every a, b, k ∈ [K]3. +Let µ(i,j) +(a,b) = [µ(i,j) +(a,b),1 +µ(i,j) +(a,b),2 +· · · +µ(i,j) +(a,b),K]⊤. +We +introduce a quantity that measures the average ability of workers in distinguishing the ground-truth pair of +top-two answers (gj, hj) from any other pair (a, b) ∈ [K]2/{(gj, hj)} for the task j ∈ [m]. We define +D +(j) := +min +(gj,hj)̸=(a,b) +1 +n +n +� +i=1 +DKL +� +µ(i,j) +(gj,hj), µ(i,j) +(a,b) +� +; +D := min +j∈[m] D +(j), +(7) +where DKL(P, Q) := � +i P(i) log(P(i)/Q(i)) is the KL-divergence between P and Q. Note that D +(j) is strictly +positive if there exist at least one worker i with pi > 0 and qj ∈ (1/2, 1) for the distribution in equation 1, +so that (gj, hj) can be distinguished from any other (a, b) ∈ [K]2/{(gj, hj)} statistically. We define D as the +minimum of D +(j) over j ∈ [m], indicating the average ability of workers in distinguishing (gj, hj) from any +other (a, b) for the most difficult task in the set of tasks. +We split the performance analysis of our algorithm into two parts. First, Theorem 1 states the perfor- +mance guarantees for Alg. 1. +Theorem 1 (Performance Guarantees for Algorithm 1). For any ϵ, δ1 > 0, if the sampling probability +s · s1 = Ω +� +1 +δ2 +1∥p∥2 +2 log K +ϵ +� +, Algorithm 1 guarantees the recovery of the ordered top-two answers (gj, hj) with +probability at least 1 − ϵ for any j ∈ [m] with qj ∈ (1/2, 1), i.e., +P +� +(ˆgj, ˆhj) = (gj, hj) +� +≥ 1 − ϵ +for all j ∈ [m] with qj ∈ (1/2, 1), +(8) +and the recovery of the confusion probability qj with +P (|ˆqj − qj| < δ1) ≥ 1 − ϵ +for all j ∈ [m], +(9) +for every sufficiently large number m of tasks and the number of workers n = O(m/ log m). +By using Theorem 1, we can also find the sufficient conditions to guarantee the recovery of paired top-two +answers for all tasks and q with an arbitrarily small ℓ∞-norm error. +Corollary 1. For any ϵ, δ1 > 0, if the sampling probability s·s1 = Ω +� +1 +δ2 +1∥p∥2 +2 log mK +ϵ +� +, Algorithm 1 guarantees +the recovery of {(gj, hj)}m +j=1 and q with probability at least 1 − ϵ as m → ∞ such that +P +� +(ˆgj, ˆhj) = (gj, hj), ∀j ∈ [m] +� +≥ 1 − ϵ +and +P (∥q − ˆq∥∞ < δ1) ≥ 1 − ϵ. +(10) +Proofs of Theorem 1 and Corollary 1 are available in Appendix §G. +We next analyze the performance of Alg. 2, which uses Alg. 1 as the first stage. Before providing the +main theorem for Alg. 2, we state a lemma charactering a sufficient condition for estimating the worker +reliability vector p from equation 5 with an arbitrarily small ℓ∞-norm error. +Lemma 1. Conditioned on (ˆgj, ˆhj) = (gj, hj) for all j ∈ [m], if s(1 − s1) = Ω +� +1 +δ2 +2m log n +ϵ +� +, the estimator ˆpi +defined in equation 5 of Alg. 2 guarantees P (∥p − ˆp∥∞ < δ2) ≥ 1 − ϵ for any ϵ > 0. +Combining Corollary 1 and Lemma 1, we can have the estimators (ˆp, ˆq) for the worker reliability vector +p and the confusion probability vector q with ℓ∞-norm error bounded by any arbitrarily small δ > 0 with +probaiblity at least 1 − 2ϵ if +s = s · s1 + s(1 − s1) = Ω +�log(mK/ϵ) +δ2∥p∥2 +2 ++ log(n/ϵ) +δ2m +� += Ω +�log(mK/ϵ) +δ2∥p∥2 +2 +� +(11) +6 + +where the last equality is from the assumption that ∥p∥2 = Θ(√n) and n = O(m/ log m). In this regime, +the sample complexity for estimating the task difficulty q is larger than that for estimating worker reliability +p. To make sure that the sampling probability s < 1, we need n = Ω(log m). +Our second theorem, Theorem 2, characterizes the sufficient condition on the sampling probability s to +guarantee the recovery of the pair of top-two answers for all tasks by equation 6 of Alg. 2, when a sufficiently +accurate estimation of (p, q) is given. +Theorem 2. Assume that there is a positive scalar ρ such that µ(i,j) +(gj,hj),c ≥ ρ for all (i, j, gj, hj, c) ∈ [n] × +[m] × [K]3. For any ϵ > 0, if (ˆp, ˆq) are given with +max{∥p − ˆp∥∞, ∥q − ˆq∥∞} ≤ δ := min +�ρ +4, +ρD +4(6 + D) +� +, +(12) +and the sampling probability s = Ω +� +log(1/ρ) log(mK2/ϵ)+D log(m/ϵ) +nD +� +, then for any ϵ > 0 the estimates of +{(gj, hj)}m +j=1 from equation 6 of Algorithm 2 guarantees +P +� +(ˆgj, ˆhj) = (gj, hj), ∀j ∈ [m] +� +≥ 1 − ϵ. +(13) +Proofs of Lemma 1 and Theorem 2 are available in Appendix §H. The assumption in Theorem 2 that +µ(i,j) +(gj,hj),c ≥ ρ for some ρ > 0 holds if pi < 1 for all i ∈ [n], i.e., there is no perfectly reliable worker. This +assumption can be easily satisfied by adding an arbitrary small random noise to the worker answers as well. +By combining the statements in Corollary 1, Lemma 1, and Theorem 2 with δ1 = δ2 = δ for δ defined in +equation 12, we get the overall performance guarantee for Alg. 2. +Corollary 2 (Performance Guarantees for Alg. 2). Alg. 2 guarantees the recovery of top-two answers for all +tasks with P +� +(ˆgj, ˆhj) = (gj, hj), ∀j ∈ [m] +� +≥ 1 − ϵ for any ϵ > 0 if s satisfies +s = Ω +�log(mK/ϵ) +δ2∥p∥2 +2 ++ log(1/ρ) log(mK2/ϵ) + D log(m/ϵ) +nD +� += Ω +�log(m/ϵ) +δ2∥p∥2 +2 ++ log(m/ϵ) +nD +� +. +(14) +In equation 14, the first term is for guaranteeing accurate estimates of p and q with ℓ∞-norm error +bounded by δ and the second term is for guaranteeing the recovery of the top-two answers from MLE with +high probability. +Since ∥p∥2 +2 = Θ(n), the two terms effectively have the same order but with different +constant scaling, depending on model-specific parameters (p, q). +Lastly, we show the optimality of convergence rates of Alg. 1 and Alg. 2 with respect to two types of +minimax errors, respectively. The proof of Theorem 3 is available in Appendix §I. +Theorem 3. (a) Let Fp be a set of p ∈ [0, 1]n such that the collective quality of workers, measured by ∥p∥2, +is parameterized by p as F¯p := {p : 1 +n∥p∥2 +2 = p}. Assume that p ≤ 2/3. If the average number ns of samples +(queries) per task is less than (1/2p) log(1/ϵ), then +min +ˆg +max +p∈Fp, g∈[K]m +1 +m +� +j∈[m] +P(ˆgj ̸= gj) ≥ ϵ. +(15) +(b) There is a universal constant c > 0 such that for any p ∈ [0, 1]n, q ∈ (1/2, 1]m, if the sampling probability +s < Ω +� +1/(nD) +� +, then +min +(ˆg,ˆh) +max +(g,h)∈[K]m×[K]m +gj̸=hj,∀j[m] +1 +m +� +j∈[m] +P((ˆgj, ˆhj) ̸= (gj, hj)) ≥ c. +(16) +From part (a) of Theorem 3, it is necessary to have s > Ω +� +(1/∥p∥2 +2) log(1/ϵ) +� +to make the minimax +error in equation 15 less than ϵ. Since Theorem 1 shows that Alg. 1 recovers (ˆgj, ˆhj) with probability at +7 + +Figure 1: Prediction error for (g, h) for four scenarios as the avg. number of queries per task changes. Our +TopTwo2 alg. achieves the best performance, near the oracle MLE for all the scenarios. +least 1 − ϵ if s > Ω +� +(1/∥p∥2 +2) log(1/ϵ) +� +when s1 = 1, we can conclude that Alg. 1 achieves the minimax +optimal rate for a fixed collective intelligence of workers, measured by ∥p∥2. From part (b) of Theorem 3, +for any (p, q), unless we have s > Ω(1/(nD)) there always exists a constant fraction of tasks for which the +recovered top-two answers are incorrect. This bound matches with our sufficient condition on s from Alg. +2 in equation 14 upto logarithmic factors, as long as δ2∥p∥2 ≳ nD, showing the minimax optimality of our +Alg. 2 for any (p, q). More discussions on the theoretical results are available at Appendix §E. +5 +Experiments +We evaluate the proposed algorithm under diverse scenarios of synthetic datasets in Sec. 5.1, and for two +applications–in identifying difficult tasks in real datasets in Sec. 5.2 and in training neural network models +with soft labels defined from the top-two plausible labels in Sec. 5.3. +5.1 +Experiments on synthetic dataset +We compare the empirical performance of Alg. 1 and Alg. 2 (referred as TopTwo1 and TopTwo2) with +other baselines: majority voting(MV), OTP-D&S and MV-D&S (Zhang et al., 2014), PGD (Ma et al., +2018), M-MSR (Ma & Olshevsky, 2020) and oracle-MLE, whose details can be found in Appx. §C. We +choose these baselines since they have the strongest established guarantees in the literature and they are +all MLE-based approaches, from which the top-two answers can be inferred. Obviously, oracle-MLE, which +uses the ground-truth model parameters, provides the best possible performance. We devise four scenarios +described in Table 1 to verify the robustness of our model for various (p, q) ranges, at (n, m) = (50, 500) with +s ∈ (0, 0.2]. The number of choices for each task is fixed as 5. Fig. 1 reports the empirical error probability +1 +m +�m +j=1 P((ˆgj, ˆhj) ̸= (gj, hj)) averaged over 30 runs, with 95% confidence intervals (shaded region). Four +columns correspond to the four scenarios, resp. The prediction errors for gj and hj are plotted in Fig. 6 of +Appx. §D.1. +Table 1: Parameters for synthetic data experiments under diverse scenarios. +Easy +Hard +Few-smart +High-variance +Worker +pi ∈ [0, 1] +pi ∈ [0, 1] +90% pi ∈ [0, 0.1] +pi ∈ [0, 1] +10% pi ∈ [0.9, 1] +Task +qj ∈ [0.9, 1] +qj ∈ (0.5, 0.6] +qj ∈ (0.5, 1] +50% qj ∈ (0.5, 0.6] +50% qj ∈ [0.9, 1.0] +8 + +MV +MV-D&S +OPT-D&S +PGD +M-MSR +ToiwoT +Top1wo2 +Oracle] +Easy +Hard +Few Smart +High Variance +0.75 +0.75 +0.7 +0.7 +0.651 +0.7 +0.9 +0.65F +(y'6)d +0.6 +0.6 +0.8 +0.65 +0.55A +0.55 +≠(y"6))d +0.7 +0.5 +0. +0.5 +0.45 +0.6 +0.45 +0.55 +0.4 +0.5 +0.4 +0.35 +0.5 +0.3 +0.4 +0.35 +2 +10 +6 +6 +8 +10 +2 +8 +10 +2 +8 +10 +Avg, # of queries per task +Avg, # of queries per task +Avg, # of queries per task +Avg, # of queries per taskWe can observe that for all the considered scenarios TopTwo2 achieves the best performance, near the +oracle MLE, in recovering (gj, hj). Depending on the scenarios, the reason TopTwo2 outperforms can be +explained differently. For the Easy scenario, since qj is close to 1, it is easy to distinguish gj from hj but +hard to distinguish hj from other labels. Our algorithm achieves the best performance in estimating hj by +a large margin (Fig. 6). For the Hard scenario, it is hard to distinguish gj and hj, but our algorithm, which +uses an accurate ˆqj, can better distinguish gj and hj. For Few-smart, our algorithm achieves the biggest +gain compared to other methods, since our algorithm can effectively distinguish few smart workers from +spammers. High-variance shows the effect of having diverse qj in a dataset. We remark that our algorithm +achieves the best performance, near that of the oracle-MLE, for all the scenarios, while the next performer +keeps changing depending on scenarios. For example, the OPT D&S is the second best performer in the +Easy scenario, while it is the worst performer in the Few-smart scenario. We also show the robustness of +our algorithm against changes in model parameters in Appendix §D. +5.2 +Experiments on real-world dataset: inferring task difficulties +We next provide experimental results using real-world data collected from MTurk and show that our algo- +rithm can be used for inferring task difficulties. We devised a color comparison task where we asked the +crowd to choose a color, among six given choices, that looks the most similar to a reference color of each +task. See Fig. 4 in Appx. §A.1 for example tasks. After randomly generating a reference color and the +six choices, we identified the ground truth and the most confusing answer for each task by measuring the +distance between colors using the CIEDE2000 color difference formula (Sharma et al., 2005). If the distance +from the reference color to the ground truth is much shorter than that to the most confusing answer, then the +task was considered easy. We designed 1000 tasks and distributed it to 200 workers, collecting 19.5 responses +on each task. After collecting the data, we subsampled it to simulate how the prediction error decreases as +the number of responses per task increases. Fig. 2a shows the performances in detecting (gj, hj), gj and +hj, averaged over 10 times of random sampling, with 95% confidence interval (shaded region). TopTwo2 +algorithm achieved the best performance in detecting (gj, hj), gj and hj in all ranges. We further examined +the correlation between the task difficulty - quantified by the distance gap between the ground truth and the +most confusing answer from the reference color - and the estimated confusion probability qj across tasks. We +selected top 50 most difficult/easiest tasks according to the estimated confusion probability qj and plotted +the histograms of the distance gap for the two groups in Fig 2b. We can see that the difficult group (blue, +having lowest qj) tends to have a smaller distance gap than those of the easy task group (red). This result +shows that our algorithm can identify difficult tasks in real datasets. +(a) The average prediction error on color comparison tasks +(b) Histogram of dist. gap +Figure 2: (a) Prediction error for (gj, hj), gj and hj (from left to right) for color comparison tasks using real +data collected from MTurk. Our TopTwo2 algorithm achieves the best performance. (b) Histogram of color +distance gap for the task groups with the highest qj (easiest tasks) and lowest qj (most difficult tasks). The +difficult task group (blue) tends to have a smaller color distance gap. +9 + +AIVIV +MV-D&S +UPT-D&S +PGD +M-MSR +Topiwo1 +TopTwo2 +P(g, h) ≠ P(g,h)) +P(g ≠g) +P(h ≠h) +0.6 +0.85 +0.85 +0.8 +0.55 +0.8 +Prediction error +0.75 +0.75 +0.5 +0.7 +0.7 +0.65 +0.65 +0.45 +0.6 +0.6 +0.4 +0.55 +0.55 +0.5 +0.35 +0.5 +10 +15 +20 +10 +15 +20 +10 +15 +20 +Avg, # of queries per task +Avg, # of queries per task +Avg, # of queries per taskmo +50.bib +Ton +50.1IOP +10 +8 +9 +Count +4 +2 +0 +0 +0.5 +1 +1.5 +2 +Color distance gap5.3 +Training neural networks with soft labels having top-two information +An appealing example where we can use the knowledge of the second best answer is in training deep neural +networks for classification tasks. Traditionally, a hard label (one ground-truth label per image) has been used +to train a classifier. In recent works, it has been shown that using a soft label (full label distribution that +reflect human perceptual uncertainty) is sometimes beneficial in obtaining a model with better generalization +capability (Peterson et al., 2019). However, obtaining an accurate full label distribution requires much higher +sample complexity than recovering only the ground-truth. For example, Peterson et al. (2019) provided a +CIFAR10H dataset with full human label distributions for 10000 instances of CIFAR10 test examples by +collecting on average 50 judgements per image, which is about 5-10 times larger than those of usual datasets +(Table 4 in Appendix A.1). +Our top-two model, on the other hand, can effectively reduce the required sample complexity, while still +guaranteeing the advantages in training models with soft labels. To demonstrate this idea, we trained two +deep neural networks, VGG-19 and ResNet18, with the soft-label vectors having the top-two structure (top2) +for CIFAR10H dataset3. We then compared the training/test results with those of the hard label (hard) and +full label distribution (full). Experimental details are in Appendix §B. Compared to the original training +with hard labels, training with top-two soft labels achieved 1.56% and 4.09% higher test accuracy in VGG-19 +and ResNet18, respectively (averaged in three runs, 150 epochs) as shown in Table 2, which is even higher +than that of the full label distribution in VGG-19. This result shows that training with the top-two soft +labels results in better generalization (test accuracy) than training with hard labels, since the top-two soft +label includes simple yet helpful side information, the most confusable class and the confusion probability. +Table 2: Comparison of performances for CIFAR10H dataset with hard/soft label training +Network +Train accuracy +Training loss +Test accuracy +Test loss +VGG-19 (hard) +97.46±0.59% +0.081±0.012 +77.64±1.54% +1.057±0.118 +VGG-19 (top2) +97.00±0.51% +0.231±0.014 +79.20±1.04% +0.754±0.050 +VGG-19 (full) +96.69±0.48% +0.282±0.010 +78.66±0.97% +0.740±0.030 +ResNet18 (hard) +98.47±0.320% +0.046±0.009 +76.49%±1.80% +1.275±0.157 +ResNet18 (top2) +98.67±0.491% +0.168±0.024 +80.58%±2.36% +0.640±0.093 +ResNet18 (full) +99.19±0.125% +0.189±0.023 +80.93%±2.66% +0.611±0.102 +6 +Discussion +We proposed a new model for multiple-choice crowdsourcing, with top-two confusable answers and varying +confusion probability over tasks. We provided an algorithm to infer the top-two answers and the confusion +probability. This work can benefit several query-based data acquisition systems such as MTurk or review +systems by providing additional information about the task such as the most plausible answer other than +the ground truth and how plausible it is, which can be used to quantify the accuracy of the ground truth or +to classify the tasks based on difficulty. The topic of confusion is getting increasing attention in the machine +learning community for designing reliable classifiers (Jin et al., 2017; Luque et al., 2019; Chang et al., 2017). +We also demonstrated possible applications of our algorithm in designing soft labels for better generalization +of neural networks. +3As in (Peterson et al., 2019), we used the original 10000 test examples of CIFAR10 for training and 50000 training examples +for testing. Thus, the final accuracy is lower than usual. 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Advances in neural information processing systems, 25, 2012. +13 + +A +Verification for the Proposed Top-Two Model +We proposed the top-two model to reflect the key attributes of seven datasets including Adult2, Dog, Web, +Flag, Food, Plot, and Color, of which the details are summarized in Appendix A.1. +Table 3 shows empirical distributions of the mean incidence of responses for the top-three dominating +answers, sorted by the dominance proportions, for the six public datasets and the Color dataset that we +collected, with the standard deviation over the tasks in the dataset. +In Fig. +3, we also plot empirical +distributions of the mean incidence of responses sorted by the dominant proportion with error bars indicating +the standard deviation. The i-th data point represents the average incidence of the i-th highest response +in each task. For example, in Adult2 dataset, the most dominating answer takes 0.8 portion of the total +answers, and the next dominating answer takes 0.14 portion of the total answers on average. +Table 3: Proportions of top-three dominating answers in public datasets +Dataset +Ground truth +2nd dominating answer +3rd dominating answer +Adult2 +0.80±0.19 +0.14±0.13 +0.04±0.07 +Dog +0.76±0.15 +0.22±0.14 +0.01±0.04 +Web +0.59±0.20 +0.25±0.12 +0.12±0.09 +Flag +0.90±0.16 +0.09±0.13 +0.01±0.03 +Food +0.80±0.18 +0.17±0.15 +0.02±0.05 +Plot +0.62±0.21 +0.30±0.16 +0.06±0.07 +Color +0.43±0.1 +0.23±0.06 +0.15±0.05 +From the table and figure, we can observe that for all the considered public datasets the top-two answers +dominate the overall answers, i.e., about 65-90% of the total answers belong to the top two. Furthermore, +the average ratio from the most dominating answer to the second one is 4:1, while that between the second +and the third is 7.5:1. There often exist overlaps in the error bars between the ground truth and the second +dominating answer, e.g., for Web, Plot, and Color datasets, but no such overlap is found between the ground +truth and the third dominating answer. What we can call a ‘confusing answer’ is an answer that has an +incidence rate comparable to that of the ground truth. +In all the considered datasets, only the second +dominating answer shows such a tendency, and thus, we can conclude that the third dominating answer +cannot be called a ‘confusing answer’, and the top-two model in equation 1 well describes the errors in +answers caused by confusion. +Moreover, from the public datasets, we also observe that the task difficulty can be quantified by the +confusion probability between the top-two answers. As an example, for the Web dataset, when we select the +easiest 500 tasks and hardest 500 tasks by ordering tasks with the ratio of correct answers, the ratio between +the ground-truth to the 2nd best answer was 10.7:1 for the easiest group, while it was 1.5:1 for the hardest +group. This observation shows that the ratio between the top-two answers indeed captures task difficulty as +does our model parameter for task difficulty qj in equation 1. +A.1 +Datasets +We collect six publicly available multi-class datasets: Adult2, Dog, Web, Flag, Food and Plot. Since these +datasets do not provide information about the most confusing answer or the task difficulty, we additionally +create a new dataset called ‘Color’, for which we can identify the most confusing answer and also quantify +the task difficulty for all the included tasks. +• Color is a dataset where the task is to find the most similar color to the reference color among six +different choices. For each task, we randomly create a reference color and then choose six choices of +colors. The distance from the reference color to the ground truth color is in between 4.5 and 5.5, the +14 + +(a) +(b) +(c) +(d) +(e) +(f) +(g) +Figure 3: Empirical distribution of the mean incidence of responses sorted by the dominant proportion, +averaged over all tasks in each dataset. The i-th data point represents the average incidence of the i-th +highest response in each task. The error bars indicate the standard deviation of the mean incidence of the +i-th dominating answer over the tasks in the dataset. +distance to the most confusing answer is in between 5.5 and 6.5, and the distance to the rest of the +choices is between 11 and 12, where the distance between the pairs of colors is measured by CIEDE2000 +(Sharma et al., 2005) color difference formulation. The tasks are ordered in terms of their difficulty +levels by measuring the gap between: the distance from the reference color to the ground truth; and +that to the most confusing answer. If the distance from the reference color to the ground truth is +much shorter than that to the most confusing answer, then the task is considered easy. Using MTurk, +we collected 19600 labels from 196 workers for 1000 tasks. Each Human Intelligence Task (HIT) is +composed of randomly selected 100 tasks, and we pay $1 to each worker who completed a HIT. Fig. 4 +shows an example task for the Color dataset. +• Adult2 (Ipeirotis et al., 2010) is a 4-class dataset where the task is to classify the web pages into four +categories (G, PG, R, X) depending on the adult level of the websites. This dataset contains 3317 +labels for 333 websites which are offered by 269 workers. +• Dog (Zhang et al., 2014) is a 4-class dataset where the task is to discriminate a breed (out of Norfolk +Terrire, Norwich Terrier, Irish Wolfhound, and Scottich Deerhound) for a given dog. This dataset +contains 7354 labels collected from 52 workers for 807 tasks. +• Web (Zhou et al., 2012) is a 5-class dataset where the task is to determine the relevance of query-URL +pairs with a 5-level rating (from 1 to 5). The dataset contains 15567 labels for the 2665 query-URL +pairs offered by 177 workers. +• Flag (Krivosheev et al., 2020) is a dataset for multiple-choice tasks where each task is to identify the +country for a given flag from 10 given choices. A total of 1600 votes are collected from 220 workers for +the 100 tasks. +15 + +Color +N +0.8 +Empirical probabili +0.6 +0.4 +0.2 +0 +1 +2 +3 +4 +5 +6 +LabelAdult2 +N +0.8 +Empirical probabili +0.6 +0.4 +0.2 +0 +1 +2 +3 +4 +LabelDog +Empirical probability +0.8 +0.6' +0.4 +0.2 +0 +1 +2 +3 +4 +LabelWeb +0.8 +Empirical probabilit +0.6 +0.4. +0.2 +0 +1 +2 +3 +4 +5 +LabelFlag +0.8 +0.6 +0.4 +0.2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +LabelFood +0.8 +0.6 +0.4 +0.2 +0 +1 +2 +3 +4 +5 +LabelPlot +0.8 +0.6 +0.4' +0.2 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Label(a) gj = 6 and hj = 5 +(b) gj = 4 and hj = 3 +(c) gj = 5 and hj = 3 +(d) gj = 6 and hj = 2 +Figure 4: Example tasks for ‘Color’ dataset where the ground truth g and the most confusing answer h are +determined by the color distance from the reference color (top). +• Food (Krivosheev et al., 2020) is a dataset for multiple-choice tasks where each task asks to identify +a picture of a given food or dish from 5 given choices. This dataset contains 1220 labels for 76 tasks +collected from 177 workers. +• Plot (Krivosheev et al., 2020) is a dataset for multiple-choice tasks where the task is to identify a +movie from a description of its plot from 10 given choices. Only workers who correctly solved the first +10 test questions can answer the rest of the tasks. A total of 1937 labels are collected from 122 workers +for 100 tasks. +Table 4 shows a summarized information for the introduced datasets. +Table 4: Dataset information +Dataset +# workers +# tasks +# labels or choices +sparsity +dtask +dworker +Adult2 +269 +333 +4 +0.037 +10.0 +12.4 +Dog +109 +807 +4 +0.092 +10.0 +74.0 +Web +176 +2653 +5 +0.033 +5.9 +88.3 +Flag +220 +100 +10 +0.073 +16.0 +7.3 +Food +177 +54 +5 +0.125 +22.1 +6.7 +Plot +122 +56 +10 +0.293 +35.7 +16.4 +Color +196 +1000 +6 +0.1 +19.5 +99.4 +B +Experimental Details for Neural Network Training in Sec. 5.3 +We show the details of the experiments in Sec. 5.3. +B.1 +Datasets +The CIFAR10H dataset (Peterson et al., 2019) consists of 511,400 human classifications by 2,571 participants +which were collected via Amazon Mechanical Turk. Each participant classified 200 images, 20 from each +16 + +2 +3 +525 +2 +3 +6(a) Images with lowest q (considered to be hard) +(b) Images with highest q (considered to be easy) +Figure 5: Training images with (a) lowest and (b) highest confusion probabilities. +category. Every 20 tasks, a trivial question is presented to prevent random guessing, and participants who +scored below 75% were excluded from the dataset. We present the images with the lowest/highest q from +the training samples in Fig 5. The image with a lower q means that the first answer and the second answer +are hard to distinguish. +B.2 +Model +We trained two simple CNN architectures, VGG-19 and ResNet-18, to show the usefulness of the second +answer and the confusion probability. +For each model, our loss function is defined as the cross-entropy +between the softmax output and the two-hot vector (in which the values are q and 1 − q for g and h, +respectively). We compare the results of our top-two label training with those of full-distribution training +and hard label (one-hot vector) training. +B.3 +Training +We train each model using 10-fold cross validation (using 90% of images for training and 10% images for +validation) and average the results across 5 runs. We run a grid search over learning rates, with the base +learning rate chosen from {0.1, 0.01, 0.001}. We find 0.1 to be optimal in all cases. We trained each model +for a maximum of 150 epochs using SGD optimizer with a momentum of 0.9 and a weight decay of 0.0001. +Our neural networks are trained using NVIDIA GeForce 3090 GPUs. +C +Baseline Methods +In this section, we explain the baseline methods with which we compare the performance of our algorithms. +To analyze the performance in recovering the top-two answers, we considered the ML-based algorithms, +including the Spectral-EM algorithm (MV-D&S and OPT-D&S) (Zhang et al., 2014), Projected +Gradient Descent (PGD) (Ma et al., 2018) and M-MSR (Ma & Olshevsky, 2020), which provide a +“score” for each label so that we can recover the top-two answers. +• Spectral-EM algorithm (MV-D&S and OPT-D&S) (Zhang et al., 2014) is a two-stage algorithm +for multi-class crowd labeling problems. These algorithms are built for the D&S model where each +worker has his/her own confusion matrix. In the first stage of the algorithm, the confusion matrix of +each worker is estimated via spectral method (OPT-D&S) or majority voting (MV-D&S), respectively, +and in the second stage, the estimates for the confusion matrices are refined by optimizing the objective +function of the D&S estimator via the Expectation Maximization (EM) algorithm. +• Projected Gradient Descent (PGD) (Ma et al., 2018) is an approach to estimate the skills of +each worker in the single-coin D&S model. The authors formulate the skill estimation problem as a +rank-one correlation-matrix completion problem. They propose a projected gradient descent method +to solve the correlation-matrix completion problem. +• M-MSR (Ma & Olshevsky, 2020) algorithm is an approach to estimate the reliability of each worker +in the multi-class D&S model. M-MSR algorithm utilizes that the rank of the response matrix is one. +17 + +Figure 6: Prediction error for (gj, hj) (top row), gj (middle) and hj (bottom) for four scenarios. +Our +algorithm (TopTwo2) achieves the best performance, near the oracle MLE for all the scenarios. +To estimate the reliability of the workers, they use update rules to find the left singular vector and +right singular vector of the response matrix. In this process, the extreme values are filtered out to +guarantee the stable convergence of the algorithm. +D +Synthetic Experiments +D.1 +Additional plots for synthetic data experiments in Sec. 5.1 +In Section 5.1, we devised four scenarios described in Table 1 to verify the robustness of our model for various +(p, q) ranges, with (n, m, s) = (50, 500, 0.2). The performance of algorithms is measured by the empirical +average error probabilities in recovering gj, hj and (gj, hj), i.e., +1 +m +�m +j=1 P(ˆgj ̸= gj), +1 +m +�m +j=1 P(ˆhj ̸= hj) and +1 +m +�m +j=1 P((ˆgj, ˆhj) ̸= (gj, hj)) and plotted in Fig. 6. We can observe that for all the considered scenarios +TopTwo2 achieves the best performance, near the oracle MLE, in recovering (gj, hj). Depending on scenarios +though, the reason TopTwo2 outperforms can be explained differently. For Easy scenario, since qj is close +to 1, it becomes easy to distinguish gj from hj but hard to distinguish hj from other labels. Our algorithm +achieves the best performance in estimating hj by a large margin. +For Hard scenario, it becomes hard +to distinguish gj and hj, but our algorithm, which uses an accurate ˆqj, can better distinguish gj and hj. +High-variance show the effect of having diverse qj in a dataset. For Few-smart, our algorithm achieves the +18 + +MVMV-D&S +←OPT-D&S +PGD +M-MSR +TopTwo1 TopTwo2 +OracleEasy +Hard +Few Smart +High Variance +0.75 +0.75 +0.7 +0.7 +0.65 +0.9] +0.7 +P(g, h) ≠ P(g,h) +0.6 +0.6 +0.8 +0.65 +0.55 +0.7 +0. +0.5 +0.5 +0.45 +0.6 +0.45 +0.4 +0.55 +0.5 +0.4 +0.35 +0.5 +0.3 +0.4 +0.35 +2 +4 +6 +8 +10 +4 +6 +8 +10 +4 +6 +8 +10 +4 +6 +8 +10 +0.1 +0.55 +0.9 +0.26 +0.8 +0.24 +0.08h +0.5 +0.7 +0.22 +0.06 +0.6* +0.45F +0.2 +0.04 +P(g +0.54 +0.18 +0.4 +0.4 +0.02 +0.16 +0.3 +0.35 +0.2 +0.14 +-0.02 +0.3 +0. +0.12 +4 +6 +8 +10 +4 +2 +6 +10 +6 +8 +10 +8 +10 +8 +0.75 +0.7 +0.85 +0.7 +0.65 +0.8 +0.65 +0.7 +0.75 +0.6T +0.6 +0.7 +<0.65 +0.55 +0.65 +0.55 +0.5 +0.6 +0.5 +0.6 +0.45 +0.55 +0.45 +0.4 +0.5 +0.55 +0.4 +0.35 +0.45 +0.5 +0.3 +0.4 +0.35 +4 +8 +10 +4 +6 +8 +10 +6 +8 +10 +2 +4 +4 +6 +8 +10 +Avg, # of queries per task +Avg, # of queries per task +Avg, # of queries per task +Avg, # of queries per taskbiggest gain compared to other methods, since our algorithm can effectively distinguish few smart workers +from spammers. We remark that even though the performance gap between TopTwo2 and the next best +performer is not significant for some cases, our algorithm always achieves the best performance, near that of +the oracle-MLE, for all the scenarios, while the next performer keeps changing depending on scenarios. For +example, the OPT D&S is the second best performer in the ‘Easy’ scenario, while it is the worst performer +in the ‘Few smart’ scenario. +D.2 +Robustness of our methods +In this section, we present a set of four additional synthetic experiments to demonstrate the robustness of +our methods, Alg. 1 and Alg. 2 (referred to as TopTwo1 and TopTwo2). In each experiment, we change a +parameter of our synthetic error model and compare the prediction error of our algorithms to the baselines: +majority voting(MV), OTP-D&S and MV-D&S Zhang et al. (2014), PGD Ma et al. (2018) and Oracle-MLE. +We measure the performance of each algorithm by the empirical average error probabilties in recovering the +ground truth gj, the most confusing answer hj and the pair of top two (gj, hj), i.e., +1 +m +�m +j=1 P(ˆgj ̸= gj), +1 +m +�m +j=1 P(ˆhj ̸= hj) and +1 +m +�m +j=1 P((ˆgj, ˆhj) ̸= (gj, hj)). Obviously, Oracle-MLE provides a lower bound for +the performance. +Changing the dimension of observed matrix: We first check the robustness of our methods against +the change of dimensions of the observation matrix A ∈ {0, 1 . . . , K}n×m with n ≤ m. We vary the number +of workers (n) or the number of tasks (m) while fixing the other dimension. The default values of n and +m are 50 and 500, respectively, and the sampling probability s is fixed as 0.1 throughout the experiments. +The worker reliability pi and the task difficulty qj is sampled uniformly at random from [0, 1] and (1/2, 1], +respectively, for all i ∈ [n] and j ∈ [m]. +In Fig. 7a and 7b, we report the results when we change n for a fixed m and s, or when we change +m for a fixed n and s, respectively. From Fig. 7a, we can see that as the number of workers increases, +the performance of every algorithm improves since the number of samples per task scales as ns for a fixed +s. Our algorithm achieves the performance close to the Oracle-MLE for all the considered range, which +implies that the worker reliabilities {pi} are well estimated with our methods. From Fig. 7b, we can see +that our algorithm achieves a robust performance against the change in the number of tasks, although the +performance gets closer to that of Oracle-MLE as the number of tasks increases. Since our method uses +SVD in the first stage, the larger dimension is beneficial for the concentration of the random perturbation +matrix with respect to the expectation of the observation matrix. This phenomenon is observed for other +baseline methods as well, which are based on the spectral method, OPT D&S, for example. +Changing the variance of worker reliability: In this experiment, we change the range of pi, the +parameter for worker skill/reliability, for i ∈ [n], with a fixed mean in order to observe the impact of +the variance of the worker reliability on the prediction error. We randomly sample pi from the window +[0.5 − x, 0.5 + x] with x varying from 0.05 to 0.25. The mean of the worker reliability is fixed as 0.5. +As shown in Fig. 7c, when the variance of the worker reliability increases, the baseline methods estimating +worker reliabilities perform better than the majority voting. +Our TopTwo2 algorithm achieves the best +performance close to Oracle-MLE, as the standard deviation increases, i.e., as the workers become more +heterogeneous. +Changing the variance of task difficulty: We also design an experiment to observe the impact of +the variance of qj, j ∈ [m], the parameter for task difficulty, on the prediction error. We randomly sample qj +from the window [0.75 − x, 0.75 + x] with x varying from 0.05 to 0.25. The mean of the worker reliability is +fixed as 0.75. If the variance of the task difficulty is small, it could be sufficient to only estimate the worker +reliability since all the tasks have almost the similar task difficulties. +As shown in Fig. 7d, when the variance of the task difficulty increases, our TopTwo2 algorithm performs +better than the other baselines. This is the evidence for the validity of our method in estimating the task +difficulty. +Changing the portion of spammers: Spammers who provide random answers always exist in crowd- +sourcing systems. +To improve the inference performance, it is important to distinguish spammers from +19 + +(a) Effect of the number of workers on the performance +(b) Effect of the number of tasks on the performance +(c) Effect of the variance of worker reliability on the performance +(d) Effect of the variance of task difficulty on the performance +(e) Effect of the portion of spammers on the performance +Figure 7: Prediction error for (gj, hj) (first column), gj (second column), and hj (third column) for five +different setups. The solid lines represent the mean prediction errors of each algorithm averaged over 10 +runs, and the shaded regions represent the standard deviations. +20 + +D((α )/(a )) +Da/a) +D( / )1（(9,10)+(9,10)) +0.8 +1(9+9) +0.9 +1(十) +1 +0.9 +0.7 +0.87 +0.8 +0.6 +★-MV +MV-D& +0.5 +0.3 +-OPT-D +PGD +0.4 +0.4 +-TopTw +0.2 +TopTw +→—Oracle +0.3 +0.1 +0.3 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +# of workers +# of workers +# of workerss +&S +01 +02D(( )/(a )) +Da/ +D(h / )1((9,1)+(9,1)) +1(9/9) +1(+) +0.8 +0.6 +0.75 +0.75 +0.7 +0.5 +0.7 +0.65 +g 0.4 +error +0.6. +0.6 +一MV + 0.2 +一MV-D& +0.5 +←OPT-D +0.5 +PGD +0.1 +0.45 +-TopTw +0.45 +-TopTw +Oracle +0.4 +0 +0.4 +200 +400 +600 +800 +1000 +200 +400 +600 +800 +1000 +200 +400 +600 +800 +1000 +# of tasks +# of tasks +# of taskss +&S +01 +02D((α )/(a )) +Da/a) +D( / )1((9,10)于(9,10)) +0.34 +1(9/9) +0.75 +0.7 +0.32 +0.7 +0.3 +0.65 +Prediction error +0.6 +一MV +P +-MV-D8 +-OPT-D +0.6 +0.22 +0.55 +PGD +-TopTw +0.2 +-TopTw +一Oracle +0.55 +0.18 +0.5 +0.05 +0.1 +0.15 +0.2 +0.25 +0.05 +0.1 +0.15 +0.2 +0.25 +0.05 +0.1 +0.15 +0.2 +0.25 +Std. of worker reliability +Std. of worker reliability +Std. of worker reliabilitys +&S +01 +02D((α )/(a )) +Da/a) +D( / )1(9+9) +0.8 +0.5 +0.75 +0.45 +0.7 +0.7 +0.4 +0.65 +error +0.6 +0.3 +一MV +0.5 +P +一MV-D +←OPT-[ +0.2 +0.45 +0.4 +-PGD +—TopTv +0.15 +0.4 +TopTv +一Oracle +0.3 +0.1 +0.35 +0.05 +0.1 +0.15 +0.2 +0.25 +0.05 +0.1 +0.15 +0.2 +0.25 +0.05 +0.1 +0.15 +0.2 +0.25 +Std. of task difficulty +Std. of task difficulty +Std. of task difficulty&S +D&S +/01 +/02D((α )/(a )) +Da/a) +D( / )1((9,1) +(9,10) +1(9≠9) +0.85 +1( +0.9 +0.8 +0.8 +0.9 +0.7 +error +0.6 +0.7 +★一MV + 0.65 +一MV-D& +0.4 +0.6 +-PGD +0.6 +0.3 +-TopTw +一Oracle +0.5 +0.2 +0.55 +0.2 +0.4 +0.6 +0.8 +0.2 +0.4 +0.6 +0.8 +0.2 +0.4 +0.6 +0.8 +Portion of spammers +Portion of spammers +Portion of spammers&S +&S +01 +102reliable workers. In our experimental setup, we define a spammer as a worker whose reliability parameter pi +is in the range [0, 0.1]. We change the portion of spammers among the workers from 0.1 to 0.9 and compare +the prediction error of our methods to those of other baseline methods. +In Fig. 7e, we can see that our algorithm achieves the best performance among all the considered baselines +except Oracle-MLE, which can exactly distinguish spammers from reliable workers. This result demonstrates +the superiority of our methods in detecting spammers compared to other methods. +D.3 +Estimating the worker reliability vector and the task difficulty vector +In this section, we examine the accuracy of our estimates for the worker reliability vector p and the task +difficulty vector q. The worker reliability is estimated by ˆp defined in equation 5 of Algorithm 2 and the +task difficulty is estimated by ˆq defined in equation 4 of Algorithm 1. To analyze the accuracy of these +estimators, we compute the mean squared error (MSE), 1 +n∥ˆp − p∥2 +2 and +1 +m∥ˆq − q∥2 +2, respectively. +To analyze the estimation accuracy for the worker reliability, we first sample pi uniformly at random from +[0, 1] for all i ∈ [n] and fix the worker reliability vector p. Then, we randomly sample the task difficulty vector +q ∈ (1/2, 1]m fifty times and then sample the observation matrices from the distribution equation 1 for each +(p, q) pair with a fixed p. For each observation matrix, we subsample the data with varying probabilities +and apply Algorithm 2 to get the estimate ˆp, which is then used to calculate the MSE of p. We report the +MSE averaged over these fifty cases. Similarly, to analyze the estimation accuracy for the task difficulty, +we randomly sample and fix a task difficulty vector q ∈ (1/2, 1]m and generate fifty different observation +matrices while varying the worker reliability vector p. We again report the MSE averaged over these fifty +cases. The number of workers and that of tasks is set to be (50, 500) for the worker reliability estimation, +and to be (100, 1000) for the task difficulty estimation. +In Fig. 8a and 8b, we plot the MSE for p and q, respectively, as the average number of queries per task +increases. We can see that both for p and q, the MSEs converge to near zero as the average number of +queries per task increases. However, estimating the task difficulty requires more number of samples as our +theory equation 11 suggests. +(a) Mean squared error 1 +n ∥ ˆp − p∥2 +2 +(b) Mean squared error +1 +m ∥ˆq − q∥2 +2 +Figure 8: Mean squared errors in estimating the worker reliability vector p (left) and the task difficulty +vector q (right), respectively. +E +Discussion of theoretical results +In this section, we present a discussion of the main theoretical results. +21 + +0.05 +0.045 +0.04 +0.035 +rror +E +0.03 +Square +0.025 +Mean +0.02 +0.015 +0.01 +0.005 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Avg.#ofqueriespertask0.5 +0.45 +0.4 +Error +0.35 +Square +0.3 +0.25 +Mean +0.2 +0.15 +0.1 +0.05 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Avg.#ofqueriespertask• Theorem 1 asserts that the sampling probability of Ω +� +1 +δ2 +1∥p∥2 +2 log K +ϵ +� +is sufficient to recover the top-two +answers (gj, hj) for any task j ∈ [m] and to estimate the confusion probability qj with accuracy of +|ˆqj − qj| < δ1 by Algorithm 1 with probability at least 1 − ϵ. Combined with Theorem 3 part (a), +we can see that this sample complexity is the minimax optimal rate for a fixed collective quality of +workers, measured by ∥p∥2 +2. +• It is also worth comparing our algorithm with the simple majority voting (MV) scheme. The MV +scheme infers the top-two answers by counting the majority of the received answers. Simple analysis +shows that the MV scheme requires the sampling probability s such that ns = Θ +� +( 1 +n +� +i pi)−2 log 1 +ϵ +� +to recover (gj, hj) with probability 1 − ϵ. Remind that Algorithm 1 requires ns = Ω +� +n +δ2 +1∥p∥2 +2 log K +ϵ +� +samples per task. Since +1 +n∥p∥2 = +1 +n +� +i p2 +i ≥ +� 1 +n +� +i pi +�2 by Cauchy-Schwarz inequality, Algorithm +1 achieves strictly better trade-offs unless pi is same for all workers i ∈ [n]. As an example, for a +spammer-hammer model where α ∈ (0, 1) fraction of workers are hammers with pi = 1 and the rest +are spammers with pi = 0, Algorithm 1 requires ns = Θ +� 1 +α log 1 +ϵ +� +samples per task, while MV requires +ns = Θ +� 1 +α2 log 1 +ϵ +� +samples per task to recover top-two answers with probability 1 − ϵ. +• Theorem 2 shows that when we have an entrywise bound on the estimated worker reliability vector +p and the task difficulty vector q, the plug-in MLE estimator, used in Algorithm 2, guarantees the +recovery of top-two answers if the sampling probability s = Ω( log(m/ϵ) +n ¯ +D +) where ¯D, which depend on +(p, q), indicates the average reliability of workers in distinguishing the top-two answers from any other +pairs for the most difficult task. Combined with Theorem 3 part (b), we can see that this sample +complexity is the minimax optimal rate for any (p, q), ignoring the logarithmic terms. +• Combining the conditions for the accurate estimation of model parameters in equation 11 and the +convergence of the plug-in MLE (Theorem 2), Corollary 2 shows the condition on the sample complexity +to guarantee the performance of Algorithm 2. +F +Proof of Proposition 1 +For each task j and label k, define four indicator functions: +Πa(j, k) :=1(gj > k, hj > k), +Πb(j, k) :=1(gj ≤ k, hj > k), +Πc(j, k) :=1(gj > k, hj ≤ k), +Πd(j, k) :=1(gj ≤ k, hj ≤ k), +(17) +which satisfy Πa(j, k) + Πb(j, k) + Πc(j, k) + Πd(j, k) = 1. For notational simplicity, we will often drop (j, k) +fron Π∗. The pmf of A(k) is given by +A(k) +ij = +� +� +� +� +� +−1 +with probability s(1 − ρ(k) +ij ), +1 +with probability sρ(k) +ij , +0 +with probability 1 − s, +(18) +where ρ(k) +ij += Πa(j, k)pi + Πb(j, k)pi(1 − qj) + Πc(j, k)piqj + (K−k)(1−pi) +K +, and its expectation is E[A(k) +ij ] = +s(2ρ(k) +ij − 1). Note that by using Πa = 1 − Πb − Πc − Πd, the probability ρ(k) +ij +can be written as ρ(k) +ij += +pi +� +qj(Πc − Πb) − (Πc + Πd) + k +K +� ++ K−k +K . Thus, by defining +r(k) +j +:= qj(Πc − Πb) − (Πc + Πd) + k +K , +(19) +22 + +the expectation of A(k) +ij +can be written as +E[A(k) +ij ] = s(2ρ(k) +ij − 1) = s +� +2pir(k) +j ++ K − 2k +K +� +, +(20) +and +E[A(k)] − s(K − 2k) +K +1n×m = 2sp(r(k))⊤. +(21) +Note that +Case I: gj > hj +Πa(j, k) = 1 where k < hj, +Πc(j, k) = 1 where hj ≤ k < gj, +Πd(j, k) = 1 where gj ≤ k; +Case II: gj < hj +Πa(j, k) = 1 where k < gj, +Πb(j, k) = 1 where gj ≤ k < hj, +Πd(j, k) = 1 where hj ≤ k. +(22) +Thus, r(k) +j +in equation 19 is equal to +Case I: gj > hj +r(k) +j += +� +� +� +� +� +k +K +where k < hj; +k +K − (1 − qj) +where hj ≤ k < gj; +k +K − 1 +where gj ≤ k, +Case II: gj < hj +r(k) +j += +� +� +� +� +� +k +K +where k < gj; +k +K − qj +where gj ≤ k < hj; +k +K − 1 +where hj ≤ k. +G +Performance Analysis of Algorithm 1 +G.1 +Proofs of Theorem 1 and Corollary 1 +In Algorithm 1, we use the data matrix A1, which is obtained by randomly splitting the original data matrix +A into A1 and A2 with probability s1 and (1 − s1), respectively. Then, the first stage of Algorithm 1 begins +with randomly splitting A1 again into two independent matrices B and C with equal probabilities, and then +converting B and C into (K − 1)-binary matrices B(k) and C(k) as explained in Sec. 2. We define X(k) +and Y (k) as X(k) := B(k) − s′(K−2k) +K +1n×m and Y (k) := C(k) − s′(K−2k) +K +1n×m where s′ = s · s1/2. We have +E[X(k)] = E[Y (k)] = s′p(r(k))⊤ from Prop. 1. For notational simplicity, we will ignore this random splitting +for a moment and just pretend that X(k) and Y (k) are sampled independently with s′ = s throughout this +section. +We first outline the proof. Based on the observation that E[X(k)] = sp(r(k))⊤, if E[X(k)] is available +we can recover p∗ = +p +∥p∥2 by SVD, and by using p∗ it is possible to recover ∥p∥2r(k), which then reveals +{(gj, hj)}m +j=1 as well as q from the relation in equation 2. To estimate p∗ from X(k), we first bound the +spectral norm of the perturbation, ∥X(k) − E[X(k)]∥2. We then use this bound and Wedin SinΘ theorem to +bound sin θ(u(k), p∗) where u(k) is the left singular vector of X(k) with the largest singular value. We trim +the abnormally large components of u(k) and denote the resulting vector by ˜u(k). After trimming, it is still +possible to show that sin θ(˜u(k), p∗) can be bounded in the same order as that of sin θ(u(k), p∗). Finally, +we provide an entrywise bound between v(k) = 2 +s(Y (k))⊤ ˜u(k) and ∥p∥2r(k) in Lemma 5, which is the main +lemma to prove Theorem 1. We state our main technical lemmas first and then prove Theorem 1. +Let us define the perturbation matrix +E := X(k) − E[X(k)] = B(k) − s(K − 2k) +K +1n×m − sp(r(k))⊤ = B(k) − E[B(k)] +(23) +where +B(k) +ij += +� +� +� +� +� +−1 +w.p. s(1 − ρ(k) +ij ), +1 +w.p. sρ(k) +ij , +0 +w.p. 1 − s, +(24) +23 + +and ρ(k) +ij = Πa(j, k)pi+Πb(j, k)pi(1−qj)+Πc(j, k)piqj+ (K−k)(1−pi) +K +for (Πa, Πb, Πc, Πd) defined in equation 17. +For the perturbation matrix E in equation 23, we have +E[Ei,j] = 0, +and +|Ei,j| ≤ 2, +1 ≤ i ≤ n, 1 ≤ j ≤ m, +(25) +and also +var(Eij) = var(B(k) +ij ) = E[(B(k) +ij )2] − (E[B(k) +ij ])2 += s − (s(ρ(k) +ij − 1/2))2 ≤ s. +(26) +Note that {Eij} are independent across all i, j. Define +ν := max +� +� +�max +i +� +j +E[E2 +i,j], max +j +� +i +E[E2 +i,j] +� +� +� ≤ max{m, n}s. +(27) +By applying the spectral norm bound to random matrices with independent entires, appeared in Bandeira +& Van Handel (2016) and summarized in Theorem 4, we can bound the spectral norm of E as below. +Lemma 2 (Spectral norm bound of E). With probability 1 − (n + m)−8, we have +∥E∥ ≤ 4 +� +s max (m, n) + ˜c +� +log(n + m) +(28) +for some constant ˜c > 0 when m ≥ n. +For some sufficiently large m, assuming n = o(m) and s = +Ω(log(n + m)/m), the spectral norm of E can be further bounded by +∥E∥ ≤ 5√sm. +(29) +Using the bounded spectral norm of E in equation 29 and applying the Wedin SinΘ theorem, summarized +in Theorem 5, we can bound the angle between u(k) and p∗. +Lemma 3. For some sufficiently large m, assuming n = o(m) and s = Ω(log(n + m)/m), we have +sin θ(u(k), p∗) ≤ Θ(1/√sn) +(30) +with probability at least 1 − (n + m)−8. +Proof. By applying the Wedin SinΘ Theorem (Theorem 5), we have +sin θ(u(k), p∗) ≤ +√ +2∥E∥ +s∥p∥2 · ∥r(k)∥2 − ∥E∥. +(31) +We have ∥p∥2 = Θ(√n) and ∥r(k)∥2 = Θ(√m) by assumptions on model parameters. By Lemma 2, for some +sufficiently large m, assuming n = o(m) and s = Ω(log(n + m)/m), we have ∥E∥ ≤ 5√sm with probability +at least 1 − (n + m)−8. Combining these bounds, we get +sin θ(u(k), p∗) ≤ +Θ(√sm) +Θ(s√mn) − Θ(√sm) = +1 +Θ (√sn). +(32) +We trim the abnormally large components of u(k) by letting it zero if u(k) +i +> 2/(η√n) and denote the +resulting vector as ˜u(k). This process is required to control the maximum entry size of ˜u(k), which is used +later in the proof. For the next lemma, we show that after the trimming process, the norm of ˜u(k) is still +close to 1 and the angle between ˜u(k) and p∗ has the same order as that of sin θ(u(k), p∗). +24 + +Lemma 4. Given ∥p∗∥2 ≥ η√n, we have +∥˜u(k)∥2 ≥ +� +1 − 50 sin2 θ(u(k), p∗), +(33) +sin θ(˜u(k), p∗) ≤ 6 +√ +2 sin θ(u(k), p∗). +(34) +The proof of Lemma 4 is provided in Section G.2. +Finally, we provide our main lemma giving the entrywise bound on the difference between v(k) = +1 +s(Y (k))⊤ ˜u(k) and ∥p∥2r(k). +Lemma 5 (Entrywise Bound). For any δ1, ϵ > 0, and any task j ∈ [m] and label index k ∈ [K], if the +sampling probability s ≥ Θ +� +1 +δ2 +1∥p∥2 +2 log 1 +ϵ +� +, then we can guarantee +P +����� +1 +s +� +Y (k) +∗j , ˜u(k)� +− ∥p∥2r(k) +j +���� < δ1∥p∥2 +� +> 1 − ϵ +(35) +as m → ∞ when n = O(m/ log m). +Proof. For notional simplicity, denote θ(˜u(k), p∗) by θ. +To prove equation 35, we show bounds on two +probabilities, +P +����� +1 +s +� +Y (k) +∗j , ˜u(k)� +− ∥˜u(k)∥2∥p∥2r(k) +j +cos θ +���� > δ1∥p∥2 +2 +� +< ϵ/2, +(36) +P +����∥˜u(k)∥2∥p∥2r(k) +j +cos θ − ∥p∥2r(k) +j +��� > δ1∥p∥2 +2 +� +< ϵ/2. +(37) +Then, the triangle inequality implies equation 35. +We first prove equation 36. Remind that we do the random splitting of the input matrix A and define +the two independent binary-converted matrices as X(k) and Y (k), for 1 ≤ k < K, which are used to estimate +˜u(k) and v(k), respectively. Thus, ˜u(k) is independent from Y (k) and this independence is used when we +bound the first and second moments of v(k) +j += +1 +s⟨Y (k) +∗j , ˜u(k)⟩. For any 1 ≤ j ≤ m, the first and second +moments of v(k) +j += 1 +s⟨Y (k) +∗j , ˜u(k)⟩ satisfy +E +�1 +s +� +Y (k) +∗j , ˜u(k)�� += ⟨p, ˜u(k)⟩r(k) +j += ∥p∥2∥˜u(k)∥2(cos θ)r(k) +j += Θ(√n) +(38) +if r(k) +j +̸= 0 by Lemma 3 and 4, and +var +�1 +s +� +Y (k) +∗j , ˜u(k)�� +≤ 1 +s2 +n +� +i=1 +(˜u(k) +i +)2E[(Y (k) +ij )2] = Θ +�1 +s +� +(39) +since E[(Y (k) +ij )2] = Θ(s) and �n +i=1(˜u(k) +i +)2 = Θ(1) by Lemma 3 and 4. Furthermore, we have max1≤i≤m |Y (k) +ij ˜u(k) +i +| ≤ +Θ +� +1 +√n +� +since ˜u(k) +i +≤ +2 +η√n. By applying the Bernstein’s inequality, we can show that +P +����� +1 +s +� +Y (k) +∗j , ˜u(k)� +− ∥˜u(k)∥2∥p∥2r(k) +j +cos θ +���� > δ1∥p∥2 +2 +� +≤ 2 exp +� +− +Θ(δ2 +1∥p∥2 +2) +Θ +� 1 +s +� ++ Θ (δ1∥p∥2/√n) +� +≤ exp +� +−Θ(sδ2 +1∥p∥2 +2) +� +(40) +where the second inequality is due to the assumption ∥p∥2 = Θ(√n). To make this probability less than ϵ +2, +it is sufficient to have s ≥ Ω +� +1 +δ2 +1∥p∥2 +2 log 1 +ϵ +� +. +25 + +We next prove equation 37 by bounding +���∥˜u(k)∥2∥p∥2r(k) +j +cos θ − ∥p∥2r(k) +j +���. By the triangle inequality, +we have +���∥˜u(k)∥2∥p∥2r(k) +j +cos θ − ∥p∥2r(k) +j +��� ≤ +���∥˜u(k)∥2∥p∥2r(k) +j +cos θ − ∥p∥2r(k) +j +cos θ +��� ++ +���∥p∥2r(k) +j +cos θ − ∥p∥2r(k) +j +��� . +(41) +Note that +1 +∥p∥2 +· +���∥˜u(k)∥2∥p∥2r(k) +j +cos θ − ∥p∥2r(k) +j +cos θ +��� = r(k) +j +cos θ +���∥˜u(k)∥2 − 1 +��� +≤ Θ(sin2 θ(u(k), p∗)) = +1 +Θ (ns), +(42) +with probability 1 − (n + m)−8 by Lemma 3 and 4, and also note that +1 +∥p∥2 +· +���∥p∥2r(k) +j +cos θ − ∥p∥2r(k) +j +��� = r(k) +j +(1 − cos θ) +≤ Θ(sin2 θ(u(k), p∗)) = +1 +Θ (ns), +(43) +with probability 1 − (n + m)−8 by Lemma 3 and 4. To make these errors of order 1/Θ (ns) less than δ1 +2 , it +is sufficient to have s ≥ Ω +� +1 +δ1n +� +. +By combining the above results, it can be guaranteed that +��� 1 +2s +� +Y (k) +∗j , ˜u(k)� +− ∥p∥2r(k) +j +��� < δ∥p∥2 with +probability at least 1 − ϵ, if the sampling probability +s ≥ max +� +Ω +� +1 +δ2 +1∥p∥2 +2 +log 1 +ϵ +� +, Ω +� 1 +δ1n +�� += Ω +� +1 +δ2 +1∥p∥2 +2 +log 1 +ϵ +� +(44) +where the last equality is due to ∥p∥2 = Θ(√n). +The condition s = Ω(log(n + m)/m) in Lemma 3 is +immediately satisfied by equation 44 when n = O(m/ log m). +Proof of Theorem 1. +By using Lemma 5, we next prove Theorem 1. By applying the union bound over +k ∈ [K], if s ≥ Θ +� +1 +δ2 +1∥p∥2 +2 log K +ϵ +� +then we have +∥p∥2(r(k) +j +− δ1) ≤ v(k) +j += 1 +s +� +Y (k) +∗j , ˜u(k)� +≤ ∥p∥2(r(k) +j ++ δ1), ∀k ∈ [K] +(45) +for any δ1 > 0 and j ∈ [m] with probability at least 1 − ϵ. +Under the condition equation 45, for any +qj ∈ (1/2, 1) and δ < min +� +2qj−1 +2 +, 1−qj +2 +� +, we can guarantee that +1 +K − qj + δ < 1 +K − (1 − qj) − δ +and +1 +K − (1 − qj) + δ < 1 +K − δ, +(46) +which implies (ˆgj, ˆhj) = (gj, hj) for (ˆgj, ˆhj) defined in equation 3. This proves equation 8 of Theorem 1. +We next prove equation 9, the accuracy guarantee in estimating the task difficulty vector q. After estimat- +ing ∥p∥2r(k) by v(k) = 1 +s(Y (k))⊤ ˜u(k), we estimate ∥p∥2 by calculating l where lj := +K +K−2 +� +k̸=ˆgj,k̸=ˆhj ∆v(k) +j +and l := +1 +m +�m +j=1 lj. +Assume that |∥p∥2 − l| ≤ ∥p∥2δ′. We will specify the required order of δ′ later. +Remind that the estimate for qj is defined as ˆqj := +1 +K − +∆v +(ˆgj ) +j +l +. Under the condition that ˆgj = gj and +|vj − ∥p∥2r(k) +j +| ≤ ∥p∥2δ1, both of which are satisfied under the conditions of Lemma 5, we have +� 1 +K − qj − 2δ1 +� +1 + δ′ +≤ +∆v(ˆgj) +j +l +≤ +� 1 +K − qj + 2δ1 +� +1 − δ′ +. +(47) +26 + +By the Taylor expansion for +1 +1−x = 1 + x + Θ(x2) as x → 0, we have +|ˆqj − qj| ≤ 2δ1 + δ′ +� 1 +K − qj + 2δ1 +� ++ Θ(δ′2) = Θ(δ1 + δ′). +(48) +Thus, both the order of δ′, which is the estimation error of ∥p∥2, and that of δ, which is the estimation error +of ∥p∥2r(k) +j +, govern the estimation accuracy of qj. We next show that we can have δ′ = Θ(δ1). By Lemma +5, we have |vj − ∥p∥2r(k) +j +| ≤ ∥p∥2δ1, which implies +∥p∥2(∆r(k) +j +− 2δ1) ≤ ∆v(k) +j +≤ ∥p∥2(∆r(k) +j ++ 2δ1). +(49) +Under the condition (ˆgj, ˆhj) = (gj, hj), since ∆r(k) +j += 1 +K for k ̸= ˆgj, ˆhj, we have +∥p∥2 − ∥p∥2 +2δ1K +K − 2 ≤ lj = +K +K − 2 +� +k̸=ˆgj,k̸=ˆhj +∆v(k) +j +≤ ∥p∥2 + ∥p∥2 +2δ1K +K − 2, +(50) +and thus δ′ = 2δ1K +K−2 = Θ(δ1). Thus, it is enough to have s = Ω +� +1 +δ2 +1∥p∥2 +2 log K +ϵ +� +to guarantee equation 9. +Proof of Corollary 1. +By using Lemma 5 and taking the union bound over all tasks j ∈ [m] as well as +k ∈ [K], we can prove Corollary 1 in a similar way as that of Theorem 1. +G.2 +Proof of Lemma 4 +We first prove equation 33, +∥˜u(k)∥2 ≥ +� +1 − 50 sin2 θ(u(k), p∗). +Let I be the set of indices 1 ≤ i ≤ n such that u(k) +i +≥ +2 +η√n. Then, we have u(k) +i +− p∗ +i ≥ +1 +η√n for all i ∈ I +since p∗ +i = pi/∥p∥2 ≤ +1 +η√n due to the assumption that ∥p∥2 +2 ≥ η2n. Thus, we have +|I| +η2n ≤ +� +i∈I +(u(k) +i +− p∗ +i )2 ≤ ∥u(k) − p∗∥2 +2. +(51) +By using the triangle inequality, we can show that +�� +i∈I +� +u(k) +i +�2 +≤ +� +� +� +�� +i∈I +� +u(k) +i +− +2 +η√n +�2 ++ +� +4|I| +η2n +≤ +� +� +� +�� +i∈I +� +p∗ +i − +2 +η√n +�2 ++ +�� +i∈I +� +u(k) +i +− p∗ +i +�2 ++ +� +4|I| +η2n +≤ +� +4|I| +η2n + +�� +i∈I +� +u(k) +i +− p∗ +i +�2 ++ +� +4|I| +η2n +≤ 5∥u(k) − p∗∥2. +(52) +Therefore, we get +1 ≥ ∥˜u(k)∥2 +2 = 1 − +� +i∈I +(u(k) +i +)2 ≥ 1 − 25∥u(k) − p∗∥2 +2. +(53) +27 + +By the law of cosine, we have +∥p∗ − u(k)∥2 +2 = sin2 θ(u(k), p∗) + (1 − cos θ(u(k), p∗))2 = 2 − 2 cos θ(u(k), p∗) += 2 +� +1 − +� +1 − sin2 θ(u(k), p∗) +� += 2 +sin2 θ(u(k), p∗) +1 + +� +1 − sin2 θ(u(k), p∗) +≤ 2 sin2 θ(u(k), p∗). +(54) +Combining equation 53 and equation 54 proves equation 33. +We next prove equation 34, +sin θ(˜u(k), p∗) ≤ 6 +√ +2 sin θ(u(k), p∗). +First, note that ∥˜u(k) − u(k)∥2 +2 = � +i∈I +� +u(k) +i +�2 +. We have +sin θ(˜u(k), p∗) ≤ ∥˜u(k) − p∥2 ≤ ∥˜u(k) − u(k)∥2 + ∥u(k) − p∗∥2 ≤ 6∥u(k) − p∗∥2 +(55) +where the last inequality is from equation 52. Combined with equation 54, we get equation 34. +H +Performance Analysis of Algorithm 2 +H.1 +Proof of Lemma 1 +In this lemma, we show that conditioned on (ˆgj, ˆhj) = (gj, hj) for all j ∈ [m], if s(1 − s1) = Ω +� +1 +δ2m log n +ϵ +� +, +the estimator ˆpi defined in equation 5, +ˆpi = +K +(K − 2) +� +� +1 +s(1 − s1) +� +� 1 +m +m +� +j=1 +1(A2 +ij = ˆgj or ˆhj) +� +� − 2 +K +� +� , +guarantees P (∥p − ˆp∥∞ < δ2) ≥ 1 − ϵ for any ϵ > 0. +Given (ˆgj, ˆhj) = (gj, hj) for all j ∈ [m], since A2 is independent of (ˆgj, ˆhj), we have +E +� +1(A2 +ij = ˆgj or ˆhj) +� += P(A2 +ij = ˆgj or ˆhj) = s(1 − s1) +�K − 2 +K +pi + 2 +K +� +, +var +� +1(A2 +ij = ˆgj or ˆhj) +� +≤ s(1 − s1). +(56) +By applying the Bernstein’s inequality, we can show that +P +� +� +������ +m +� +j=1 +� +1(A2 +ij = ˆgj or ˆhj) − s(1 − s1) +�K − 2 +K +pi + 2 +K +�������� +> (K − 2)ms(1 − s1)δ2 +K +� +� +≤ exp +� +� +�− +1 +2 +� +(K−2)ms(1−s1)δ2 +K +�2 +ms(1 − s1) + 1 +3 +(K−2)ms(1−s1)δ2 +K +� +� +� ≤ exp +� +−Θ +� +ms(1 − s1)δ2 +2 +�� +. +(57) +Thus, if the sampling probability satisfies +s(1 − s1) = Ω +� 1 +mδ2 +2 +log 1 +ϵ +� +, +(58) +then we can guarantee that P(|ˆpi − pi| < δ2) ≥ 1 − ϵ. By taking the union bound over i ∈ [n], if the sampling +probability satisfies +s(1 − s1) = Ω +� 1 +mδ2 +2 +log n +ϵ +� +, +(59) +then we can guarantee that P (∥ˆp − p∥∞ < δ2) ≥ 1 − ϵ. +28 + +H.2 +Proof of Theorem 2 +To prove this theorem, we use similar proof techniques from Zhang et al. (2014). Since the work in Zhang +et al. (2014) focuses on the recovery of only the ground-truth label for each task, we generalize the techniques +to recover not only the ground-truth label but also the most confusing answer. +We first introduce some notations. Let µ(i,j) +(a,b),k denote the probability that a worker i ∈ [n] gives label +k ∈ [K] for the assigned task j ∈ [m] of which the top-two answers are (gj, hj) = (a, b). Let µ(i,j) +(a,b) = +[µ(i,j) +(a,b),1 µ(i,j) +(a,b),2 +· · · +µ(i,j) +(a,b),K]⊤. We introduce a quantity that measures the average ability of workers in +distinguishing the ground-truth pair of top-two answers (gj, hj) from any other pair (a, b) ∈ [K]2/{(gj, hj)} +for the task j ∈ [m]. We define +D +(j) := +min +(gj,hj)̸=(a,b) +1 +n +n +� +i=1 +DKL +� +µ(i,j) +(gj,hj), µ(i,j) +(a,b) +� +; +D := min +j∈[m] D +(j), +(60) +where DKL(P, Q) := � +i P(i) log(P(i)/Q(i)) is the KL-divergence between P and Q. Note that D +(j) is strictly +positive if qj ∈ (1/2, 1) and there exists at least one worker i with pi > 0 for the distribution equation 1, so +that (gj, hj) can be distinguished from any other (a, b) ∈ [K]2/{(gj, hj)} statistically. We define D as the +minimum of D +(j) over j ∈ [m], indicating the average ability of workers in distinguishing (gj, hj) from any +other (a, b) for the most difficult task in the set. +Let us define an event that will be shown holding with high probability, +E : +n +� +i=1 +K +� +k=1 +1(Aij = k) log +� +�µ(i,j) +(gj,hj),k +µ(i,j) +(a,b),k +� +� ≥ nsD/2 for all j ∈ [m] and (a, b) ∈ [K] × [K]\(gj, hj). +(61) +Define +li := +K +� +k=1 +1(Aij = k) log +� +µ(i,j) +(gj,hj),k/µ(i,j) +(a,b),k +� +. +(62) +We can see that l1, . . . , ln are mutually independent on any value of (gj, hj), and each li belongs to the +interval [0, log(1/ρ)] where µ(i,j) +(gj,hj),c ≥ ρ for all (i, j, gj, hj, c) ∈ [n] × [m] × [K]3. We can easily show that +E +� n +� +i=1 +li +�����(gj, hj) +� += +n +� +i=1 +sDKL +� +µ(i,j) +(gj,hj), µ(i,j) +(a,b) +� +. +(63) +We define +D := +n +� +i=1 +DKL +� +µ(i,j) +(gj,hj), µ(i,j) +(a,b) +� +. +(64) +The following lemma shows that the second moment of li is bounded above by the KL-divergence between +the label distribution under (gj, hj) pair and the label distribution under (a, b) pair. +Lemma 6. Conditioning on any value of (gj, hj), we have +E +� +l2 +i |(gj, hj) +� +≤ 2 log(1/ρ) +1 − ρ +sDKL +� +µ(i,j) +(gj,hj), µ(i,j) +(a,b) +� +. +(65) +The proof of this lemma can be obtained by following the proof of the similar result, Lemma 4 of Zhang +et al. (2014). +29 + +According to Lemma 6, the aggregated second moment of li is bounded by +E +� n +� +i=1 +l2 +i +�����(gj, hj) +� +≤ 2 log(1/ρ) +1 − ρ +n +� +i=1 +sDKL +� +µ(i,j) +(gj,hj), µ(i,j) +(a,b) +� += 2 log(1/ρ) +1 − ρ +sD. +(66) +Thus, applying the Bernstein’s inequality, we have +P +� n +� +i=1 +li ≥ sD/2 +�����(gj, hj) +� +≥ 1 − exp +� +− +1 +2(sD/2)2 +2 log(1/ρ) +1−ρ +sD + 1 +3(2 log(1/ρ))(sD/2) +� +. +(67) +Since ρ ≤ 1/2 and D ≥ nD +(j) ≥ nD, combining the above inequality with union bound over j ∈ [m], we +have +P [E] ≥ 1 − mK2 exp +� +− +nsD +33 log(1/ρ) +� +. +(68) +The maximum likelihood estimator finds a pair of (a, b) ∈ [K]2, a ̸= b, maximizing +(ˆgj, ˆhj) = +arg max +(a,b)∈[K]2,a̸=b +n +� +i=1 +P(Aij|p, qj, (a, b)) += +arg max +(a,b)∈[K]2,a̸=b +n +� +i=1 +log P(Aij|p, qj, (a, b)) += +arg max +(a,b)∈[K]2,a̸=b +n +� +i=1 +K +� +k=1 +1(Aij = k) log µ(i,j) +(a,b),k. +(69) +The plug-in MLE in equation 6, on the other hand, finds a pair of (a, b) ∈ [K]2, a ̸= b, maximizing +(ˆgj, ˆhj) = +arg max +(a,b)∈[K]2,a̸=b +n +� +i=1 +K +� +k=1 +1(Aij = k) log ˆµ(i,j) +(a,b),k +(70) +where ˆµ(i,j) +(a,b),k is the estimated probability that a worker i ∈ [n] gives label k ∈ [K] for the assigned task +j ∈ [m] of which the top two answers are (gj, hj) = (a, b) assuming pi = ˆpi from equation 5 and qj = ˆqj from +equation 4 in the distribution equation 1. Thus, for the plug-in MLE to correctly find the ground-truth top +two answers (gj, hj), we need to satisfy the following event: +n +� +i=1 +K +� +k=1 +1(Aij = k) log +� +ˆµ(i,j) +(gj,hj),k/ˆµ(i,j) +(a,b),k +� +≥ 0 for all (a, b) ∈ [K] × [K]\(gj, hj). +(71) +For any arbitrary (a, b) ̸= (gj, hj), consider the quantity +Q(a,b) := +n +� +i=1 +K +� +k=1 +1(Aij = k) log +� +ˆµ(i,j) +(gj,hj),k/ˆµ(i,j) +(a,b),k +� +, +(72) +which can be written as +Q(a,b) = +n +� +i=1 +K +� +k=1 +1(Aij = k) log +µ(i,j) +(gj,hj),k +µ(i,j) +(a,b),k ++ +n +� +i=1 +K +� +k=1 +1(Aij = k) +� +�log +� +� ˆµ(i,j) +(gj,hj),k +µ(i,j) +(gj,hj),k +� +� − log +� +� ˆµ(i,j) +(a,b),k +µ(i,j) +(a,b),k +� +� +� +� . +(73) +30 + +Assuming that there exist ρ > δ3 such that +µ(i,j) +(a,b),k ≥ ρ and |ˆµ(i,j) +(a,b),k − µ(i,j) +(a,b),k| ≤ δ3 for all i ∈ [n], j ∈ [m], (a, b) ∈ [K]2, +(74) +we have +max +i∈[n],k∈[K] +� +�log +� +� ˆµ(i,j) +(gj,hj),k +µ(i,j) +(gj,hj),k +� +� − log +� +� ˆµ(i,j) +(a,b),k +µ(i,j) +(a,b),k +� +� +� +� ≤ 2 log +� +ρ +ρ − δ3 +� +. +(75) +By the Bernstein’s inequality, we also have +P +������ +n +� +i=1 +K +� +k=1 +1(Aij = k) − ns +����� > ns/2 +� +≤ exp +� +− +1 +2(ns/2)2 +ns + 1 +3(ns/2) +� += exp +� +−3ns +28 +� +. +(76) +By taking the union bound over j ∈ [m], we have +P +������ +n +� +i=1 +K +� +k=1 +1(Aij = k) − ns +����� > ns/2 for any j ∈ [m] +� +≤ m exp +� +−3ns +28 +� +. +(77) +Under the intersection of the event +����n +i=1 +�K +k=1 1(Aij = k) − ns +��� ≤ ns/2 for all j ∈ [m] and the event E, we +can guarantee +Q(a,b) = +n +� +i=1 +K +� +k=1 +1(Aij = k) log +µ(i,j) +(gj,hj),k +µ(i,j) +(a,b),k ++ +n +� +i=1 +K +� +k=1 +1(Aij = k) +� +�log +� +� ˆµ(i,j) +(gj,hj),k +µ(i,j) +(gj,hj),k +� +� − log +� +� ˆµ(i,j) +(a,b),k +µ(i,j) +(a,b),k +� +� +� +� +≥ nsD +2 +− 3ns log +� +ρ +ρ − δ3 +� +≥ ns +�D +2 − +3δ3 +ρ − δ3 +� +> 0 +(78) +for every j ∈ [m] where the last inequality holds if +δ3 < ρ +D +6 + D. +(79) +In summary, under that the event +����n +i=1 +�K +k=1 1(Aij = k) − ns +��� ≤ ns/2 for all j ∈ [m] and the event E hold, +if we have δ3 such that +|ˆµ(i,j) +(a,b),k − µ(i,j) +(a,b),k| ≤ δ3 for all i ∈ [n], j ∈ [m], (a, b) ∈ [K]2 +(80) +and +δ3 < ρ and +δ3 < ρ +D +6 + D, +(81) +then we can guarantee that the plug-in MLE in equation 70 successfully recovers the pair of top two (gj, hj) +for all the tasks j ∈ [m]. To make the right-hand side of equation 68 and equation 77 less than ϵ/2, it is +sufficient to have +s = Ω +�log(1/ρ) log(mK2/ϵ) + D log(m/ϵ) +nD +� +. +(82) +Lastly, when we have +max{∥p − ˆp∥∞, ∥q − ˆq∥∞} ≤ δ, +(83) +we can guarantee that +|ˆµ(i,j) +(a,b),k − µ(i,j) +(a,b),k| ≤ 4δ := δ3. +(84) +Thus, it is sufficient to guarantee equation 83 with +δ < min +�ρ +4, +ρD +4(6 + D) +� +. +(85) +31 + +I +Proof of Theorem 3 +I.1 +Proof of part (a) +To prove this minimax bound, we use the similar arguments from Karger et al. (2014). In particular, we +consider a spammer-hammer model such that +pi = +� +0, for 1 ≤ i ≤ ⌊(1 − p)n⌋ +1, otherwise. +(86) +Assume that total lj workers randomly sampled from [n] provide answers for the task j. Under the spammer- +hammer model, the oracle estimator makes a mistake on task j with probability (K − 1)/K if it is only +assigned to spammers. When lj is the number of assignments, we have +P(ˆgj ̸= gj) = K − 1 +K +(1 − p)lj. +(87) +By convexity and using Jensen’s inequality, the average probability of error is lower bounded by +1 +m +� +j∈[m] +P(ˆgj ̸= gj) ≥ K − 1 +K +(1 − p)l +(88) +where +1 +m +� +i∈[m] li ≤ l. By assuming p ≤ 2/3, we have (1 − p) ≥ e−(p+p2). Thus, +min +ˆg +max +p∈Fp, g∈[K]m +1 +m +� +j∈[m] +P(ˆgj ̸= gj) ≥ K − 1 +K +e−(p+p2)l ≥ K − 1 +K +e−2pl. +(89) +The inequality in equation 89 implies that if l is less than +1 +2p log +� K−1 +Kϵ +� +, then no algorithm can make the +minimax error in equation 89 less than ϵ. Since the average number of queries per task in our model is ns, +it implies that it is necessary to have s = Ω +� +1 +∥p∥2 +2 log 1 +ϵ +� +. +I.2 +Proof of part (b) +To prove the second part of the theorem, we use proof techniques from Zhang et al. (2014), but generalizes +the results for pair of top two answers. We assume that jc ∈ [m], (gc, hc) ∈ [K]2 and (ac, bc) ∈ [K]2 are the +task index and the pairs of labels such that +D = 1 +n +n +� +i=1 +DKL +� +µ(i,jc) +(gc,hc), µ(i,jc) +(ac,bc) +� +(90) +for D defined in equation 60. +Let Q be a uniform distribution over the set {(gc, hc), (ac, bc)}m. For any (ˆg, ˆh), we have +max +(v,u)∈[K]m×[K]m +vj̸=uj,∀j[m] +E +� +� +m +� +j=1 +1((ˆgj, ˆhj) ̸= (gj, hj)) +���(g, h) = (v, u) +� +� +≥ +m +� +j=1 +� +(v,u)∈{(gc,hc),(ac,bc)}m +Q((v, u))E +� +1((ˆgj, ˆhj) ̸= (gj, hj)) +���(g, h) = (v, u) +� +(91) +Let A := {Aij : i ∈ [n], j ∈ [m]} be the set of observations. Define two probability measures P0 and P1, such +that P0 is the measure of A conditioned on (gj, hj) = (gc, hc), while P1 is that on (gj, hj) = (ac, bc). Then, +32 + +we can have +� +(v,u)∈{(gc,hc),(ac,bc)}m +Q((v, u))E +� +1((ˆgj, ˆhj) ̸= (gj, hj)) +���(g, h) = (v, u) +� += Q((gj, hj) = (gc, hc))P0((ˆgj, ˆhj) ̸= (gc, hc)) + Q((gj, hj) = (ac, bc))P1((ˆgj, ˆhj) ̸= (ac, bc)) +≥ 1 +2 − 1 +2∥P0 − P1∥TV +≥ 1 +2 − 1 +4 +� +DKL(P0, P1). +(92) +where the second to the last inequality is by Le Cam’s method and the last inequality is by Pinsker’s +inequality.4 +Conditioned on (gj, hj), the set of random variables Aj := {Aij : i ∈ [n]} are independent of A\Aj for +both P0 and P1, and thus +DKL(P0, P1) = DKL(P0(Aj), P1(Aj)) + DKL(P0(A\Aj), P1(A\Aj)) = DKL(P0(Aj), P1(Aj)) +(93) +where P(X) denote the distribution of X with respect to the probability measure P. Given (gj, hj), since +A1j, . . . , Anj are independent, we can show that +DKL(P0(Aj), P1(Aj)) = +n +� +i=1 +DKL(P0(Aij), P1(Aij)) += +n +� +i=1 +� +(1 − s) log 1 − s +1 − s + sDKL +� +µ(i,j) +(gc,hc), µ(i,j) +(ac,bc) +�� +≥ snD. +(94) +Combining equation 91– equation 94, we have +max +(v,u)∈[K]m×[K]m +vj̸=uj,∀j[m] +E +� +� 1 +m +m +� +j=1 +1((ˆgj, ˆhj) ̸= (gj, hj)) +���(g, h) = (v, u) +� +� +≥ 1 +2 − 1 +4 +� +snD. +(95) +Thus, if s ≤ +1 +4nD, then the above inequality is lower bounded by 3/8. This completes the proof. +J +Useful Inequalities +In this section, we summarize the useful inequalities used in the proof of the main results. +The following inequality, which appeared in Bandeira & Van Handel (2016) provides a non-asymptotic +spectral norm bound for random matrices with independent random entries. +Theorem 4 (Spectral norm bound of a random matrice with independent entries). Consider a random +matrix X ∈ Rn×m, whose entries are independently generated and obey +E[Xi,j] = 0, +and +|Xi,j| ≤ B, +1 ≤ i ≤ n, 1 ≤ j ≤ m. +(96) +Define +ν := max +� +� +�max +i +� +j +E[X2 +i,j], max +j +� +i +E[X2 +i,j] +� +� +� . +(97) +4The total variation distance between probability distributions P and Q defined on a set X is defined as the maximum +difference between probabilities they assign on subsets of X: ∥P − Q∥TV := supA⊂X |P(A) − Q(A)|. +33 + +Then there exists some universal constant c > 0 such that for any t > 0, +P +� +∥X∥ ≥ 4√ν + t +� +≤ (n + m) exp +� +− t2 +cB2 +� +. +(98) +We also present a useful corollary of Theorem 4, which can be shown from equation 98 by setting ˜c = +√ +9c +and t = B +� +9c log(n + m). +Corollary 3 (Corollary of Theorem 4). If E[X2 +i,j] ≤ σ2 for all i, j and satisfying conditions in Theorem 4, +then we have +∥X∥ ≤ 4σ +� +max(m, n) + ˜cB +� +log(n + m) +(99) +with probability 1 − (n + m)−8 for some constant ˜c > 0. +We next summarize the eigenspace perturbation theory for asymmetric matrices with singular value +composition (SVD). Suppose X := [X0, X1] and Z := [Z0, Z1] are orthonormal matrices. When we define +the distance between two subspaces X0 and Z0 by +dist(X0, Z0) := ∥X0X⊤ +0 − Z0Z⊤ +0 ∥, +(100) +then we have +dist(X0, Z0) = ∥X⊤ +0 Z1∥ = ∥Z⊤ +0 X1∥. +(101) +Given ∥X⊤ +0 Z0∥ ≤ 1, we write SVD of X⊤ +0 Z0 ∈ Rr×r as X⊤ +0 Z0 := U cos ΘV ⊤ where cos Θ = diag(cos θ1, . . . , cos θr). +We call {θ1, . . . , θr} principal angles between X0 and Z0. Then, we have +∥X⊤ +0 Z1∥ = ∥ sin Θ∥ = max{| sin θ1|, · · · , | sin θr|}. +(102) +Let M ∗ and M = M ∗ + E be two matrices in Rn×m with n ≤ m, whose SVD are represented by +M ∗ = �n +i=1 σ∗ +i u∗ +i v∗ +i +⊤ and M = �n +i=1 σiuivi⊤, where σ1 ≥ · · · ≥ σn (resp. σ∗ +1 ≥ · · · ≥ σ∗ +n). Let us define +U0 := [u1, · · · , ur] ∈ Rn×r, +V0 := [v1, · · · , vr] ∈ Rm×r. +(103) +The matrices U ∗ +0 and V ∗ +0 are defined analogously. +Theorem 5 (Wedin sin Θ Theorem). If ∥E∥ < σ∗ +r − σ∗ +r+1, then one has +max{∥dist(U0, U ∗ +0 )∥, ∥dist(V0, V ∗ +0 )∥} ≤ +√ +2∥E∥ +σ∗r − σ∗ +r+1 − ∥E∥, +(104) +where U ∗ +0 (V ∗ +0 ) and U0 (V0) are subspaces spanned by the largest r left (right) singular vectors of M ∗ and +M, respecively. +Lastly, we also write down two useful concentration inequalities. +Theorem 6 (Hoeffding). Let X1, X2, . . . , Xn be independent random variables such that Xi ∈ [ai, bi] for +1 ≤ i ≤ n. Then, we have +P +������ +n +� +i=1 +(Xi − E[Xi]) +����� > t +� +≤ 2 exp +� +− +2t2 +�n +i=1(bi − ai)2 +� +. +(105) +Theorem 7 (Bernstein). Let X1, X2, . . . , Xn be independent random variables such that Xi ∈ [ai, bi] for +1 ≤ i ≤ n. Let C := max1≤i≤n(bi − ai) and σ2 = �n +i=1 var(Xi). Then we have +P +������ +n +� +i=1 +(Xi − E[Xi]) +����� > t +� +≤ 2 exp +� +− +t2/2 +σ2 + C · t/3 +� +. +(106) +34 + diff --git a/wdAyT4oBgHgl3EQfOfZu/content/tmp_files/load_file.txt b/wdAyT4oBgHgl3EQfOfZu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f5e64b3725bd9b51ddf3d2a497c2e3b242db4f2f --- /dev/null +++ b/wdAyT4oBgHgl3EQfOfZu/content/tmp_files/load_file.txt @@ -0,0 +1,1476 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf,len=1475 +page_content='Recovering Top-Two Answers and Confusion Probability in Multi-Choice Crowdsourcing Hyeonsu Jeong∗ Hye Won Chung† Abstract Crowdsourcing has emerged as an effective platform to label a large volume of data in a cost- and time- efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Most previous works have focused on designing an efficient algorithm to recover only the ground-truth labels of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In this paper, we consider multi-choice crowdsourced labeling with the goal of recovering not only the ground truth but also the most confusing answer and the confusion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The most confusing answer provides useful information about the task by revealing the most plausible answer other than the ground truth and how plausible it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To theoretically analyze such scenarios, we propose a model where there are top-two plausible answers for each task, distinguished from the rest of choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Task difficulty is quantified by the confusion probability between the top two, and worker reliability is quantified by the probability of giving an answer among the top two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Under this model, we propose a two-stage inference algorithm to infer the top-two answers as well as the confusion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We show that our algorithm achieves the minimax optimal convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We conduct both synthetic and real-data experiments and demonstrate that our algorithm outperforms other recent algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We also show the applicability of our algorithms in inferring the difficulty of tasks and training neural networks with the soft labels composed of the top-two most plausible classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 Introduction Crowdsourcing has been widely adopted to solve a large number of tasks in a time- and cost-efficient manner with the aid of human workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In this paper, we consider ‘multiple-choice’ tasks where a worker is asked to provide a single answer among multiple choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Some examples are as follows: 1) Using crowdsourcing platforms such as MTurk, we solve object counting or classification tasks on a large collection of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Answers can be noisy either due to the difficulty of the scene or due to unreliable workers who provide random guesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2) Scores are collected from reviewers for papers submitted at a conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For certain papers, scores can vary widely among reviewers, either due to the paper’s inherent nature (clear pros and cons) or due to the reviewer’s subjective interpretation of the scoring scale (Stelmakh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In the above scenarios, responses provided by human workers may not be consistent among themselves not only due to the existence of unreliable workers but also due to the inherent difficulty of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In particular, for multiple-choice tasks, there could exist plausible answers other than the ground truth, which we call confusing answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 For tasks with confusing answers, even reliable workers may provide wrong answers due to confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Thus, we need to decompose the two different causes of wrong answers: low reliability of workers and confusion due to task difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Most previous models for multi-choice crowdsourcing, however, fall short of modeling the confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For example, in the single-coin Dawid-Skene model (Dawid & Skene, 1979), which is the most widely studied crowdsourcing model in the literature, it is assumed that a worker is associated with a single skill parameter ∗School of Electrical Engineering, KAIST, Daejeon, 34141, Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' email: hsjeong1121@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='kr †School of Electrical Engineering, KAIST, Daejeon, 34141, Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' email: hwchung@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='kr 1This phenomenon is evident on public datasets: for ‘Web’ dataset (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2012), which has five labels, the most dominating top-two answers take 80% of the overall answers and the ratio between the top two is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='00006v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='HC] 29 Dec 2022 fixed across all tasks, which models the probability of giving a correct answer for every task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Under this model, any algorithm that infers the worker skill would count a confused labeling as the worker’s error and lower its accuracy estimate for the worker, which results in a wrong estimate for their true skill level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To model the effect of confusion in multi-choice crowdsourcing problems, we propose a new model under which each task can have a confusing answer other than the ground truth, with a varying confusion probability across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The task difficulty is quantified by the confusion probability, and the worker skill is modeled by the probability of giving an answer among the top two, to distinguish reliable workers from pure spammers who just provide random guesses among possible choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We justify the proposed top-two model with public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Under this new model, we aim to recover both the ground truth and the most confusing answer with the confusion probability, indicating how plausible the recovered ground truth is compared to the most confusing answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We provide an efficient two-stage inference algorithm to recover the top-two plausible answers and the confusion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The first stage of our algorithm uses the spectral method to get an initial estimate for top-two answers as well as the confusion probability, and the second stage uses this initial estimate to estimate the worker reliabilities and to refine the estimates for the top-two answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our algorithm achieves the minimax optimal convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We then perform experiments where we compare our method to recent crowdsourcing algorithms on both synthetic and real datasets, and show that our method outperforms other methods in recovering top-two answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This result demonstrates that our model better explains the real-world datasets including errors from confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our key contributions can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Top-two model: We propose a new model for multi-choice crowdsourcing tasks where each task has top-two answers and the difficulty of the task is quantified by the confusion probability between the top-two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We justify the proposed model by analyzing six public datasets, and showing that the top-two structure explains well the real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Inference algorithm and its applicaitons: We propose a two-stage algorithm that recovers the top-two answers and the confusion probability of each task at the minimax optimal convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We demonstrate the potential applications of our algorithm not only in crowdsourced labeling but also in quantifying task difficulty and training neural networks for classification with soft labels including the top-two information and the task difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Related works In crowdsourcing (Welinder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Liu & Wang, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Demartini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Aydin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Demartini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2012), one of the most widely studied models is the Dawid-Skene (D&S) model (Dawid & Skene, 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In this model, each worker is associated with a single confusion matrix fixed across all tasks, which models the probability of giving a label b ∈ [K] for the true label a ∈ [K] for K-ary classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In the single-coin D&S model, the model is further simplified such that each worker possesses a fixed skill level regardless of the true label or the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Under the D&S model, various methods were proposed to estimate the confusion matrix or skill of each worker by spectral method (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Dalvi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Karger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2013), belief propagation or iterative algorithms (Karger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Li & Yu, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Ok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2016), or rank-1 matrix completion (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Ma & Olshevsky, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Ibrahim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The estimated skill can be used to infer the ground-truth answer by approximating the maximum likelihood (ML)-type estimators (Gao & Zhou, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Karger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Li & Yu, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Raykar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Smyth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Ipeirotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Berend & Kontorovich, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In contrast to the D&S models, our model allows the worker to have different probability of error caused by confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Thus, our algorithm needs to estimate not only the worker skill but also the task difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Since the number of tasks is often much larger than the number of workers in practice, estimating the task difficulties is much more challenging than estimating worker skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We provide a statistically-efficient algorithm to estimate the task difficulties and use this estimate to infer the top-two answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We also remark that there are some recent attempts to model task difficulties (Khetan & Oh, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Shah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Krivosheev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Shah & Lee, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Bachrach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Tian & Zhu, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' However, these works are either restricted to binary tasks (Khetan & Oh, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Shah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2 Shah & Lee, 2018) or focus on grouping confusable classes (Krivosheev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Tian & Zhu, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our result, on the other hand, applies to any set of multi-choice tasks, where the choices of each task are not necessarily restricted to a fixed set of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For a vector x, xi represents the i-th component of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For a matrix M, Mij refers to the (i, j)th entry of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For any vector x, its ℓ2 and ℓ∞-norm are denoted by ∥x∥2 and ∥x∥∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We follow the standard definitions of asymptotic notations, Θ(·), O(·), o(·), and Ω(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2 Model and Problem Setup We consider a crowdsourcing model to infer the top-two most plausible answers among K choices for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' There are n workers and m tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For each task j ∈ [m] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' , m}, we denote the correct answer by gj ∈ [K] and the next plausible, or the most confusing answer by hj ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We call the pair (gj, hj) the top-two answers for task j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let p ∈ [0, 1]n and q ∈ (1/2, 1]m be parameters modeling the reliability of workers and difficulty of tasks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For every pair of (i, j), the j-th task is assigned to the i-th worker independently with probability s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We use a matrix A ∈ Rn×m to represent the responses of workers, where Aij = 0 if the j-th task is not assigned to the i-th worker, and if it is assigned, and Aij is equal to the received label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The distribution of Aij is specified by the worker reliability pi and task difficulty qj as follows: Aij = � � � � � � � � � gj, with prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' s � piqj + 1−pi K � , hj, with prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' s � pi(1 − qj) + 1−pi K � , each b ∈ [K]\\{gj, hj}, with prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' s � 1−pi K � , 0, with prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (1) Here pi stands for the reliability of the i-th worker, in giving the answer from the most plausible top two (gj, hj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' If pi = 0, the worker is considered a spammer who provides random answers among K choices, and a larger value of pi indicates a higher worker reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The parameter qj represents the inherent difficulty of the task j in distinguishing between the top two answers: for an easy task, qj is closer to 1, and for a hard task, qj is closer to 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We call qj the confusion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our goal is to recover top-two answers (gj, hj) for all j ∈ [m] with high probability with the minimum possible sampling probability s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We assume that the model parameters (p, q) are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We propose the top-two model to reflect common attributes of public crowdsourcing datasets, summarized in Appendix §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The most important observation is that the top-two answers dominate the overall answers, and only the second-dominating answer has an incidence rate comparable to that of the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In other words, the standard deviation in the incidence rate of the second dominating answer has an overlap with that of the ground truth, but not the third-, or fourth-dominating answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This indicates that assuming a unique ‘confusing answer’ is sufficient to model the confusion stemming from task difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' More details are available in Appendix §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Binary conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The K-ary task can be decomposed into (K − 1)-binary tasks (Karger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2013): define A(k) for 1 ≤ k < K such that the (i, j)-th entry A(k) ij indicates whether the answer Aij is larger than k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', A(k) ij = −1 if 1 ≤ Aij ≤ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' A(k) ij = 1 if k < Aij ≤ K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' and A(k) ij = 0 if Aij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We show that E[A(k)] is rank-1 and the singular value decomposition (SVD) of E[A(k)] can reveal the top-two answers {(gj, hj)}m j=1 and the confusion probability vector q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For every 1 ≤ k < K, the binary-mapped matrix A(k) ∈ {−1, 0, 1}n×m satisfies E[A(k)] − 3 s(K−2k) K 1n×m = 2sp(r(k))⊤, where r(k) = [r(k) 1 · · r(k) m ]⊤ is defined as Case I: gj > hj r(k) j := � � � � � k K where k < hj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' k K − (1 − qj) where hj ≤ k < gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' k K − 1 where gj ≤ k, Case II: gj < hj r(k) j := � � � � � k K where k < gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' k K − qj where gj ≤ k < hj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' k K − 1 where hj ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By defining ∆r(k) j := r(k) j − r(k−1) j for k ∈ [K] with r(0) j := 0 and r(K) j := 0 for all j, we have ∆r(k) j = � � � � � 1 K − qj where k = gj, 1 K − (1 − qj) where k = hj, 1 K otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (2) Note that ∆r(k) j has its minimum at k = gj and the second smallest value at k = hj for qj ∈ (1/2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' If one can specify gj, the task difficulty qj can also be revealed from 1 K − ∆r(gj) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In the next section, we use this structure of r(k) for k ∈ [K] to infer the top two answers and the confusion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 3 Proposed Algorithm Our algorithm consists of two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In Stage 1, we compute an initial estimate on top-two answers and the confusion probability q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In Stage 2, we estimate the worker reliability vector p by using the result of the first stage, and use the estimated p and q to refine our estimates for the top two answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Assume that we randomly split the original response matrix A into A1 and A2 with probability s1 and 1 − s1, respectively, and use only A1 for stage 1 and (A1, A2) for stage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 Stage 1: Initial estimates using SVD The first stage begins with randomly splitting A1 again into two independent matrices B and C with equal probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We then convert B and C into (K − 1)-binary matrices B(k) and C(k) as explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Define X(k) and Y (k) as X(k) := B(k) − s′(K−2k) K 1n×m and Y (k) := C(k) − s′(K−2k) K 1n×m for s′ = s · s1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We have E[X(k)] = E[Y (k)] = s′p(r(k))⊤ from Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We use X(k) and Y (k) to estimate p∗ := p/∥p∥2 and ∥p∥2r(k), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The estimators are denoted by u(k) and v(k), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We define u(k) as the left singular vector of X(k) with the largest singular value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Sign ambiguity of the singular vector is resolved by defining u(k) as the one between {u(k), −u(k)} in which at least half of the entries are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' After trimming abnormally large components of u(k) and defining the trimmed vector as ˜u(k), we calculate v(k) := 1 s′ (Y (k))⊤ ˜u(k), which is an estimate for ∥p∥2r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By using v(k) for 1 ≤ k < K, we get estimates for top-two answers (ˆgj, ˆhj) based on the observation in equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Lastly, we estimate ∥p∥2 and use v(k)/∥p∥2 ≈ r(k) to estimate the confusion probability vector q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' See Algorithm 1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 Stage 2: Plug-in Maximum Likelihood Estimator (MLE) The second stage uses the result of Stage 1 to estimate the worker reliability vector p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We first propose an estimate for the worker reliability vector p by using the estimated top-two answers {(gj, hj)}m j=1 from Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We randomly split the original response matrix A into A1 and A2 with probability s1 and 2We assume that η√n ≤ ∥p∥2 ≤ √n for some η > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', there are only o(n) spammers (pi = 0), and ∥r(k)∥2 = Θ(√m) for every k ∈ [K], which can be easily satisfied except exceptional cases from equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 4 Algorithm 1 Spectral Method for Initial Estimation (TopTwo1 Algorithm) 1: Input: data matrix A1 ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' , K}n×m and parameter η > 0 where η√n ≤ ∥p∥2 ≤ √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2: Randomly split (with equal probabilities) and convert A1 into binary matrices X(k) ∈ {−1, 0, 1}n×m and Y (k) ∈ {−1, 0, 1}n×m for 1 ≤ k < K as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 3: Let u(k) be the leading normalized left singular vector of X(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Trim the abnormally large components of u(k) by letting it be zero if u(k) i > 2 η√n and denote the resulting vector as ˜u(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 4: Calculate the estimate of ∥p∥r(k) by v(k) := 1 s′ (Y (k))⊤ ˜u(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Assume v(0) := 0 and v(K) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5: For k ∈ [K], calculate ∆v(k) j := v(k) j − v(k−1) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Estimate the top-two answers for j ∈ [m] by ˆgj := arg min k∈[K] ∆v(k) j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' ˆhj := arg min k̸=ˆgj,k∈[K] ∆v(k) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (3) 6: Estimate ∥p∥2 by defining lj := K K−2 � k̸=ˆgj,k̸=ˆhj ∆v(k) j and l := 1 m �m j=1 lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 7: Estimate qj for j ∈ [m] by defining ˆqj := 1/K − ∆v(ˆgj) j /l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (4) 8: Output: estimated top-two answers {(ˆgj, ˆhj)}m j=1 and confusion probability vector ˆq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Algorithm 2 Plug-in MLE (TopTwo2 Algorithm) 1: Input: data matrix A ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' , K}n×m and the sample splitting rate s1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2: Randomly split A into A1 and A2 by defining A1 := A ◦ S and A2 = A ◦ (1n×m − S) where S is an n × m matrix whose entries are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' with Bern(s1) and ◦ is an entrywise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 3: Apply Algorithm 1 to A1 to yield estimates for top-two answers {(ˆgj, ˆhj)}m j=1 and confusion probability vector ˆq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 4: By using {(ˆgj, ˆhj)}m j=1 and A2, calculate the estimate ˆp as in equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5: By using the whole A and (ˆp, ˆq), find the plug-in MLE estimates (ˆgMLE j , ˆhMLE j ) by arg max a,b∈[K]2,a̸=b n � i=1 log �K ˆpiˆqj 1 − ˆpi + 1 � 1(Aij = a) + log �K ˆpi(1 − ˆqj) 1 − ˆpi + 1 � 1(Aij = b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (6) 6: Output: estimated top-two answers {(ˆgMLE j , ˆhMLE j )}m j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 − s1, respectively, and use A1 only for Algorithm 1 and A2 only for calculating the estimator ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our estimate for the worker reliability pi is defined as ˆpi = K (K − 2) � � 1 s(1 − s1) � � 1 m m � j=1 1(A2 ij = ˆgj or ˆhj) � � − 2 K � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (5) Our plug-in MLE uses the estimated (ˆp, ˆq) in the place of (p, q) at the oracle MLE, which finds (ˆgj, ˆhj) ∈ [K]2\\{(1, 1), (1, 2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' , (K, K)} such that (ˆgj, ˆhj) := arg max(a,b)∈[K]2,a̸=b �n i=1 log P(Aij|p, qj, (a, b)) as in equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Details can be found in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The time complexity of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2 is O(m2 log m + nmK2), since the SVD in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 can be computed via power iterations within O(m2 log m) steps (Boutsidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2015), and the step for finding the pair of answers maximizing equation 6 requires O(nmK2) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5 4 Performance Analysis To state our main theoretical results, we first need to introduce some notation and assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let µ(i,j) (a,b),k denote the probability that a worker i ∈ [n] gives label k ∈ [K] for the assigned task j ∈ [m] of which the top-two answers are (gj, hj) = (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Note that µ(i,j) (a,b),k can be written in terms of (pi, qj) from the distribution in equation 1 for every a, b, k ∈ [K]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let µ(i,j) (a,b) = [µ(i,j) (a,b),1 µ(i,j) (a,b),2 · · µ(i,j) (a,b),K]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We introduce a quantity that measures the average ability of workers in distinguishing the ground-truth pair of top-two answers (gj, hj) from any other pair (a, b) ∈ [K]2/{(gj, hj)} for the task j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We define D (j) := min (gj,hj)̸=(a,b) 1 n n � i=1 DKL � µ(i,j) (gj,hj), µ(i,j) (a,b) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' D := min j∈[m] D (j), (7) where DKL(P, Q) := � i P(i) log(P(i)/Q(i)) is the KL-divergence between P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Note that D (j) is strictly positive if there exist at least one worker i with pi > 0 and qj ∈ (1/2, 1) for the distribution in equation 1, so that (gj, hj) can be distinguished from any other (a, b) ∈ [K]2/{(gj, hj)} statistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We define D as the minimum of D (j) over j ∈ [m], indicating the average ability of workers in distinguishing (gj, hj) from any other (a, b) for the most difficult task in the set of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We split the performance analysis of our algorithm into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' First, Theorem 1 states the perfor- mance guarantees for Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Theorem 1 (Performance Guarantees for Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For any ϵ, δ1 > 0, if the sampling probability s · s1 = Ω � 1 δ2 1∥p∥2 2 log K ϵ � , Algorithm 1 guarantees the recovery of the ordered top-two answers (gj, hj) with probability at least 1 − ϵ for any j ∈ [m] with qj ∈ (1/2, 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', P � (ˆgj, ˆhj) = (gj, hj) � ≥ 1 − ϵ for all j ∈ [m] with qj ∈ (1/2, 1), (8) and the recovery of the confusion probability qj with P (|ˆqj − qj| < δ1) ≥ 1 − ϵ for all j ∈ [m], (9) for every sufficiently large number m of tasks and the number of workers n = O(m/ log m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By using Theorem 1, we can also find the sufficient conditions to guarantee the recovery of paired top-two answers for all tasks and q with an arbitrarily small ℓ∞-norm error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For any ϵ, δ1 > 0, if the sampling probability s·s1 = Ω � 1 δ2 1∥p∥2 2 log mK ϵ � , Algorithm 1 guarantees the recovery of {(gj, hj)}m j=1 and q with probability at least 1 − ϵ as m → ∞ such that P � (ˆgj, ˆhj) = (gj, hj), ∀j ∈ [m] � ≥ 1 − ϵ and P (∥q − ˆq∥∞ < δ1) ≥ 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (10) Proofs of Theorem 1 and Corollary 1 are available in Appendix §G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We next analyze the performance of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2, which uses Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 as the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Before providing the main theorem for Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2, we state a lemma charactering a sufficient condition for estimating the worker reliability vector p from equation 5 with an arbitrarily small ℓ∞-norm error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Conditioned on (ˆgj, ˆhj) = (gj, hj) for all j ∈ [m], if s(1 − s1) = Ω � 1 δ2 2m log n ϵ � , the estimator ˆpi defined in equation 5 of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2 guarantees P (∥p − ˆp∥∞ < δ2) ≥ 1 − ϵ for any ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Combining Corollary 1 and Lemma 1, we can have the estimators (ˆp, ˆq) for the worker reliability vector p and the confusion probability vector q with ℓ∞-norm error bounded by any arbitrarily small δ > 0 with probaiblity at least 1 − 2ϵ if s = s · s1 + s(1 − s1) = Ω �log(mK/ϵ) δ2∥p∥2 2 + log(n/ϵ) δ2m � = Ω �log(mK/ϵ) δ2∥p∥2 2 � (11) 6 where the last equality is from the assumption that ∥p∥2 = Θ(√n) and n = O(m/ log m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In this regime, the sample complexity for estimating the task difficulty q is larger than that for estimating worker reliability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To make sure that the sampling probability s < 1, we need n = Ω(log m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our second theorem, Theorem 2, characterizes the sufficient condition on the sampling probability s to guarantee the recovery of the pair of top-two answers for all tasks by equation 6 of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2, when a sufficiently accurate estimation of (p, q) is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Assume that there is a positive scalar ρ such that µ(i,j) (gj,hj),c ≥ ρ for all (i, j, gj, hj, c) ∈ [n] × [m] × [K]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For any ϵ > 0, if (ˆp, ˆq) are given with max{∥p − ˆp∥∞, ∥q − ˆq∥∞} ≤ δ := min �ρ 4, ρD 4(6 + D) � , (12) and the sampling probability s = Ω � log(1/ρ) log(mK2/ϵ)+D log(m/ϵ) nD � , then for any ϵ > 0 the estimates of {(gj, hj)}m j=1 from equation 6 of Algorithm 2 guarantees P � (ˆgj, ˆhj) = (gj, hj), ∀j ∈ [m] � ≥ 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (13) Proofs of Lemma 1 and Theorem 2 are available in Appendix §H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The assumption in Theorem 2 that µ(i,j) (gj,hj),c ≥ ρ for some ρ > 0 holds if pi < 1 for all i ∈ [n], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', there is no perfectly reliable worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This assumption can be easily satisfied by adding an arbitrary small random noise to the worker answers as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By combining the statements in Corollary 1, Lemma 1, and Theorem 2 with δ1 = δ2 = δ for δ defined in equation 12, we get the overall performance guarantee for Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Corollary 2 (Performance Guarantees for Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2 guarantees the recovery of top-two answers for all tasks with P � (ˆgj, ˆhj) = (gj, hj), ∀j ∈ [m] � ≥ 1 − ϵ for any ϵ > 0 if s satisfies s = Ω �log(mK/ϵ) δ2∥p∥2 2 + log(1/ρ) log(mK2/ϵ) + D log(m/ϵ) nD � = Ω �log(m/ϵ) δ2∥p∥2 2 + log(m/ϵ) nD � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (14) In equation 14, the first term is for guaranteeing accurate estimates of p and q with ℓ∞-norm error bounded by δ and the second term is for guaranteeing the recovery of the top-two answers from MLE with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Since ∥p∥2 2 = Θ(n), the two terms effectively have the same order but with different constant scaling, depending on model-specific parameters (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Lastly, we show the optimality of convergence rates of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 and Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2 with respect to two types of minimax errors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The proof of Theorem 3 is available in Appendix §I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (a) Let Fp be a set of p ∈ [0, 1]n such that the collective quality of workers, measured by ∥p∥2, is parameterized by p as F¯p := {p : 1 n∥p∥2 2 = p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Assume that p ≤ 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' If the average number ns of samples (queries) per task is less than (1/2p) log(1/ϵ), then min ˆg max p∈Fp, g∈[K]m 1 m � j∈[m] P(ˆgj ̸= gj) ≥ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (15) (b) There is a universal constant c > 0 such that for any p ∈ [0, 1]n, q ∈ (1/2, 1]m, if the sampling probability s < Ω � 1/(nD) � , then min (ˆg,ˆh) max (g,h)∈[K]m×[K]m gj̸=hj,∀j[m] 1 m � j∈[m] P((ˆgj, ˆhj) ̸= (gj, hj)) ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (16) From part (a) of Theorem 3, it is necessary to have s > Ω � (1/∥p∥2 2) log(1/ϵ) � to make the minimax error in equation 15 less than ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Since Theorem 1 shows that Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 recovers (ˆgj, ˆhj) with probability at 7 Figure 1: Prediction error for (g, h) for four scenarios as the avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' number of queries per task changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our TopTwo2 alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' achieves the best performance, near the oracle MLE for all the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' least 1 − ϵ if s > Ω � (1/∥p∥2 2) log(1/ϵ) � when s1 = 1, we can conclude that Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 achieves the minimax optimal rate for a fixed collective intelligence of workers, measured by ∥p∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' From part (b) of Theorem 3, for any (p, q), unless we have s > Ω(1/(nD)) there always exists a constant fraction of tasks for which the recovered top-two answers are incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This bound matches with our sufficient condition on s from Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2 in equation 14 upto logarithmic factors, as long as δ2∥p∥2 ≳ nD, showing the minimax optimality of our Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2 for any (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' More discussions on the theoretical results are available at Appendix §E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5 Experiments We evaluate the proposed algorithm under diverse scenarios of synthetic datasets in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1, and for two applications–in identifying difficult tasks in real datasets in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 and in training neural network models with soft labels defined from the top-two plausible labels in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 Experiments on synthetic dataset We compare the empirical performance of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 and Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2 (referred as TopTwo1 and TopTwo2) with other baselines: majority voting(MV), OTP-D&S and MV-D&S (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2014), PGD (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2018), M-MSR (Ma & Olshevsky, 2020) and oracle-MLE, whose details can be found in Appx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' §C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We choose these baselines since they have the strongest established guarantees in the literature and they are all MLE-based approaches, from which the top-two answers can be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Obviously, oracle-MLE, which uses the ground-truth model parameters, provides the best possible performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We devise four scenarios described in Table 1 to verify the robustness of our model for various (p, q) ranges, at (n, m) = (50, 500) with s ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The number of choices for each task is fixed as 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 reports the empirical error probability 1 m �m j=1 P((ˆgj, ˆhj) ̸= (gj, hj)) averaged over 30 runs, with 95% confidence intervals (shaded region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Four columns correspond to the four scenarios, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The prediction errors for gj and hj are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 6 of Appx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' §D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Table 1: Parameters for synthetic data experiments under diverse scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Easy Hard Few-smart High-variance Worker pi ∈ [0, 1] pi ∈ [0, 1] 90% pi ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1] pi ∈ [0, 1] 10% pi ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='9, 1] Task qj ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='9, 1] qj ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6] qj ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5, 1] 50% qj ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6] 50% qj ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='0] 8 MV MV-D&S OPT-D&S PGD M-MSR ToiwoT Top1wo2 Oracle] Easy Hard Few Smart High Variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='651 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='55A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='55 ≠(y"6))d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='45 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='35 2 10 6 6 8 10 2 8 10 2 8 10 Avg, # of queries per task Avg, # of queries per task Avg, # of queries per task Avg, # of queries per taskWe can observe that for all the considered scenarios TopTwo2 achieves the best performance, near the oracle MLE, in recovering (gj, hj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Depending on the scenarios, the reason TopTwo2 outperforms can be explained differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For the Easy scenario, since qj is close to 1, it is easy to distinguish gj from hj but hard to distinguish hj from other labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our algorithm achieves the best performance in estimating hj by a large margin (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For the Hard scenario, it is hard to distinguish gj and hj, but our algorithm, which uses an accurate ˆqj, can better distinguish gj and hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For Few-smart, our algorithm achieves the biggest gain compared to other methods, since our algorithm can effectively distinguish few smart workers from spammers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' High-variance shows the effect of having diverse qj in a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We remark that our algorithm achieves the best performance, near that of the oracle-MLE, for all the scenarios, while the next performer keeps changing depending on scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For example, the OPT D&S is the second best performer in the Easy scenario, while it is the worst performer in the Few-smart scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We also show the robustness of our algorithm against changes in model parameters in Appendix §D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 Experiments on real-world dataset: inferring task difficulties We next provide experimental results using real-world data collected from MTurk and show that our algo- rithm can be used for inferring task difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We devised a color comparison task where we asked the crowd to choose a color, among six given choices, that looks the most similar to a reference color of each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 4 in Appx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 for example tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' After randomly generating a reference color and the six choices, we identified the ground truth and the most confusing answer for each task by measuring the distance between colors using the CIEDE2000 color difference formula (Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' If the distance from the reference color to the ground truth is much shorter than that to the most confusing answer, then the task was considered easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We designed 1000 tasks and distributed it to 200 workers, collecting 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 responses on each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' After collecting the data, we subsampled it to simulate how the prediction error decreases as the number of responses per task increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2a shows the performances in detecting (gj, hj), gj and hj, averaged over 10 times of random sampling, with 95% confidence interval (shaded region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' TopTwo2 algorithm achieved the best performance in detecting (gj, hj), gj and hj in all ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We further examined the correlation between the task difficulty - quantified by the distance gap between the ground truth and the most confusing answer from the reference color - and the estimated confusion probability qj across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We selected top 50 most difficult/easiest tasks according to the estimated confusion probability qj and plotted the histograms of the distance gap for the two groups in Fig 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We can see that the difficult group (blue, having lowest qj) tends to have a smaller distance gap than those of the easy task group (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This result shows that our algorithm can identify difficult tasks in real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (a) The average prediction error on color comparison tasks (b) Histogram of dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' gap Figure 2: (a) Prediction error for (gj, hj), gj and hj (from left to right) for color comparison tasks using real data collected from MTurk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our TopTwo2 algorithm achieves the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (b) Histogram of color distance gap for the task groups with the highest qj (easiest tasks) and lowest qj (most difficult tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The difficult task group (blue) tends to have a smaller color distance gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 9 AIVIV MV-D&S UPT-D&S PGD M-MSR Topiwo1 TopTwo2 P(g, h) ≠ P(g,h)) P(g ≠g) P(h ≠h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 Prediction error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 10 15 20 10 15 20 10 15 20 Avg, # of queries per task Avg, # of queries per task Avg, # of queries per taskmo 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='bib Ton 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1IOP 10 8 9 Count 4 2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 2 Color distance gap5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 Training neural networks with soft labels having top-two information An appealing example where we can use the knowledge of the second best answer is in training deep neural networks for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Traditionally, a hard label (one ground-truth label per image) has been used to train a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In recent works, it has been shown that using a soft label (full label distribution that reflect human perceptual uncertainty) is sometimes beneficial in obtaining a model with better generalization capability (Peterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' However, obtaining an accurate full label distribution requires much higher sample complexity than recovering only the ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For example, Peterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (2019) provided a CIFAR10H dataset with full human label distributions for 10000 instances of CIFAR10 test examples by collecting on average 50 judgements per image, which is about 5-10 times larger than those of usual datasets (Table 4 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our top-two model, on the other hand, can effectively reduce the required sample complexity, while still guaranteeing the advantages in training models with soft labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To demonstrate this idea, we trained two deep neural networks, VGG-19 and ResNet18, with the soft-label vectors having the top-two structure (top2) for CIFAR10H dataset3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We then compared the training/test results with those of the hard label (hard) and full label distribution (full).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Experimental details are in Appendix §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Compared to the original training with hard labels, training with top-two soft labels achieved 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='56% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='09% higher test accuracy in VGG-19 and ResNet18, respectively (averaged in three runs, 150 epochs) as shown in Table 2, which is even higher than that of the full label distribution in VGG-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This result shows that training with the top-two soft labels results in better generalization (test accuracy) than training with hard labels, since the top-two soft label includes simple yet helpful side information, the most confusable class and the confusion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Table 2: Comparison of performances for CIFAR10H dataset with hard/soft label training Network Train accuracy Training loss Test accuracy Test loss VGG-19 (hard) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='46±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='023 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='93%±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='66% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='611±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='102 6 Discussion We proposed a new model for multiple-choice crowdsourcing, with top-two confusable answers and varying confusion probability over tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We provided an algorithm to infer the top-two answers and the confusion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This work can benefit several query-based data acquisition systems such as MTurk or review systems by providing additional information about the task such as the most plausible answer other than the ground truth and how plausible it is, which can be used to quantify the accuracy of the ground truth or to classify the tasks based on difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The topic of confusion is getting increasing attention in the machine learning community for designing reliable classifiers (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2017;' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Advances in neural information processing systems, 25, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 13 A Verification for the Proposed Top-Two Model We proposed the top-two model to reflect the key attributes of seven datasets including Adult2, Dog, Web, Flag, Food, Plot, and Color, of which the details are summarized in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Table 3 shows empirical distributions of the mean incidence of responses for the top-three dominating answers, sorted by the dominance proportions, for the six public datasets and the Color dataset that we collected, with the standard deviation over the tasks in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 3, we also plot empirical distributions of the mean incidence of responses sorted by the dominant proportion with error bars indicating the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The i-th data point represents the average incidence of the i-th highest response in each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For example, in Adult2 dataset, the most dominating answer takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 portion of the total answers, and the next dominating answer takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='14 portion of the total answers on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Table 3: Proportions of top-three dominating answers in public datasets Dataset Ground truth 2nd dominating answer 3rd dominating answer Adult2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='80±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='07 Dog 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='04 Web 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='09 Flag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='03 Food 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='80±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='05 Plot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='06±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='07 Color 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='05 From the table and figure, we can observe that for all the considered public datasets the top-two answers dominate the overall answers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', about 65-90% of the total answers belong to the top two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Furthermore, the average ratio from the most dominating answer to the second one is 4:1, while that between the second and the third is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' There often exist overlaps in the error bars between the ground truth and the second dominating answer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', for Web, Plot, and Color datasets, but no such overlap is found between the ground truth and the third dominating answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' What we can call a ‘confusing answer’ is an answer that has an incidence rate comparable to that of the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In all the considered datasets, only the second dominating answer shows such a tendency, and thus, we can conclude that the third dominating answer cannot be called a ‘confusing answer’, and the top-two model in equation 1 well describes the errors in answers caused by confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Moreover, from the public datasets, we also observe that the task difficulty can be quantified by the confusion probability between the top-two answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' As an example, for the Web dataset, when we select the easiest 500 tasks and hardest 500 tasks by ordering tasks with the ratio of correct answers, the ratio between the ground-truth to the 2nd best answer was 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7:1 for the easiest group, while it was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5:1 for the hardest group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This observation shows that the ratio between the top-two answers indeed captures task difficulty as does our model parameter for task difficulty qj in equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 Datasets We collect six publicly available multi-class datasets: Adult2, Dog, Web, Flag, Food and Plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Since these datasets do not provide information about the most confusing answer or the task difficulty, we additionally create a new dataset called ‘Color’, for which we can identify the most confusing answer and also quantify the task difficulty for all the included tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Color is a dataset where the task is to find the most similar color to the reference color among six different choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For each task, we randomly create a reference color and then choose six choices of colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The distance from the reference color to the ground truth color is in between 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5, the 14 (a) (b) (c) (d) (e) (f) (g) Figure 3: Empirical distribution of the mean incidence of responses sorted by the dominant proportion, averaged over all tasks in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The i-th data point represents the average incidence of the i-th highest response in each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The error bars indicate the standard deviation of the mean incidence of the i-th dominating answer over the tasks in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' distance to the most confusing answer is in between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5, and the distance to the rest of the choices is between 11 and 12, where the distance between the pairs of colors is measured by CIEDE2000 (Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2005) color difference formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The tasks are ordered in terms of their difficulty levels by measuring the gap between: the distance from the reference color to the ground truth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' and that to the most confusing answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' If the distance from the reference color to the ground truth is much shorter than that to the most confusing answer, then the task is considered easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Using MTurk, we collected 19600 labels from 196 workers for 1000 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Each Human Intelligence Task (HIT) is composed of randomly selected 100 tasks, and we pay $1 to each worker who completed a HIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 4 shows an example task for the Color dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Adult2 (Ipeirotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2010) is a 4-class dataset where the task is to classify the web pages into four categories (G, PG, R, X) depending on the adult level of the websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This dataset contains 3317 labels for 333 websites which are offered by 269 workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Dog (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2014) is a 4-class dataset where the task is to discriminate a breed (out of Norfolk Terrire, Norwich Terrier, Irish Wolfhound, and Scottich Deerhound) for a given dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This dataset contains 7354 labels collected from 52 workers for 807 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Web (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2012) is a 5-class dataset where the task is to determine the relevance of query-URL pairs with a 5-level rating (from 1 to 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The dataset contains 15567 labels for the 2665 query-URL pairs offered by 177 workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Flag (Krivosheev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2020) is a dataset for multiple-choice tasks where each task is to identify the country for a given flag from 10 given choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' A total of 1600 votes are collected from 220 workers for the 100 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 15 Color N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 Empirical probabili 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0 1 2 3 4 5 6 LabelAdult2 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 Empirical probabili 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0 1 2 3 4 LabelDog Empirical probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content="6' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0 1 2 3 4 LabelWeb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 Empirical probabilit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0 1 2 3 4 5 LabelFlag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0 1 2 3 4 5 6 7 8 9 10 LabelFood 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0 1 2 3 4 5 LabelPlot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content="4' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0 1 2 3 4 5 6 7 8 9 10 Label(a) gj = 6 and hj = 5 (b) gj = 4 and hj = 3 (c) gj = 5 and hj = 3 (d) gj = 6 and hj = 2 Figure 4: Example tasks for ‘Color’ dataset where the ground truth g and the most confusing answer h are determined by the color distance from the reference color (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Food (Krivosheev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2020) is a dataset for multiple-choice tasks where each task asks to identify a picture of a given food or dish from 5 given choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This dataset contains 1220 labels for 76 tasks collected from 177 workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Plot (Krivosheev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2020) is a dataset for multiple-choice tasks where the task is to identify a movie from a description of its plot from 10 given choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Only workers who correctly solved the first 10 test questions can answer the rest of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' A total of 1937 labels are collected from 122 workers for 100 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Table 4 shows a summarized information for the introduced datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Table 4: Dataset information Dataset # workers # tasks # labels or choices sparsity dtask dworker Adult2 269 333 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='037 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 Dog 109 807 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='092 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='0 Web 176 2653 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='033 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 Flag 220 100 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='073 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 Food 177 54 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='125 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 Plot 122 56 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='293 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 Color 196 1000 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 B Experimental Details for Neural Network Training in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 We show the details of the experiments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 Datasets The CIFAR10H dataset (Peterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2019) consists of 511,400 human classifications by 2,571 participants which were collected via Amazon Mechanical Turk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Each participant classified 200 images, 20 from each 16 2 3 525 2 3 6(a) Images with lowest q (considered to be hard) (b) Images with highest q (considered to be easy) Figure 5: Training images with (a) lowest and (b) highest confusion probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Every 20 tasks, a trivial question is presented to prevent random guessing, and participants who scored below 75% were excluded from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We present the images with the lowest/highest q from the training samples in Fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The image with a lower q means that the first answer and the second answer are hard to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 Model We trained two simple CNN architectures, VGG-19 and ResNet-18, to show the usefulness of the second answer and the confusion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For each model, our loss function is defined as the cross-entropy between the softmax output and the two-hot vector (in which the values are q and 1 − q for g and h, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We compare the results of our top-two label training with those of full-distribution training and hard label (one-hot vector) training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 Training We train each model using 10-fold cross validation (using 90% of images for training and 10% images for validation) and average the results across 5 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We run a grid search over learning rates, with the base learning rate chosen from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='001}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We find 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 to be optimal in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We trained each model for a maximum of 150 epochs using SGD optimizer with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='9 and a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our neural networks are trained using NVIDIA GeForce 3090 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' C Baseline Methods In this section, we explain the baseline methods with which we compare the performance of our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To analyze the performance in recovering the top-two answers, we considered the ML-based algorithms, including the Spectral-EM algorithm (MV-D&S and OPT-D&S) (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2014), Projected Gradient Descent (PGD) (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2018) and M-MSR (Ma & Olshevsky, 2020), which provide a “score” for each label so that we can recover the top-two answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Spectral-EM algorithm (MV-D&S and OPT-D&S) (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2014) is a two-stage algorithm for multi-class crowd labeling problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' These algorithms are built for the D&S model where each worker has his/her own confusion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In the first stage of the algorithm, the confusion matrix of each worker is estimated via spectral method (OPT-D&S) or majority voting (MV-D&S), respectively, and in the second stage, the estimates for the confusion matrices are refined by optimizing the objective function of the D&S estimator via the Expectation Maximization (EM) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Projected Gradient Descent (PGD) (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 2018) is an approach to estimate the skills of each worker in the single-coin D&S model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The authors formulate the skill estimation problem as a rank-one correlation-matrix completion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' They propose a projected gradient descent method to solve the correlation-matrix completion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' M-MSR (Ma & Olshevsky, 2020) algorithm is an approach to estimate the reliability of each worker in the multi-class D&S model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' M-MSR algorithm utilizes that the rank of the response matrix is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 17 Figure 6: Prediction error for (gj, hj) (top row), gj (middle) and hj (bottom) for four scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our algorithm (TopTwo2) achieves the best performance, near the oracle MLE for all the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To estimate the reliability of the workers, they use update rules to find the left singular vector and right singular vector of the response matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In this process, the extreme values are filtered out to guarantee the stable convergence of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' D Synthetic Experiments D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 Additional plots for synthetic data experiments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1, we devised four scenarios described in Table 1 to verify the robustness of our model for various (p, q) ranges, with (n, m, s) = (50, 500, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The performance of algorithms is measured by the empirical average error probabilities in recovering gj, hj and (gj, hj), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 1 m �m j=1 P(ˆgj ̸= gj), 1 m �m j=1 P(ˆhj ̸= hj) and 1 m �m j=1 P((ˆgj, ˆhj) ̸= (gj, hj)) and plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We can observe that for all the considered scenarios TopTwo2 achieves the best performance, near the oracle MLE, in recovering (gj, hj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Depending on scenarios though, the reason TopTwo2 outperforms can be explained differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For Easy scenario, since qj is close to 1, it becomes easy to distinguish gj from hj but hard to distinguish hj from other labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our algorithm achieves the best performance in estimating hj by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For Hard scenario, it becomes hard to distinguish gj and hj, but our algorithm, which uses an accurate ˆqj, can better distinguish gj and hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' High-variance show the effect of having diverse qj in a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For Few-smart, our algorithm achieves the 18 MVMV-D&S ←OPT-D&S PGD M-MSR TopTwo1 TopTwo2 OracleEasy Hard Few Smart High Variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='35 4 8 10 4 6 8 10 6 8 10 2 4 4 6 8 10 Avg, # of queries per task Avg, # of queries per task Avg, # of queries per task Avg, # of queries per taskbiggest gain compared to other methods, since our algorithm can effectively distinguish few smart workers from spammers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We remark that even though the performance gap between TopTwo2 and the next best performer is not significant for some cases, our algorithm always achieves the best performance, near that of the oracle-MLE, for all the scenarios, while the next performer keeps changing depending on scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For example, the OPT D&S is the second best performer in the ‘Easy’ scenario, while it is the worst performer in the ‘Few smart’ scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 Robustness of our methods In this section, we present a set of four additional synthetic experiments to demonstrate the robustness of our methods, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 and Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2 (referred to as TopTwo1 and TopTwo2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In each experiment, we change a parameter of our synthetic error model and compare the prediction error of our algorithms to the baselines: majority voting(MV), OTP-D&S and MV-D&S Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (2014), PGD Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (2018) and Oracle-MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We measure the performance of each algorithm by the empirical average error probabilties in recovering the ground truth gj, the most confusing answer hj and the pair of top two (gj, hj), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', 1 m �m j=1 P(ˆgj ̸= gj), 1 m �m j=1 P(ˆhj ̸= hj) and 1 m �m j=1 P((ˆgj, ˆhj) ̸= (gj, hj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Obviously, Oracle-MLE provides a lower bound for the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Changing the dimension of observed matrix: We first check the robustness of our methods against the change of dimensions of the observation matrix A ∈ {0, 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' , K}n×m with n ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We vary the number of workers (n) or the number of tasks (m) while fixing the other dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The default values of n and m are 50 and 500, respectively, and the sampling probability s is fixed as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 throughout the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The worker reliability pi and the task difficulty qj is sampled uniformly at random from [0, 1] and (1/2, 1], respectively, for all i ∈ [n] and j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 7a and 7b, we report the results when we change n for a fixed m and s, or when we change m for a fixed n and s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 7a, we can see that as the number of workers increases, the performance of every algorithm improves since the number of samples per task scales as ns for a fixed s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our algorithm achieves the performance close to the Oracle-MLE for all the considered range, which implies that the worker reliabilities {pi} are well estimated with our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 7b, we can see that our algorithm achieves a robust performance against the change in the number of tasks, although the performance gets closer to that of Oracle-MLE as the number of tasks increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Since our method uses SVD in the first stage, the larger dimension is beneficial for the concentration of the random perturbation matrix with respect to the expectation of the observation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This phenomenon is observed for other baseline methods as well, which are based on the spectral method, OPT D&S, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Changing the variance of worker reliability: In this experiment, we change the range of pi, the parameter for worker skill/reliability, for i ∈ [n], with a fixed mean in order to observe the impact of the variance of the worker reliability on the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We randomly sample pi from the window [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 − x, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 + x] with x varying from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='05 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The mean of the worker reliability is fixed as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 7c, when the variance of the worker reliability increases, the baseline methods estimating worker reliabilities perform better than the majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Our TopTwo2 algorithm achieves the best performance close to Oracle-MLE, as the standard deviation increases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=', as the workers become more heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Changing the variance of task difficulty: We also design an experiment to observe the impact of the variance of qj, j ∈ [m], the parameter for task difficulty, on the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We randomly sample qj from the window [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75 − x, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75 + x] with x varying from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='05 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The mean of the worker reliability is fixed as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' If the variance of the task difficulty is small, it could be sufficient to only estimate the worker reliability since all the tasks have almost the similar task difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 7d, when the variance of the task difficulty increases, our TopTwo2 algorithm performs better than the other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This is the evidence for the validity of our method in estimating the task difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Changing the portion of spammers: Spammers who provide random answers always exist in crowd- sourcing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To improve the inference performance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' it is important to distinguish spammers from 19 (a) Effect of the number of workers on the performance (b) Effect of the number of tasks on the performance (c) Effect of the variance of worker reliability on the performance (d) Effect of the variance of task difficulty on the performance (e) Effect of the portion of spammers on the performance Figure 7: Prediction error for (gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' hj) (first column),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' gj (second column),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' and hj (third column) for five different setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The solid lines represent the mean prediction errors of each algorithm averaged over 10 runs, and the shaded regions represent the standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 20 D((α )/(a )) Da/a) D( / )1（(9,10)+(9,10)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 1(9+9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='9 1(十) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 ★-MV MV-D& 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 OPT-D PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 TopTw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 TopTw →—Oracle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 # of workers # of workers # of workerss &S 01 02D(( )/(a )) Da/ D(h / )1((9,1)+(9,1)) 1(9/9) 1(+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='65 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 一MV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 一MV-D& 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 ←OPT-D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='45 TopTw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='45 TopTw Oracle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 200 400 600 800 1000 200 400 600 800 1000 200 400 600 800 1000 # of tasks # of tasks # of taskss &S 01 02D((α )/(a )) Da/a) D( / )1((9,10)于(9,10)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='34 1(9/9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='65 Prediction error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 一MV P MV-D8 OPT-D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='55 PGD TopTw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 TopTw 一Oracle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='25 Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' of worker reliability Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' of worker reliability Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' of worker reliabilitys &S 01 02D((α )/(a )) Da/a) D( / )1(9+9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 一MV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 P 一MV-D ←OPT-[ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 PGD —TopTv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 TopTv 一Oracle 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='25 Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' of task difficulty Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' of task difficulty Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' of task difficulty&S D&S /01 /02D((α )/(a )) Da/a) D( / )1((9,1) (9,10) 1(9≠9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='85 1( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='7 ★一MV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='65 一MV-D& 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 TopTw 一Oracle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='8 Portion of spammers Portion of spammers Portion of spammers&S &S 01 102reliable workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In our experimental setup, we define a spammer as a worker whose reliability parameter pi is in the range [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We change the portion of spammers among the workers from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='9 and compare the prediction error of our methods to those of other baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 7e, we can see that our algorithm achieves the best performance among all the considered baselines except Oracle-MLE, which can exactly distinguish spammers from reliable workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This result demonstrates the superiority of our methods in detecting spammers compared to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 Estimating the worker reliability vector and the task difficulty vector In this section, we examine the accuracy of our estimates for the worker reliability vector p and the task difficulty vector q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The worker reliability is estimated by ˆp defined in equation 5 of Algorithm 2 and the task difficulty is estimated by ˆq defined in equation 4 of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To analyze the accuracy of these estimators, we compute the mean squared error (MSE), 1 n∥ˆp − p∥2 2 and 1 m∥ˆq − q∥2 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To analyze the estimation accuracy for the worker reliability, we first sample pi uniformly at random from [0, 1] for all i ∈ [n] and fix the worker reliability vector p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Then, we randomly sample the task difficulty vector q ∈ (1/2, 1]m fifty times and then sample the observation matrices from the distribution equation 1 for each (p, q) pair with a fixed p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For each observation matrix, we subsample the data with varying probabilities and apply Algorithm 2 to get the estimate ˆp, which is then used to calculate the MSE of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We report the MSE averaged over these fifty cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Similarly, to analyze the estimation accuracy for the task difficulty, we randomly sample and fix a task difficulty vector q ∈ (1/2, 1]m and generate fifty different observation matrices while varying the worker reliability vector p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We again report the MSE averaged over these fifty cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The number of workers and that of tasks is set to be (50, 500) for the worker reliability estimation, and to be (100, 1000) for the task difficulty estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 8a and 8b, we plot the MSE for p and q, respectively, as the average number of queries per task increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We can see that both for p and q, the MSEs converge to near zero as the average number of queries per task increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' However, estimating the task difficulty requires more number of samples as our theory equation 11 suggests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (a) Mean squared error 1 n ∥ ˆp − p∥2 2 (b) Mean squared error 1 m ∥ˆq − q∥2 2 Figure 8: Mean squared errors in estimating the worker reliability vector p (left) and the task difficulty vector q (right), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' E Discussion of theoretical results In this section, we present a discussion of the main theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='035 rror E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='03 Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='025 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='005 0 5 10 15 20 25 30 35 40 45 50 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='#ofqueriespertask0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='35 Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='25 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='05 10 20 30 40 50 60 70 80 90 100 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='#ofqueriespertask• Theorem 1 asserts that the sampling probability of Ω � 1 δ2 1∥p∥2 2 log K ϵ � is sufficient to recover the top-two answers (gj, hj) for any task j ∈ [m] and to estimate the confusion probability qj with accuracy of |ˆqj − qj| < δ1 by Algorithm 1 with probability at least 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Combined with Theorem 3 part (a), we can see that this sample complexity is the minimax optimal rate for a fixed collective quality of workers, measured by ∥p∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' It is also worth comparing our algorithm with the simple majority voting (MV) scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The MV scheme infers the top-two answers by counting the majority of the received answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Simple analysis shows that the MV scheme requires the sampling probability s such that ns = Θ � ( 1 n � i pi)−2 log 1 ϵ � to recover (gj, hj) with probability 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Remind that Algorithm 1 requires ns = Ω � n δ2 1∥p∥2 2 log K ϵ � samples per task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Since 1 n∥p∥2 = 1 n � i p2 i ≥ � 1 n � i pi �2 by Cauchy-Schwarz inequality, Algorithm 1 achieves strictly better trade-offs unless pi is same for all workers i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' As an example, for a spammer-hammer model where α ∈ (0, 1) fraction of workers are hammers with pi = 1 and the rest are spammers with pi = 0, Algorithm 1 requires ns = Θ � 1 α log 1 ϵ � samples per task, while MV requires ns = Θ � 1 α2 log 1 ϵ � samples per task to recover top-two answers with probability 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Theorem 2 shows that when we have an entrywise bound on the estimated worker reliability vector p and the task difficulty vector q, the plug-in MLE estimator, used in Algorithm 2, guarantees the recovery of top-two answers if the sampling probability s = Ω( log(m/ϵ) n ¯ D ) where ¯D, which depend on (p, q), indicates the average reliability of workers in distinguishing the top-two answers from any other pairs for the most difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Combined with Theorem 3 part (b), we can see that this sample complexity is the minimax optimal rate for any (p, q), ignoring the logarithmic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Combining the conditions for the accurate estimation of model parameters in equation 11 and the convergence of the plug-in MLE (Theorem 2), Corollary 2 shows the condition on the sample complexity to guarantee the performance of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' F Proof of Proposition 1 For each task j and label k, define four indicator functions: Πa(j, k) :=1(gj > k, hj > k), Πb(j, k) :=1(gj ≤ k, hj > k), Πc(j, k) :=1(gj > k, hj ≤ k), Πd(j, k) :=1(gj ≤ k, hj ≤ k), (17) which satisfy Πa(j, k) + Πb(j, k) + Πc(j, k) + Πd(j, k) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For notational simplicity, we will often drop (j, k) fron Π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The pmf of A(k) is given by A(k) ij = � � � � � −1 with probability s(1 − ρ(k) ij ), 1 with probability sρ(k) ij , 0 with probability 1 − s, (18) where ρ(k) ij = Πa(j, k)pi + Πb(j, k)pi(1 − qj) + Πc(j, k)piqj + (K−k)(1−pi) K , and its expectation is E[A(k) ij ] = s(2ρ(k) ij − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Note that by using Πa = 1 − Πb − Πc − Πd, the probability ρ(k) ij can be written as ρ(k) ij = pi � qj(Πc − Πb) − (Πc + Πd) + k K � + K−k K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Thus, by defining r(k) j := qj(Πc − Πb) − (Πc + Πd) + k K , (19) 22 the expectation of A(k) ij can be written as E[A(k) ij ] = s(2ρ(k) ij − 1) = s � 2pir(k) j + K − 2k K � , (20) and E[A(k)] − s(K − 2k) K 1n×m = 2sp(r(k))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (21) Note that Case I: gj > hj Πa(j, k) = 1 where k < hj, Πc(j, k) = 1 where hj ≤ k < gj, Πd(j, k) = 1 where gj ≤ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Case II: gj < hj Πa(j, k) = 1 where k < gj, Πb(j, k) = 1 where gj ≤ k < hj, Πd(j, k) = 1 where hj ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (22) Thus, r(k) j in equation 19 is equal to Case I: gj > hj r(k) j = � � � � � k K where k < hj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' k K − (1 − qj) where hj ≤ k < gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' k K − 1 where gj ≤ k, Case II: gj < hj r(k) j = � � � � � k K where k < gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' k K − qj where gj ≤ k < hj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' k K − 1 where hj ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' G Performance Analysis of Algorithm 1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 Proofs of Theorem 1 and Corollary 1 In Algorithm 1, we use the data matrix A1, which is obtained by randomly splitting the original data matrix A into A1 and A2 with probability s1 and (1 − s1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Then, the first stage of Algorithm 1 begins with randomly splitting A1 again into two independent matrices B and C with equal probabilities, and then converting B and C into (K − 1)-binary matrices B(k) and C(k) as explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We define X(k) and Y (k) as X(k) := B(k) − s′(K−2k) K 1n×m and Y (k) := C(k) − s′(K−2k) K 1n×m where s′ = s · s1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We have E[X(k)] = E[Y (k)] = s′p(r(k))⊤ from Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For notational simplicity, we will ignore this random splitting for a moment and just pretend that X(k) and Y (k) are sampled independently with s′ = s throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We first outline the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Based on the observation that E[X(k)] = sp(r(k))⊤, if E[X(k)] is available we can recover p∗ = p ∥p∥2 by SVD, and by using p∗ it is possible to recover ∥p∥2r(k), which then reveals {(gj, hj)}m j=1 as well as q from the relation in equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To estimate p∗ from X(k), we first bound the spectral norm of the perturbation, ∥X(k) − E[X(k)]∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We then use this bound and Wedin SinΘ theorem to bound sin θ(u(k), p∗) where u(k) is the left singular vector of X(k) with the largest singular value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We trim the abnormally large components of u(k) and denote the resulting vector by ˜u(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' After trimming, it is still possible to show that sin θ(˜u(k), p∗) can be bounded in the same order as that of sin θ(u(k), p∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Finally, we provide an entrywise bound between v(k) = 2 s(Y (k))⊤ ˜u(k) and ∥p∥2r(k) in Lemma 5, which is the main lemma to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We state our main technical lemmas first and then prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let us define the perturbation matrix E := X(k) − E[X(k)] = B(k) − s(K − 2k) K 1n×m − sp(r(k))⊤ = B(k) − E[B(k)] (23) where B(k) ij = � � � � � −1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' s(1 − ρ(k) ij ), 1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' sρ(k) ij , 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 1 − s, (24) 23 and ρ(k) ij = Πa(j, k)pi+Πb(j, k)pi(1−qj)+Πc(j, k)piqj+ (K−k)(1−pi) K for (Πa, Πb, Πc, Πd) defined in equation 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For the perturbation matrix E in equation 23, we have E[Ei,j] = 0, and |Ei,j| ≤ 2, 1 ≤ i ≤ n, 1 ≤ j ≤ m, (25) and also var(Eij) = var(B(k) ij ) = E[(B(k) ij )2] − (E[B(k) ij ])2 = s − (s(ρ(k) ij − 1/2))2 ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (26) Note that {Eij} are independent across all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Define ν := max � � �max i � j E[E2 i,j], max j � i E[E2 i,j] � � � ≤ max{m, n}s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (27) By applying the spectral norm bound to random matrices with independent entires, appeared in Bandeira & Van Handel (2016) and summarized in Theorem 4, we can bound the spectral norm of E as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Lemma 2 (Spectral norm bound of E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' With probability 1 − (n + m)−8, we have ∥E∥ ≤ 4 � s max (m, n) + ˜c � log(n + m) (28) for some constant ˜c > 0 when m ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For some sufficiently large m, assuming n = o(m) and s = Ω(log(n + m)/m), the spectral norm of E can be further bounded by ∥E∥ ≤ 5√sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (29) Using the bounded spectral norm of E in equation 29 and applying the Wedin SinΘ theorem, summarized in Theorem 5, we can bound the angle between u(k) and p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For some sufficiently large m, assuming n = o(m) and s = Ω(log(n + m)/m), we have sin θ(u(k), p∗) ≤ Θ(1/√sn) (30) with probability at least 1 − (n + m)−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By applying the Wedin SinΘ Theorem (Theorem 5), we have sin θ(u(k), p∗) ≤ √ 2∥E∥ s∥p∥2 · ∥r(k)∥2 − ∥E∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (31) We have ∥p∥2 = Θ(√n) and ∥r(k)∥2 = Θ(√m) by assumptions on model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By Lemma 2, for some sufficiently large m, assuming n = o(m) and s = Ω(log(n + m)/m), we have ∥E∥ ≤ 5√sm with probability at least 1 − (n + m)−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Combining these bounds, we get sin θ(u(k), p∗) ≤ Θ(√sm) Θ(s√mn) − Θ(√sm) = 1 Θ (√sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (32) We trim the abnormally large components of u(k) by letting it zero if u(k) i > 2/(η√n) and denote the resulting vector as ˜u(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This process is required to control the maximum entry size of ˜u(k), which is used later in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For the next lemma, we show that after the trimming process, the norm of ˜u(k) is still close to 1 and the angle between ˜u(k) and p∗ has the same order as that of sin θ(u(k), p∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 24 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Given ∥p∗∥2 ≥ η√n, we have ∥˜u(k)∥2 ≥ � 1 − 50 sin2 θ(u(k), p∗), (33) sin θ(˜u(k), p∗) ≤ 6 √ 2 sin θ(u(k), p∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (34) The proof of Lemma 4 is provided in Section G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Finally, we provide our main lemma giving the entrywise bound on the difference between v(k) = 1 s(Y (k))⊤ ˜u(k) and ∥p∥2r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Lemma 5 (Entrywise Bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For any δ1, ϵ > 0, and any task j ∈ [m] and label index k ∈ [K], if the sampling probability s ≥ Θ � 1 δ2 1∥p∥2 2 log 1 ϵ � , then we can guarantee P ����� 1 s � Y (k) ∗j , ˜u(k)� − ∥p∥2r(k) j ���� < δ1∥p∥2 � > 1 − ϵ (35) as m → ∞ when n = O(m/ log m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For notional simplicity, denote θ(˜u(k), p∗) by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To prove equation 35, we show bounds on two probabilities, P ����� 1 s � Y (k) ∗j , ˜u(k)� − ∥˜u(k)∥2∥p∥2r(k) j cos θ ���� > δ1∥p∥2 2 � < ϵ/2, (36) P ����∥˜u(k)∥2∥p∥2r(k) j cos θ − ∥p∥2r(k) j ��� > δ1∥p∥2 2 � < ϵ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (37) Then, the triangle inequality implies equation 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We first prove equation 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Remind that we do the random splitting of the input matrix A and define the two independent binary-converted matrices as X(k) and Y (k), for 1 ≤ k < K, which are used to estimate ˜u(k) and v(k), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Thus, ˜u(k) is independent from Y (k) and this independence is used when we bound the first and second moments of v(k) j = 1 s⟨Y (k) ∗j , ˜u(k)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For any 1 ≤ j ≤ m, the first and second moments of v(k) j = 1 s⟨Y (k) ∗j , ˜u(k)⟩ satisfy E �1 s � Y (k) ∗j , ˜u(k)�� = ⟨p, ˜u(k)⟩r(k) j = ∥p∥2∥˜u(k)∥2(cos θ)r(k) j = Θ(√n) (38) if r(k) j ̸= 0 by Lemma 3 and 4, and var �1 s � Y (k) ∗j , ˜u(k)�� ≤ 1 s2 n � i=1 (˜u(k) i )2E[(Y (k) ij )2] = Θ �1 s � (39) since E[(Y (k) ij )2] = Θ(s) and �n i=1(˜u(k) i )2 = Θ(1) by Lemma 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Furthermore, we have max1≤i≤m |Y (k) ij ˜u(k) i | ≤ Θ � 1 √n � since ˜u(k) i ≤ 2 η√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By applying the Bernstein’s inequality, we can show that P ����� 1 s � Y (k) ∗j , ˜u(k)� − ∥˜u(k)∥2∥p∥2r(k) j cos θ ���� > δ1∥p∥2 2 � ≤ 2 exp � − Θ(δ2 1∥p∥2 2) Θ � 1 s � + Θ (δ1∥p∥2/√n) � ≤ exp � −Θ(sδ2 1∥p∥2 2) � (40) where the second inequality is due to the assumption ∥p∥2 = Θ(√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To make this probability less than ϵ 2, it is sufficient to have s ≥ Ω � 1 δ2 1∥p∥2 2 log 1 ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 25 We next prove equation 37 by bounding ���∥˜u(k)∥2∥p∥2r(k) j cos θ − ∥p∥2r(k) j ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By the triangle inequality, we have ���∥˜u(k)∥2∥p∥2r(k) j cos θ − ∥p∥2r(k) j ��� ≤ ���∥˜u(k)∥2∥p∥2r(k) j cos θ − ∥p∥2r(k) j cos θ ��� + ���∥p∥2r(k) j cos θ − ∥p∥2r(k) j ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (41) Note that 1 ∥p∥2 ���∥˜u(k)∥2∥p∥2r(k) j cos θ − ∥p∥2r(k) j cos θ ��� = r(k) j cos θ ���∥˜u(k)∥2 − 1 ��� ≤ Θ(sin2 θ(u(k), p∗)) = 1 Θ (ns), (42) with probability 1 − (n + m)−8 by Lemma 3 and 4, and also note that 1 ∥p∥2 ���∥p∥2r(k) j cos θ − ∥p∥2r(k) j ��� = r(k) j (1 − cos θ) ≤ Θ(sin2 θ(u(k), p∗)) = 1 Θ (ns), (43) with probability 1 − (n + m)−8 by Lemma 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To make these errors of order 1/Θ (ns) less than δ1 2 , it is sufficient to have s ≥ Ω � 1 δ1n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By combining the above results, it can be guaranteed that ��� 1 2s � Y (k) ∗j , ˜u(k)� − ∥p∥2r(k) j ��� < δ∥p∥2 with probability at least 1 − ϵ, if the sampling probability s ≥ max � Ω � 1 δ2 1∥p∥2 2 log 1 ϵ � , Ω � 1 δ1n �� = Ω � 1 δ2 1∥p∥2 2 log 1 ϵ � (44) where the last equality is due to ∥p∥2 = Θ(√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The condition s = Ω(log(n + m)/m) in Lemma 3 is immediately satisfied by equation 44 when n = O(m/ log m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By using Lemma 5, we next prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By applying the union bound over k ∈ [K], if s ≥ Θ � 1 δ2 1∥p∥2 2 log K ϵ � then we have ∥p∥2(r(k) j − δ1) ≤ v(k) j = 1 s � Y (k) ∗j , ˜u(k)� ≤ ∥p∥2(r(k) j + δ1), ∀k ∈ [K] (45) for any δ1 > 0 and j ∈ [m] with probability at least 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Under the condition equation 45, for any qj ∈ (1/2, 1) and δ < min � 2qj−1 2 , 1−qj 2 � , we can guarantee that 1 K − qj + δ < 1 K − (1 − qj) − δ and 1 K − (1 − qj) + δ < 1 K − δ, (46) which implies (ˆgj, ˆhj) = (gj, hj) for (ˆgj, ˆhj) defined in equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This proves equation 8 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We next prove equation 9, the accuracy guarantee in estimating the task difficulty vector q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' After estimat- ing ∥p∥2r(k) by v(k) = 1 s(Y (k))⊤ ˜u(k), we estimate ∥p∥2 by calculating l where lj := K K−2 � k̸=ˆgj,k̸=ˆhj ∆v(k) j and l := 1 m �m j=1 lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Assume that |∥p∥2 − l| ≤ ∥p∥2δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We will specify the required order of δ′ later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Remind that the estimate for qj is defined as ˆqj := 1 K − ∆v (ˆgj ) j l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Under the condition that ˆgj = gj and |vj − ∥p∥2r(k) j | ≤ ∥p∥2δ1, both of which are satisfied under the conditions of Lemma 5, we have � 1 K − qj − 2δ1 � 1 + δ′ ≤ ∆v(ˆgj) j l ≤ � 1 K − qj + 2δ1 � 1 − δ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (47) 26 By the Taylor expansion for 1 1−x = 1 + x + Θ(x2) as x → 0, we have |ˆqj − qj| ≤ 2δ1 + δ′ � 1 K − qj + 2δ1 � + Θ(δ′2) = Θ(δ1 + δ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (48) Thus, both the order of δ′, which is the estimation error of ∥p∥2, and that of δ, which is the estimation error of ∥p∥2r(k) j , govern the estimation accuracy of qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We next show that we can have δ′ = Θ(δ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By Lemma 5, we have |vj − ∥p∥2r(k) j | ≤ ∥p∥2δ1, which implies ∥p∥2(∆r(k) j − 2δ1) ≤ ∆v(k) j ≤ ∥p∥2(∆r(k) j + 2δ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (49) Under the condition (ˆgj, ˆhj) = (gj, hj), since ∆r(k) j = 1 K for k ̸= ˆgj, ˆhj, we have ∥p∥2 − ∥p∥2 2δ1K K − 2 ≤ lj = K K − 2 � k̸=ˆgj,k̸=ˆhj ∆v(k) j ≤ ∥p∥2 + ∥p∥2 2δ1K K − 2, (50) and thus δ′ = 2δ1K K−2 = Θ(δ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Thus, it is enough to have s = Ω � 1 δ2 1∥p∥2 2 log K ϵ � to guarantee equation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By using Lemma 5 and taking the union bound over all tasks j ∈ [m] as well as k ∈ [K], we can prove Corollary 1 in a similar way as that of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 Proof of Lemma 4 We first prove equation 33, ∥˜u(k)∥2 ≥ � 1 − 50 sin2 θ(u(k), p∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let I be the set of indices 1 ≤ i ≤ n such that u(k) i ≥ 2 η√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Then, we have u(k) i − p∗ i ≥ 1 η√n for all i ∈ I since p∗ i = pi/∥p∥2 ≤ 1 η√n due to the assumption that ∥p∥2 2 ≥ η2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Thus, we have |I| η2n ≤ � i∈I (u(k) i − p∗ i )2 ≤ ∥u(k) − p∗∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (51) By using the triangle inequality, we can show that �� i∈I � u(k) i �2 ≤ � � � �� i∈I � u(k) i − 2 η√n �2 + � 4|I| η2n ≤ � � � �� i∈I � p∗ i − 2 η√n �2 + �� i∈I � u(k) i − p∗ i �2 + � 4|I| η2n ≤ � 4|I| η2n + �� i∈I � u(k) i − p∗ i �2 + � 4|I| η2n ≤ 5∥u(k) − p∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (52) Therefore, we get 1 ≥ ∥˜u(k)∥2 2 = 1 − � i∈I (u(k) i )2 ≥ 1 − 25∥u(k) − p∗∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (53) 27 By the law of cosine, we have ∥p∗ − u(k)∥2 2 = sin2 θ(u(k), p∗) + (1 − cos θ(u(k), p∗))2 = 2 − 2 cos θ(u(k), p∗) = 2 � 1 − � 1 − sin2 θ(u(k), p∗) � = 2 sin2 θ(u(k), p∗) 1 + � 1 − sin2 θ(u(k), p∗) ≤ 2 sin2 θ(u(k), p∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (54) Combining equation 53 and equation 54 proves equation 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We next prove equation 34, sin θ(˜u(k), p∗) ≤ 6 √ 2 sin θ(u(k), p∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' First, note that ∥˜u(k) − u(k)∥2 2 = � i∈I � u(k) i �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We have sin θ(˜u(k), p∗) ≤ ∥˜u(k) − p∥2 ≤ ∥˜u(k) − u(k)∥2 + ∥u(k) − p∗∥2 ≤ 6∥u(k) − p∗∥2 (55) where the last inequality is from equation 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Combined with equation 54, we get equation 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' H Performance Analysis of Algorithm 2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 Proof of Lemma 1 In this lemma, we show that conditioned on (ˆgj, ˆhj) = (gj, hj) for all j ∈ [m], if s(1 − s1) = Ω � 1 δ2m log n ϵ � , the estimator ˆpi defined in equation 5, ˆpi = K (K − 2) � � 1 s(1 − s1) � � 1 m m � j=1 1(A2 ij = ˆgj or ˆhj) � � − 2 K � � , guarantees P (∥p − ˆp∥∞ < δ2) ≥ 1 − ϵ for any ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Given (ˆgj, ˆhj) = (gj, hj) for all j ∈ [m], since A2 is independent of (ˆgj, ˆhj), we have E � 1(A2 ij = ˆgj or ˆhj) � = P(A2 ij = ˆgj or ˆhj) = s(1 − s1) �K − 2 K pi + 2 K � , var � 1(A2 ij = ˆgj or ˆhj) � ≤ s(1 − s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (56) By applying the Bernstein’s inequality, we can show that P � � ������ m � j=1 � 1(A2 ij = ˆgj or ˆhj) − s(1 − s1) �K − 2 K pi + 2 K �������� > (K − 2)ms(1 − s1)δ2 K � � ≤ exp � � �− 1 2 � (K−2)ms(1−s1)δ2 K �2 ms(1 − s1) + 1 3 (K−2)ms(1−s1)δ2 K � � � ≤ exp � −Θ � ms(1 − s1)δ2 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (57) Thus, if the sampling probability satisfies s(1 − s1) = Ω � 1 mδ2 2 log 1 ϵ � , (58) then we can guarantee that P(|ˆpi − pi| < δ2) ≥ 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By taking the union bound over i ∈ [n], if the sampling probability satisfies s(1 − s1) = Ω � 1 mδ2 2 log n ϵ � , (59) then we can guarantee that P (∥ˆp − p∥∞ < δ2) ≥ 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 28 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 Proof of Theorem 2 To prove this theorem, we use similar proof techniques from Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Since the work in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (2014) focuses on the recovery of only the ground-truth label for each task, we generalize the techniques to recover not only the ground-truth label but also the most confusing answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We first introduce some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let µ(i,j) (a,b),k denote the probability that a worker i ∈ [n] gives label k ∈ [K] for the assigned task j ∈ [m] of which the top-two answers are (gj, hj) = (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let µ(i,j) (a,b) = [µ(i,j) (a,b),1 µ(i,j) (a,b),2 · · µ(i,j) (a,b),K]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We introduce a quantity that measures the average ability of workers in distinguishing the ground-truth pair of top-two answers (gj, hj) from any other pair (a, b) ∈ [K]2/{(gj, hj)} for the task j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We define D (j) := min (gj,hj)̸=(a,b) 1 n n � i=1 DKL � µ(i,j) (gj,hj), µ(i,j) (a,b) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' D := min j∈[m] D (j), (60) where DKL(P, Q) := � i P(i) log(P(i)/Q(i)) is the KL-divergence between P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Note that D (j) is strictly positive if qj ∈ (1/2, 1) and there exists at least one worker i with pi > 0 for the distribution equation 1, so that (gj, hj) can be distinguished from any other (a, b) ∈ [K]2/{(gj, hj)} statistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We define D as the minimum of D (j) over j ∈ [m], indicating the average ability of workers in distinguishing (gj, hj) from any other (a, b) for the most difficult task in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let us define an event that will be shown holding with high probability, E : n � i=1 K � k=1 1(Aij = k) log � �µ(i,j) (gj,hj),k µ(i,j) (a,b),k � � ≥ nsD/2 for all j ∈ [m] and (a, b) ∈ [K] × [K]\\(gj, hj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (61) Define li := K � k=1 1(Aij = k) log � µ(i,j) (gj,hj),k/µ(i,j) (a,b),k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (62) We can see that l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' , ln are mutually independent on any value of (gj, hj), and each li belongs to the interval [0, log(1/ρ)] where µ(i,j) (gj,hj),c ≥ ρ for all (i, j, gj, hj, c) ∈ [n] × [m] × [K]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We can easily show that E � n � i=1 li �����(gj, hj) � = n � i=1 sDKL � µ(i,j) (gj,hj), µ(i,j) (a,b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (63) We define D := n � i=1 DKL � µ(i,j) (gj,hj), µ(i,j) (a,b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (64) The following lemma shows that the second moment of li is bounded above by the KL-divergence between the label distribution under (gj, hj) pair and the label distribution under (a, b) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Conditioning on any value of (gj, hj), we have E � l2 i |(gj, hj) � ≤ 2 log(1/ρ) 1 − ρ sDKL � µ(i,j) (gj,hj), µ(i,j) (a,b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (65) The proof of this lemma can be obtained by following the proof of the similar result, Lemma 4 of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 29 According to Lemma 6, the aggregated second moment of li is bounded by E � n � i=1 l2 i �����(gj, hj) � ≤ 2 log(1/ρ) 1 − ρ n � i=1 sDKL � µ(i,j) (gj,hj), µ(i,j) (a,b) � = 2 log(1/ρ) 1 − ρ sD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (66) Thus, applying the Bernstein’s inequality, we have P � n � i=1 li ≥ sD/2 �����(gj, hj) � ≥ 1 − exp � − 1 2(sD/2)2 2 log(1/ρ) 1−ρ sD + 1 3(2 log(1/ρ))(sD/2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (67) Since ρ ≤ 1/2 and D ≥ nD (j) ≥ nD, combining the above inequality with union bound over j ∈ [m], we have P [E] ≥ 1 − mK2 exp � − nsD 33 log(1/ρ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (68) The maximum likelihood estimator finds a pair of (a, b) ∈ [K]2, a ̸= b, maximizing (ˆgj, ˆhj) = arg max (a,b)∈[K]2,a̸=b n � i=1 P(Aij|p, qj, (a, b)) = arg max (a,b)∈[K]2,a̸=b n � i=1 log P(Aij|p, qj, (a, b)) = arg max (a,b)∈[K]2,a̸=b n � i=1 K � k=1 1(Aij = k) log µ(i,j) (a,b),k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (69) The plug-in MLE in equation 6, on the other hand, finds a pair of (a, b) ∈ [K]2, a ̸= b, maximizing (ˆgj, ˆhj) = arg max (a,b)∈[K]2,a̸=b n � i=1 K � k=1 1(Aij = k) log ˆµ(i,j) (a,b),k (70) where ˆµ(i,j) (a,b),k is the estimated probability that a worker i ∈ [n] gives label k ∈ [K] for the assigned task j ∈ [m] of which the top two answers are (gj, hj) = (a, b) assuming pi = ˆpi from equation 5 and qj = ˆqj from equation 4 in the distribution equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Thus, for the plug-in MLE to correctly find the ground-truth top two answers (gj, hj), we need to satisfy the following event: n � i=1 K � k=1 1(Aij = k) log � ˆµ(i,j) (gj,hj),k/ˆµ(i,j) (a,b),k � ≥ 0 for all (a, b) ∈ [K] × [K]\\(gj, hj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (71) For any arbitrary (a, b) ̸= (gj, hj), consider the quantity Q(a,b) := n � i=1 K � k=1 1(Aij = k) log � ˆµ(i,j) (gj,hj),k/ˆµ(i,j) (a,b),k � , (72) which can be written as Q(a,b) = n � i=1 K � k=1 1(Aij = k) log µ(i,j) (gj,hj),k µ(i,j) (a,b),k + n � i=1 K � k=1 1(Aij = k) � �log � � ˆµ(i,j) (gj,hj),k µ(i,j) (gj,hj),k � � − log � � ˆµ(i,j) (a,b),k µ(i,j) (a,b),k � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (73) 30 Assuming that there exist ρ > δ3 such that µ(i,j) (a,b),k ≥ ρ and |ˆµ(i,j) (a,b),k − µ(i,j) (a,b),k| ≤ δ3 for all i ∈ [n], j ∈ [m], (a, b) ∈ [K]2, (74) we have max i∈[n],k∈[K] � �log � � ˆµ(i,j) (gj,hj),k µ(i,j) (gj,hj),k � � − log � � ˆµ(i,j) (a,b),k µ(i,j) (a,b),k � � � � ≤ 2 log � ρ ρ − δ3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (75) By the Bernstein’s inequality, we also have P ������ n � i=1 K � k=1 1(Aij = k) − ns ����� > ns/2 � ≤ exp � − 1 2(ns/2)2 ns + 1 3(ns/2) � = exp � −3ns 28 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (76) By taking the union bound over j ∈ [m], we have P ������ n � i=1 K � k=1 1(Aij = k) − ns ����� > ns/2 for any j ∈ [m] � ≤ m exp � −3ns 28 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (77) Under the intersection of the event ����n i=1 �K k=1 1(Aij = k) − ns ��� ≤ ns/2 for all j ∈ [m] and the event E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' we can guarantee Q(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='b) = n � i=1 K � k=1 1(Aij = k) log µ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='j) (gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='hj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='k µ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='j) (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='k + n � i=1 K � k=1 1(Aij = k) � �log � � ˆµ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='j) (gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='hj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='k µ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='j) (gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='hj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='k � � − log � � ˆµ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='j) (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='k µ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='j) (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='k � � � � ≥ nsD 2 − 3ns log � ρ ρ − δ3 � ≥ ns �D 2 − 3δ3 ρ − δ3 � > 0 (78) for every j ∈ [m] where the last inequality holds if δ3 < ρ D 6 + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (79) In summary, under that the event ����n i=1 �K k=1 1(Aij = k) − ns ��� ≤ ns/2 for all j ∈ [m] and the event E hold, if we have δ3 such that |ˆµ(i,j) (a,b),k − µ(i,j) (a,b),k| ≤ δ3 for all i ∈ [n], j ∈ [m], (a, b) ∈ [K]2 (80) and δ3 < ρ and δ3 < ρ D 6 + D, (81) then we can guarantee that the plug-in MLE in equation 70 successfully recovers the pair of top two (gj, hj) for all the tasks j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' To make the right-hand side of equation 68 and equation 77 less than ϵ/2, it is sufficient to have s = Ω �log(1/ρ) log(mK2/ϵ) + D log(m/ϵ) nD � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (82) Lastly, when we have max{∥p − ˆp∥∞, ∥q − ˆq∥∞} ≤ δ, (83) we can guarantee that |ˆµ(i,j) (a,b),k − µ(i,j) (a,b),k| ≤ 4δ := δ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (84) Thus, it is sufficient to guarantee equation 83 with δ < min �ρ 4, ρD 4(6 + D) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (85) 31 I Proof of Theorem 3 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='1 Proof of part (a) To prove this minimax bound, we use the similar arguments from Karger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' In particular, we consider a spammer-hammer model such that pi = � 0, for 1 ≤ i ≤ ⌊(1 − p)n⌋ 1, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (86) Assume that total lj workers randomly sampled from [n] provide answers for the task j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Under the spammer- hammer model, the oracle estimator makes a mistake on task j with probability (K − 1)/K if it is only assigned to spammers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' When lj is the number of assignments, we have P(ˆgj ̸= gj) = K − 1 K (1 − p)lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (87) By convexity and using Jensen’s inequality, the average probability of error is lower bounded by 1 m � j∈[m] P(ˆgj ̸= gj) ≥ K − 1 K (1 − p)l (88) where 1 m � i∈[m] li ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' By assuming p ≤ 2/3, we have (1 − p) ≥ e−(p+p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Thus, min ˆg max p∈Fp, g∈[K]m 1 m � j∈[m] P(ˆgj ̸= gj) ≥ K − 1 K e−(p+p2)l ≥ K − 1 K e−2pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (89) The inequality in equation 89 implies that if l is less than 1 2p log � K−1 Kϵ � , then no algorithm can make the minimax error in equation 89 less than ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Since the average number of queries per task in our model is ns, it implies that it is necessary to have s = Ω � 1 ∥p∥2 2 log 1 ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='2 Proof of part (b) To prove the second part of the theorem, we use proof techniques from Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (2014), but generalizes the results for pair of top two answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We assume that jc ∈ [m], (gc, hc) ∈ [K]2 and (ac, bc) ∈ [K]2 are the task index and the pairs of labels such that D = 1 n n � i=1 DKL � µ(i,jc) (gc,hc), µ(i,jc) (ac,bc) � (90) for D defined in equation 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let Q be a uniform distribution over the set {(gc, hc), (ac, bc)}m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' For any (ˆg, ˆh), we have max (v,u)∈[K]m×[K]m vj̸=uj,∀j[m] E � � m � j=1 1((ˆgj, ˆhj) ̸= (gj, hj)) ���(g, h) = (v, u) � � ≥ m � j=1 � (v,u)∈{(gc,hc),(ac,bc)}m Q((v, u))E � 1((ˆgj, ˆhj) ̸= (gj, hj)) ���(g, h) = (v, u) � (91) Let A := {Aij : i ∈ [n], j ∈ [m]} be the set of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Define two probability measures P0 and P1, such that P0 is the measure of A conditioned on (gj, hj) = (gc, hc), while P1 is that on (gj, hj) = (ac, bc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Then, 32 we can have � (v,u)∈{(gc,hc),(ac,bc)}m Q((v, u))E � 1((ˆgj, ˆhj) ̸= (gj, hj)) ���(g, h) = (v, u) � = Q((gj, hj) = (gc, hc))P0((ˆgj, ˆhj) ̸= (gc, hc)) + Q((gj, hj) = (ac, bc))P1((ˆgj, ˆhj) ̸= (ac, bc)) ≥ 1 2 − 1 2∥P0 − P1∥TV ≥ 1 2 − 1 4 � DKL(P0, P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (92) where the second to the last inequality is by Le Cam’s method and the last inequality is by Pinsker’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content='4 Conditioned on (gj, hj), the set of random variables Aj := {Aij : i ∈ [n]} are independent of A\\Aj for both P0 and P1, and thus DKL(P0, P1) = DKL(P0(Aj), P1(Aj)) + DKL(P0(A\\Aj), P1(A\\Aj)) = DKL(P0(Aj), P1(Aj)) (93) where P(X) denote the distribution of X with respect to the probability measure P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Given (gj, hj), since A1j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' , Anj are independent, we can show that DKL(P0(Aj), P1(Aj)) = n � i=1 DKL(P0(Aij), P1(Aij)) = n � i=1 � (1 − s) log 1 − s 1 − s + sDKL � µ(i,j) (gc,hc), µ(i,j) (ac,bc) �� ≥ snD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (94) Combining equation 91– equation 94, we have max (v,u)∈[K]m×[K]m vj̸=uj,∀j[m] E � � 1 m m � j=1 1((ˆgj, ˆhj) ̸= (gj, hj)) ���(g, h) = (v, u) � � ≥ 1 2 − 1 4 � snD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (95) Thus, if s ≤ 1 4nD, then the above inequality is lower bounded by 3/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' J Useful Inequalities In this section, we summarize the useful inequalities used in the proof of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' The following inequality, which appeared in Bandeira & Van Handel (2016) provides a non-asymptotic spectral norm bound for random matrices with independent random entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Theorem 4 (Spectral norm bound of a random matrice with independent entries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Consider a random matrix X ∈ Rn×m, whose entries are independently generated and obey E[Xi,j] = 0, and |Xi,j| ≤ B, 1 ≤ i ≤ n, 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (96) Define ν := max � � �max i � j E[X2 i,j], max j � i E[X2 i,j] � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (97) 4The total variation distance between probability distributions P and Q defined on a set X is defined as the maximum difference between probabilities they assign on subsets of X: ∥P − Q∥TV := supA⊂X |P(A) − Q(A)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' 33 Then there exists some universal constant c > 0 such that for any t > 0, P � ∥X∥ ≥ 4√ν + t � ≤ (n + m) exp � − t2 cB2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (98) We also present a useful corollary of Theorem 4, which can be shown from equation 98 by setting ˜c = √ 9c and t = B � 9c log(n + m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Corollary 3 (Corollary of Theorem 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' If E[X2 i,j] ≤ σ2 for all i, j and satisfying conditions in Theorem 4, then we have ∥X∥ ≤ 4σ � max(m, n) + ˜cB � log(n + m) (99) with probability 1 − (n + m)−8 for some constant ˜c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We next summarize the eigenspace perturbation theory for asymmetric matrices with singular value composition (SVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Suppose X := [X0, X1] and Z := [Z0, Z1] are orthonormal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' When we define the distance between two subspaces X0 and Z0 by dist(X0, Z0) := ∥X0X⊤ 0 − Z0Z⊤ 0 ∥, (100) then we have dist(X0, Z0) = ∥X⊤ 0 Z1∥ = ∥Z⊤ 0 X1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (101) Given ∥X⊤ 0 Z0∥ ≤ 1, we write SVD of X⊤ 0 Z0 ∈ Rr×r as X⊤ 0 Z0 := U cos ΘV ⊤ where cos Θ = diag(cos θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' , cos θr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' We call {θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' , θr} principal angles between X0 and Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Then, we have ∥X⊤ 0 Z1∥ = ∥ sin Θ∥ = max{| sin θ1|, · · · , | sin θr|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (102) Let M ∗ and M = M ∗ + E be two matrices in Rn×m with n ≤ m, whose SVD are represented by M ∗ = �n i=1 σ∗ i u∗ i v∗ i ⊤ and M = �n i=1 σiuivi⊤, where σ1 ≥ · · · ≥ σn (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' σ∗ 1 ≥ · · · ≥ σ∗ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let us define U0 := [u1, · · · , ur] ∈ Rn×r, V0 := [v1, · · · , vr] ∈ Rm×r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (103) The matrices U ∗ 0 and V ∗ 0 are defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Theorem 5 (Wedin sin Θ Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' If ∥E∥ < σ∗ r − σ∗ r+1, then one has max{∥dist(U0, U ∗ 0 )∥, ∥dist(V0, V ∗ 0 )∥} ≤ √ 2∥E∥ σ∗r − σ∗ r+1 − ∥E∥, (104) where U ∗ 0 (V ∗ 0 ) and U0 (V0) are subspaces spanned by the largest r left (right) singular vectors of M ∗ and M, respecively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Lastly, we also write down two useful concentration inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Theorem 6 (Hoeffding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' , Xn be independent random variables such that Xi ∈ [ai, bi] for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Then, we have P ������ n � i=1 (Xi − E[Xi]) ����� > t � ≤ 2 exp � − 2t2 �n i=1(bi − ai)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (105) Theorem 7 (Bernstein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' , Xn be independent random variables such that Xi ∈ [ai, bi] for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Let C := max1≤i≤n(bi − ai) and σ2 = �n i=1 var(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' Then we have P ������ n � i=1 (Xi − E[Xi]) ����� > t � ≤ 2 exp � − t2/2 σ2 + C · t/3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} +page_content=' (106) 34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAyT4oBgHgl3EQfOfZu/content/2301.00006v1.pdf'} diff 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Kao Department of Electrical +Engineering and Computer Science, and Institute for Advanced Materials and +Manufacturing, University of Tennessee, Knoxville, TN 37996 +E-mail: Adrian.DelMaestro@utk.edu +Sang Wook Kim +Department of Physics and Materials Science Program, University of Vermont, +Burlington, VT 05405 +Nicholas P. Bigelow +Department of Physics and Astronomy, Institute of Optics, Center for Coherence and +Quantum Optics, University of Rochester, Rochester, NY 14627 +Robert J. Thompson +Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 +Valeri N. Kotov +Department of Physics and Materials Science Program, University of Vermont, +Burlington, VT 05405 +E-mail: Valeri.Kotov@uvm.edu +Abstract. +Novel two-dimensional (2D) atomically flat materials, such as graphene +and transition-metal dichalcogenides, exhibit unconventional Dirac electronic spectra. +We propose to effectively engineer their interactions with cold atoms in microgravity, +leading to a synergy between complex electronic and atomic collective quantum +phases and phenomena. Dirac materials are susceptible to manipulation and quantum +engineering via changes in their electronic properties by application of strain, doping +with carriers, adjustment of their dielectric environment, etc. +Consequently the +interaction of atoms with such materials, namely the van der Waals / Casimir-Polder +interaction, can be effectively manipulated, leading to the potential observation of +physical effects such as Quantum Reflection off atomically thin materials and confined +Bose-Einstein Condensate (BEC) frequency shifts. +arXiv:2301.00494v1 [cond-mat.quant-gas] 2 Jan 2023 + +Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity +2 +Figure 1. +Left: Atoms near a two-dimensional free-standing graphene surface in +microgravity. Right: A comparison of the energy and length scales where interaction +effects between atoms and surfaces may compete with gravity. +1. Introduction & Conceptual Overview of Relevance to NASA’s Mission +Novel 2D Dirac materials, which range from semimetals to semiconductors, form a +unique class of two-dimensional solids where electrons effectively have a relativistic-like +dispersion (massless or massive under certain conditions), with details that are strongly +material- and environment-dependent [1, 2, 3, 4, 5]. +This makes them susceptible +to manipulation and quantum engineering by exploring their atomically flat nature +and sensitivity of the electronic motion to the lattice structure, electronic density as +well as various external factors. Similar modifications can be realized in other novel +2D materials “beyond graphene” [2, 4, 6] widely extending the possibilities for the +realization of novel phenomena. +The main driver of unconventional physics is the van der Waals (VDW) / +Casimir-Polder (CP) interaction between neutral atoms and 2D Dirac materials +[7, 8, 9, 10, 11, 12]. Here, the unique nature of electron motion causes this interaction +to have a well-defined crossover, at the scale of several hundred nanometers, between +the non-relativistic (VDW) and the relativistic, vacuum fluctuation (Casimir-Polder) +components. +In general, VDW/CP interactions in atomically flat, graphene-based +materials have proven to be of tremendous interest and importance as they reflect +the unique two-dimensional nature of such systems and the inherent possibility for +effective manipulation and functionalization [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. +Therefore this approach offers a unique, materials-based way to study weak dispersion +forces which are of fundamental importance in Nature (see Fig. 1 where ultracold +coherent atoms are released at low momenta near a tunable atomically flat surface), +with an impact akin to previous groundbreaking studies on the effects of microgravity +on critical phenomena [25]. +To expose the underlying emergent low energy quantum behavior, the competing +interactions and length scales involved (see Fig. 1) necessitate the microgravity +environment of the current and future Cold Atom Laboratory (CAL) missions on the +International Space Station. +CAL has already achieved great success in producing +trapped Bose-Einstein condensates (BECs) in microgravity [26] and it may be possible + +Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity +3 +to explore fundamentally new physical phenomena during the planned BECCAL (Bose- +Einstein Condensate Cold Atom Laboratory) mission [27] and well beyond, as envisaged +by the NASA Fundamental Physics Program. This approach represents our vision: to +leverage ground and future NASA space-based missions in microgravity for the discovery +and engineering of fundamental and exotic physical phenomena at the interface of atomic +matter and two-dimensional quantum materials. +The quantum mechanical behavior of nature (statistics, indistinguishability, +Heisenberg uncertainty) is clearly on display in two fundamental physical phenomena: +Quantum Reflection of particles above attractive potential tails, with no classical +analog, and the formation of Bose-Einstein condensates (BEC) – a macroscopically large +coherent quantum phase of matter where the participating atoms have settled into their +wave-like ground state. In addition to the BEC’s contribution to our understanding +of fundamental physics, new discoveries resulting from this research is the foundation +for a wealth of applications and new technologies in precision metrology [28] and +quantum information processing [29, 30]. For example, cold atoms have already led +to the development of chip-scale atomic clocks with major performance enhancements +over legacy technologies [31] and the global positioning system is predicated on the +precision timing resulting from such atomic clocks. Consequently, new technological +improvements based on cold atom science have the potential to make major and broad- +reaching impacts on society. +Answering fundamental questions related to properties of quantum systems of +atoms and molecules, with the aim of designing, for example, sensors using quantum +properties, will require a new generation of experiments, technologies, and discoveries +that necessitate moving to colder temperatures, lower densities, and longer coherence +times. This was broadly recognized by the NASA Fundamental Physics program as +detailed in the previous decadal research planning report of the National Research +Council: +“Recapturing a Future for Space Exploration, Life and Physical Sciences +Research for a New Era” and renewed during the recent process in 2021. The resulting +conceptualization and successful deployment in 2018 of the Cold Atom Laboratory +(CAL) [32, 26] now provides the ability to study BECs [26] and macroscopic coherent +quantum phenomena in the persistent free fall conditions of low Earth orbit. +This +microgravity environment is exciting and offers the promise of dramatically reducing +the forces required to confine ultracold samples of atoms, while simultaneously dealing +with the problems of gravitational sag and allowing for long running experiments where +the atoms remain nearly fixed relatively to the apparatus. Long coherence times can be +used to do high precision matter wave interferometry experiments [33, 34] while opening +up the possibility to study exotic geometries not possible in terrestrial labs, e.g. atomic +bubbles [35]. +These recent successes build on a long history of NASA exploiting a +microgravity environment to do fundamental physics. This is especially true in the area +of universality and critical phenomena [25, 36], including the most precise confirmation +of the existence and value of the anomalous critical exponent of the 3DXY universality +class [37] which forms the experimental underpinning of the modern renormalization + +Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity +4 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Dimensionless Wavevector k/k0 +0.2 +0.4 +0.6 +0.8 +1.0 +Refelction Coefficient R +5 +10 +15 +k/k0 +−7.5 +−5.0 +log(R) +pristine +uniaxial strain +Figure 2. +The quantum reflection coefficient for Na atoms (with energy E and +mass M) near pristine and (uniaxially) strained graphene at low atomic wavenumber +k = +√ +2ME +ℏ +, in units of k0 = 1/β4, where β4 = +√2MC4 +ℏ +[8]. The inset shows the high +momentum part. +group framework – one of the most important and successful theoretical methods in +physics. +In the rest of the paper we explore two main effects which are quite sensitive to the +VDW/CP interactions with atomically flat quantum materials: (1) quantum reflection of +atoms, and (2) trapped Bose-Einstein condensates near 2D materials. These phenomena +explore a new set of “knobs” to manipulate cold atoms and BECs in a microgravity +environment through the extreme tunability of 2D materials. +2. Quantum Reflection of Atoms +One of the most fundamental quantum phenomena with no classical analogue is above- +barrier Quantum Reflection (QR). Scattering of atoms off VDW/CP potential tails +[38, 39] can provide an extremely sensitive probe of the strength of these fundamental +interactions. QR at low atomic energies has been studied previously for bulk materials, +for example by using BECs [40, 41] or specular reflection of narrow cold atomic beams +[42, 43]. Remarkably, high-energy QR can also be accessed experimentally [44]. Keeping +in mind potential experiments in a microgravity environment, the use of BECs is the +most promising direction. The QR in such experiments tends to saturate below unity due +to atomic interaction effects driven by collective excitations of the quantum gas during +the reflection. In order to benefit from the unique low atomic velocities achievable with +BECs [45], a release from a shallow trap of few Hz in necessary. Such traps are, however, +heavily distorted by the gravitational pull on Earth. A unique feature of 2D materials, +potentially acting as atomic mirrors, is that accurate theoretical predictions can be +made for the VDW interactions and from there the QR, for a variety of materials with +different levels of functionalization (i.e. under different external factors such as strain +(Fig. 2), carrier density, etc.). At very low atomic velocities the quantum reflection + +Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity +5 +tends to the maximum value of unity, and the material surface acts as a perfect atomic +mirror. The atom-surface interactions can be accurately measured as a phase shift in an +atom interferometer where a cold atomic sample with a small drift velocity parallel to +a surface is interrogated by four π/4 pulses such that one branch of the interferometer +can spend long times in the vicinity of a surface of interest. These quantum sensors, as +planned for BECCAL for example, become extremely sensitive when the drift times are +stretched to several seconds as made possible by a microgravity operation. +Let us outline the main theoretical steps that lead to the QR predictions for a +particular 2D material – graphene. For simplicity, we only write down explicitly the +VDW (non-relativistic part) of the atom-surface interaction potential which has the +form: +Uvdw(z) = − ℏ +2π +� ∞ +0 +dξα(iξ) 2 +� ∞ +0 +dkk2e−2kz +� 2π +0 +dφ +2π +|V (k)Π(k, iξ)| +1 − V (k)Π(k, iξ). +(1) +Here α(iω) = +α0ω2 +0 +ω2 +0+ω2 is the atomic polarizability and the relevant parameters for example +for Na atoms are: α0 = 162.6 a.u., ω0 = 2.15 eV, where the atomic unit of polarizability +is 1 a.u. = 1.4818 × 10−4 nm3. +V (k) = 2πe2/|k| is the Coulomb potential. +The +polarization function for atomically thin graphene, allowing for strain effects which +makes the Dirac cone anisotropic, with different velocities vx, vy, is given by the formula: +Π(q, iω) = − +1 +4vxvy +v2 +xq2 +x + v2 +yq2 +y +�v2 +xq2 +x + v2 +yq2 +y + ω2. +(2) +For unstrained graphene vx = vy = v = 6.6 eV˚A. We assume strain is in the y +(armchair) direction leading to decrease of the electron velocity vy in that direction +(while the velocity in the perpendicular (x) direction remains practically unchanged). +It is convenient to introduce the ratio vy/vx < 1 which reflects the strain (relative +increase in lattice spacing); this ratio is perturbatively proportional to strain for small +values but exhibits non-linear behavior for larger deformations. For example vy/vx = 0.2 +corresponds to 34% strain, vy/vx = 0.4 corresponds to 25% strain, and vy/vx = 0.75 +corresponds to 10% strain. These results are derived from band structure calculations +of deformed graphene as summarized in Refs. [6, 8]. +It should be noted that the non-relativistic form, Eq. (1), in itself contains a +crossover from −C3/z3 to −C4/z4 behavior in the potential (with log corrections) which +reflects the motion of Dirac quasiparticles in graphene. Relativistic corrections are then +taken into account as described in [8, 10, 12]; we do not present the corresponding fully +relativistic formulas. While we use the full expressions, in practice, for distances up to +∼ 100 nm, we find that the relativistic effects are quite small and become gradually +more pronounced only as the distance increases. +The calculation of the QR coefficient R for Na follows the methodology outlined +in [38]. For this atom our calculations [8] show that the −C4/z4 tail is dominant and +the potential is fitted to the form U(z) = − C4 +z4 ≡ − ℏ2 +2M +β2 +4 +z4 which defines the length scale + +Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity +6 +3000 +3500 +4000 +4500 +5000 +5500 +distance of BEC to 2D material, nm +0 +0.001 +0.002 +0.003 +relative frequency shift +Dirac Insulator (MoS2) +Dirac Semimetal (Graphene) +Strained Graphene +Frequency Shift for trapped Rb BEC +Figure 3. +Bose-Einstein Condensates near 2D surfaces. Left: Schematic diagram of +confined BEC near material showing the modification of the trapping potential. Right: +Calculated relative change of the frequency of center of mass oscillations, (ω0 − ω)/ω0, +for different 2D materials, versus distance d to the surface. +β4 = +√2MC4 +ℏ +. Then in the asymptotic limits of low and high atomic energies one finds +the behavior [38]: +R ≈ 1 − 2(β4k), +β4k ≪ 1 ; +R ∼ e−1.694√β4k, +β4k ≫ 1. +(3) +Figure 2 shows theoretical predictions for suspended pristine graphene and +uniaxially strained sample (corresponding, for illustration purposes, to fairly large +strain 34% (vy/vx = 0.2)). +Since strain leads to larger graphene polarizability and +thus enhanced attractive potential, it decreases QR relative to pristine graphene. The +opposite tendency, i.e. an increase of QR at given energy is expected to take place +when physical factors that lower the 2D material polarizability are at play. For example +this can occur in relatively small graphene samples which exhibit an effective finite-size +electronic gap, or in 2D materials, such as dichalcogenides, with an intrinsic gap. +Because the phenomenon of Quantum Reflection is highly sensitive to the electronic +motion in atomically thin materials and can in principle be accurately calculated in a +variety of physical situations, and for different atoms and 2D materials, we believe it +offers a promising route towards characterization of weak VDW/CP forces, especially +in a microgravity environment. +3. Trapped BECs near 2D Materials as Ultrasensitive Force Sensors +The study of trapped BECs near material surfaces offers another opportunity to +utilize the unique capabilities of BECCAL and a microgravity environment for the +manipulation and thus precision measurement of the VDW/CP force. The cold atom +trapping potential is modified due to the attractive force of the material atoms which can +result in a noticeable change in the BEC condensate’s center of mass oscillation frequency +[46, 47] which is protected from gravitational sagging effects (see Fig. 3). Advances in +atom-on-chip techniques [48, 49], and in particular the utilization of 2D materials such + +pristine trap +surface modified trap +move trap +near surface +oscillating BEC +in microgravityQuantum Atomic Matter Near Two-Dimensional Materials in Microgravity +7 +as graphene, will make it possible to place the BEC even closer to the material (of +order hundreds of nanometers), without suffering any losses and maintaining high atom +lifetimes. +Experiments in microgravity can produce ultra-coherent condensates [26] +necessary to push the boundaries of quantum force measurement. +We have performed calculations for Rb condensates near graphene (pristine and +strained) and MoS2 (2D insulator). +The atom-material potential U(z) in the case +of graphene is calculated following the procedure described in the previous section. +The parameters characterizing the polarizability of Rb are: α0 = 318.6 a.u., ω0 = +1.67 eV. The polarization function of MoS2, which is characterized by massive Dirac +quasiparticles (i.e. an electronic gap) can be taken as: +Π(q, iω) = −|q|2 +π +�m +˜q2 + 1 +2˜q +� +1 − 4m2 +˜q2 +� +tan−1 � ˜q +2m +�� +, ˜q ≡ +� +v2|q|2 + ω2, m = ∆/2.(4) +The values for MoS2 are: ∆ = 1.66 eV, v = 3.51 eV˚A. A summary of material parameters +for members of the dichalcogenide family can be found in Refs. [3, 50]. +The center of mass frequency ω is smaller than the value ω0 (determined by the +parameters of the BEC confining potential, harmonic trap) due to the attractive nature +of the VDW/CP potential between the condensate atoms and the surface. Following +the approach of Ref. [46], the equation governing the behavior of ω is: +ω2 − ω2 +0 = 1 +M +� Rz +−Rz +n0(z)∂2U(z) +∂z2 +dz, +(5) +where M is the mass of the Rb atom and n0(z) is the effective BEC density along the +z direction (1D “column” density) as described in [46, 51]. The quantity Rz ≈ 2 µm is +the Thomas-Fermi radius of the condensate. +Figure 3 shows the calculated frequency change as a function of the distance d +between the 2D material and the center of the harmonic trap. +One can take the +optimistic viewpoint that the BEC can maintain its structure much closer to the surface +(than in previous bulk cases) as suggested in [49]; in addition, in a microgravity +environment the characteristic Rz is expected to be smaller. +Our most important +observations are: +(1) At a given distance the frequency change is about an order +or magnitude smaller than for bulk samples [46], as expected for atomically thin +configuration. (2) The frequency change is very sensitive to the type of material since +it reflects the change in 2D polarization properties (in the plot, strain makes graphene +more polarizable while MoS2 is a gapped Dirac material and thus less polarizable). +To conclude, theoretically-predicted frequency changes of BECs near 2D Dirac +materials in microgravity show extraordinary sensitivity to material parameters and +therefore this set-up can be used as a powerful and ultrasensitive probe of the nature +and strength of VDW/CP interactions. + +Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity +8 +4. Outlook: Beyond Conventional Materials Science → Functional +Intelligent Materials +The phenomena discussed in this paper are in principle possible within the present level +of our theoretical understanding and current NASA technologies. Looking beyond the +near-term we can explore some exciting and transformational trends. It is quite possible +that the above effects can be further “designed” and engineered, in the following sense. +Given the wide variety of currently known 2D materials, it is potentially feasible to +construct materials, using artificial intelligence [52] with specific properties, optimized +in such a way that their interaction with atoms has the correct strength for a given +desired functionality. This could provide unprecedented control over the fundamental +van der Waals / Casimir-Polder force, never previously achieved in a theoretical or +laboratory setting. The NASA fundamental physics program can play a decisive role in +this process through its unique ability to provide an accessible microgravity laboratory +able to probe the quantum effects of atoms near 2D materials. +5. Acknowledgements +This work was supported, in part, under NASA grant number 80NSSC19M0143. 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Phys. 91(4) 045002 URL https://link.aps.org/doi/10.1103/RevModPhys. +91.045002 + diff --git a/x9AyT4oBgHgl3EQfnvh8/content/tmp_files/load_file.txt b/x9AyT4oBgHgl3EQfnvh8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..31abc3344aa1cdcf831b85c3498f07a913aacda1 --- /dev/null +++ b/x9AyT4oBgHgl3EQfnvh8/content/tmp_files/load_file.txt @@ -0,0 +1,462 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf,len=461 +page_content='Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity Adrian Del Maestro Department of Physics and Astronomy, Min H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Kao Department of Electrical Engineering and Computer Science, and Institute for Advanced Materials and Manufacturing, University of Tennessee, Knoxville, TN 37996 E-mail: Adrian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='DelMaestro@utk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='edu Sang Wook Kim Department of Physics and Materials Science Program, University of Vermont, Burlington, VT 05405 Nicholas P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Bigelow Department of Physics and Astronomy, Institute of Optics, Center for Coherence and Quantum Optics, University of Rochester, Rochester, NY 14627 Robert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Thompson Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 Valeri N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Kotov Department of Physics and Materials Science Program, University of Vermont, Burlington, VT 05405 E-mail: Valeri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='Kotov@uvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Novel two-dimensional (2D) atomically flat materials, such as graphene and transition-metal dichalcogenides, exhibit unconventional Dirac electronic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' We propose to effectively engineer their interactions with cold atoms in microgravity, leading to a synergy between complex electronic and atomic collective quantum phases and phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Dirac materials are susceptible to manipulation and quantum engineering via changes in their electronic properties by application of strain, doping with carriers, adjustment of their dielectric environment, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Consequently the interaction of atoms with such materials, namely the van der Waals / Casimir-Polder interaction, can be effectively manipulated, leading to the potential observation of physical effects such as Quantum Reflection off atomically thin materials and confined Bose-Einstein Condensate (BEC) frequency shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='00494v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='quant-gas] 2 Jan 2023 Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Left: Atoms near a two-dimensional free-standing graphene surface in microgravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Right: A comparison of the energy and length scales where interaction effects between atoms and surfaces may compete with gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Introduction & Conceptual Overview of Relevance to NASA’s Mission Novel 2D Dirac materials, which range from semimetals to semiconductors, form a unique class of two-dimensional solids where electrons effectively have a relativistic-like dispersion (massless or massive under certain conditions), with details that are strongly material- and environment-dependent [1, 2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' This makes them susceptible to manipulation and quantum engineering by exploring their atomically flat nature and sensitivity of the electronic motion to the lattice structure, electronic density as well as various external factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Similar modifications can be realized in other novel 2D materials “beyond graphene” [2, 4, 6] widely extending the possibilities for the realization of novel phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The main driver of unconventional physics is the van der Waals (VDW) / Casimir-Polder (CP) interaction between neutral atoms and 2D Dirac materials [7, 8, 9, 10, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Here, the unique nature of electron motion causes this interaction to have a well-defined crossover, at the scale of several hundred nanometers, between the non-relativistic (VDW) and the relativistic, vacuum fluctuation (Casimir-Polder) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' In general, VDW/CP interactions in atomically flat, graphene-based materials have proven to be of tremendous interest and importance as they reflect the unique two-dimensional nature of such systems and the inherent possibility for effective manipulation and functionalization [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Therefore this approach offers a unique, materials-based way to study weak dispersion forces which are of fundamental importance in Nature (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' 1 where ultracold coherent atoms are released at low momenta near a tunable atomically flat surface), with an impact akin to previous groundbreaking studies on the effects of microgravity on critical phenomena [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' To expose the underlying emergent low energy quantum behavior, the competing interactions and length scales involved (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' 1) necessitate the microgravity environment of the current and future Cold Atom Laboratory (CAL) missions on the International Space Station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' CAL has already achieved great success in producing trapped Bose-Einstein condensates (BECs) in microgravity [26] and it may be possible Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity 3 to explore fundamentally new physical phenomena during the planned BECCAL (Bose- Einstein Condensate Cold Atom Laboratory) mission [27] and well beyond, as envisaged by the NASA Fundamental Physics Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' This approach represents our vision: to leverage ground and future NASA space-based missions in microgravity for the discovery and engineering of fundamental and exotic physical phenomena at the interface of atomic matter and two-dimensional quantum materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The quantum mechanical behavior of nature (statistics, indistinguishability, Heisenberg uncertainty) is clearly on display in two fundamental physical phenomena: Quantum Reflection of particles above attractive potential tails, with no classical analog, and the formation of Bose-Einstein condensates (BEC) – a macroscopically large coherent quantum phase of matter where the participating atoms have settled into their wave-like ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' In addition to the BEC’s contribution to our understanding of fundamental physics, new discoveries resulting from this research is the foundation for a wealth of applications and new technologies in precision metrology [28] and quantum information processing [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' For example, cold atoms have already led to the development of chip-scale atomic clocks with major performance enhancements over legacy technologies [31] and the global positioning system is predicated on the precision timing resulting from such atomic clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Consequently, new technological improvements based on cold atom science have the potential to make major and broad- reaching impacts on society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Answering fundamental questions related to properties of quantum systems of atoms and molecules, with the aim of designing, for example, sensors using quantum properties, will require a new generation of experiments, technologies, and discoveries that necessitate moving to colder temperatures, lower densities, and longer coherence times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' This was broadly recognized by the NASA Fundamental Physics program as detailed in the previous decadal research planning report of the National Research Council: “Recapturing a Future for Space Exploration, Life and Physical Sciences Research for a New Era” and renewed during the recent process in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The resulting conceptualization and successful deployment in 2018 of the Cold Atom Laboratory (CAL) [32, 26] now provides the ability to study BECs [26] and macroscopic coherent quantum phenomena in the persistent free fall conditions of low Earth orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' This microgravity environment is exciting and offers the promise of dramatically reducing the forces required to confine ultracold samples of atoms, while simultaneously dealing with the problems of gravitational sag and allowing for long running experiments where the atoms remain nearly fixed relatively to the apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Long coherence times can be used to do high precision matter wave interferometry experiments [33, 34] while opening up the possibility to study exotic geometries not possible in terrestrial labs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' atomic bubbles [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' These recent successes build on a long history of NASA exploiting a microgravity environment to do fundamental physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' This is especially true in the area of universality and critical phenomena [25, 36], including the most precise confirmation of the existence and value of the anomalous critical exponent of the 3DXY universality class [37] which forms the experimental underpinning of the modern renormalization Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='5 Dimensionless Wavevector k/k0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='0 Refelction Coefficient R 5 10 15 k/k0 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='0 log(R) pristine uniaxial strain Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The quantum reflection coefficient for Na atoms (with energy E and mass M) near pristine and (uniaxially) strained graphene at low atomic wavenumber k = √ 2ME ℏ , in units of k0 = 1/β4, where β4 = √2MC4 ℏ [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The inset shows the high momentum part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' group framework – one of the most important and successful theoretical methods in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' In the rest of the paper we explore two main effects which are quite sensitive to the VDW/CP interactions with atomically flat quantum materials: (1) quantum reflection of atoms, and (2) trapped Bose-Einstein condensates near 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' These phenomena explore a new set of “knobs” to manipulate cold atoms and BECs in a microgravity environment through the extreme tunability of 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Quantum Reflection of Atoms One of the most fundamental quantum phenomena with no classical analogue is above- barrier Quantum Reflection (QR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Scattering of atoms off VDW/CP potential tails [38, 39] can provide an extremely sensitive probe of the strength of these fundamental interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' QR at low atomic energies has been studied previously for bulk materials, for example by using BECs [40, 41] or specular reflection of narrow cold atomic beams [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Remarkably, high-energy QR can also be accessed experimentally [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Keeping in mind potential experiments in a microgravity environment, the use of BECs is the most promising direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The QR in such experiments tends to saturate below unity due to atomic interaction effects driven by collective excitations of the quantum gas during the reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' In order to benefit from the unique low atomic velocities achievable with BECs [45], a release from a shallow trap of few Hz in necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Such traps are, however, heavily distorted by the gravitational pull on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' A unique feature of 2D materials, potentially acting as atomic mirrors, is that accurate theoretical predictions can be made for the VDW interactions and from there the QR, for a variety of materials with different levels of functionalization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' under different external factors such as strain (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' 2), carrier density, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' At very low atomic velocities the quantum reflection Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity 5 tends to the maximum value of unity, and the material surface acts as a perfect atomic mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The atom-surface interactions can be accurately measured as a phase shift in an atom interferometer where a cold atomic sample with a small drift velocity parallel to a surface is interrogated by four π/4 pulses such that one branch of the interferometer can spend long times in the vicinity of a surface of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' These quantum sensors, as planned for BECCAL for example, become extremely sensitive when the drift times are stretched to several seconds as made possible by a microgravity operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Let us outline the main theoretical steps that lead to the QR predictions for a particular 2D material – graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' For simplicity, we only write down explicitly the VDW (non-relativistic part) of the atom-surface interaction potential which has the form: Uvdw(z) = − ℏ 2π � ∞ 0 dξα(iξ) 2 � ∞ 0 dkk2e−2kz � 2π 0 dφ 2π |V (k)Π(k, iξ)| 1 − V (k)Π(k, iξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' (1) Here α(iω) = α0ω2 0 ω2 0+ω2 is the atomic polarizability and the relevant parameters for example for Na atoms are: α0 = 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='6 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=', ω0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='15 eV, where the atomic unit of polarizability is 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='4818 × 10−4 nm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' V (k) = 2πe2/|k| is the Coulomb potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The polarization function for atomically thin graphene, allowing for strain effects which makes the Dirac cone anisotropic, with different velocities vx, vy, is given by the formula: Π(q, iω) = − 1 4vxvy v2 xq2 x + v2 yq2 y �v2 xq2 x + v2 yq2 y + ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' (2) For unstrained graphene vx = vy = v = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='6 eV˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' We assume strain is in the y (armchair) direction leading to decrease of the electron velocity vy in that direction (while the velocity in the perpendicular (x) direction remains practically unchanged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' It is convenient to introduce the ratio vy/vx < 1 which reflects the strain (relative increase in lattice spacing);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' this ratio is perturbatively proportional to strain for small values but exhibits non-linear behavior for larger deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' For example vy/vx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='2 corresponds to 34% strain, vy/vx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='4 corresponds to 25% strain, and vy/vx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='75 corresponds to 10% strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' These results are derived from band structure calculations of deformed graphene as summarized in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' [6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' It should be noted that the non-relativistic form, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' (1), in itself contains a crossover from −C3/z3 to −C4/z4 behavior in the potential (with log corrections) which reflects the motion of Dirac quasiparticles in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Relativistic corrections are then taken into account as described in [8, 10, 12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' we do not present the corresponding fully relativistic formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' While we use the full expressions, in practice, for distances up to ∼ 100 nm, we find that the relativistic effects are quite small and become gradually more pronounced only as the distance increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The calculation of the QR coefficient R for Na follows the methodology outlined in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' For this atom our calculations [8] show that the −C4/z4 tail is dominant and the potential is fitted to the form U(z) = − C4 z4 ≡ − ℏ2 2M β2 4 z4 which defines the length scale Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity 6 3000 3500 4000 4500 5000 5500 distance of BEC to 2D material, nm 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='003 relative frequency shift Dirac Insulator (MoS2) Dirac Semimetal (Graphene) Strained Graphene Frequency Shift for trapped Rb BEC Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Bose-Einstein Condensates near 2D surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Left: Schematic diagram of confined BEC near material showing the modification of the trapping potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Right: Calculated relative change of the frequency of center of mass oscillations, (ω0 − ω)/ω0, for different 2D materials, versus distance d to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' β4 = √2MC4 ℏ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Then in the asymptotic limits of low and high atomic energies one finds the behavior [38]: R ≈ 1 − 2(β4k), β4k ≪ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' R ∼ e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='694√β4k, β4k ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' (3) Figure 2 shows theoretical predictions for suspended pristine graphene and uniaxially strained sample (corresponding, for illustration purposes, to fairly large strain 34% (vy/vx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Since strain leads to larger graphene polarizability and thus enhanced attractive potential, it decreases QR relative to pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The opposite tendency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' an increase of QR at given energy is expected to take place when physical factors that lower the 2D material polarizability are at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' For example this can occur in relatively small graphene samples which exhibit an effective finite-size electronic gap, or in 2D materials, such as dichalcogenides, with an intrinsic gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Because the phenomenon of Quantum Reflection is highly sensitive to the electronic motion in atomically thin materials and can in principle be accurately calculated in a variety of physical situations, and for different atoms and 2D materials, we believe it offers a promising route towards characterization of weak VDW/CP forces, especially in a microgravity environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Trapped BECs near 2D Materials as Ultrasensitive Force Sensors The study of trapped BECs near material surfaces offers another opportunity to utilize the unique capabilities of BECCAL and a microgravity environment for the manipulation and thus precision measurement of the VDW/CP force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The cold atom trapping potential is modified due to the attractive force of the material atoms which can result in a noticeable change in the BEC condensate’s center of mass oscillation frequency [46, 47] which is protected from gravitational sagging effects (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Advances in atom-on-chip techniques [48, 49], and in particular the utilization of 2D materials such pristine trap surface modified trap move trap near surface oscillating BEC in microgravityQuantum Atomic Matter Near Two-Dimensional Materials in Microgravity 7 as graphene, will make it possible to place the BEC even closer to the material (of order hundreds of nanometers), without suffering any losses and maintaining high atom lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Experiments in microgravity can produce ultra-coherent condensates [26] necessary to push the boundaries of quantum force measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' We have performed calculations for Rb condensates near graphene (pristine and strained) and MoS2 (2D insulator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The atom-material potential U(z) in the case of graphene is calculated following the procedure described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The parameters characterizing the polarizability of Rb are: α0 = 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='6 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=', ω0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='67 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The polarization function of MoS2, which is characterized by massive Dirac quasiparticles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' an electronic gap) can be taken as: Π(q, iω) = −|q|2 π �m ˜q2 + 1 2˜q � 1 − 4m2 ˜q2 � tan−1 � ˜q 2m �� , ˜q ≡ � v2|q|2 + ω2, m = ∆/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' (4) The values for MoS2 are: ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='66 eV, v = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content='51 eV˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' A summary of material parameters for members of the dichalcogenide family can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' [3, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The center of mass frequency ω is smaller than the value ω0 (determined by the parameters of the BEC confining potential, harmonic trap) due to the attractive nature of the VDW/CP potential between the condensate atoms and the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Following the approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' [46], the equation governing the behavior of ω is: ω2 − ω2 0 = 1 M � Rz −Rz n0(z)∂2U(z) ∂z2 dz, (5) where M is the mass of the Rb atom and n0(z) is the effective BEC density along the z direction (1D “column” density) as described in [46, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The quantity Rz ≈ 2 µm is the Thomas-Fermi radius of the condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Figure 3 shows the calculated frequency change as a function of the distance d between the 2D material and the center of the harmonic trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' One can take the optimistic viewpoint that the BEC can maintain its structure much closer to the surface (than in previous bulk cases) as suggested in [49];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' in addition, in a microgravity environment the characteristic Rz is expected to be smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Our most important observations are: (1) At a given distance the frequency change is about an order or magnitude smaller than for bulk samples [46], as expected for atomically thin configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' (2) The frequency change is very sensitive to the type of material since it reflects the change in 2D polarization properties (in the plot, strain makes graphene more polarizable while MoS2 is a gapped Dirac material and thus less polarizable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' To conclude, theoretically-predicted frequency changes of BECs near 2D Dirac materials in microgravity show extraordinary sensitivity to material parameters and therefore this set-up can be used as a powerful and ultrasensitive probe of the nature and strength of VDW/CP interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Quantum Atomic Matter Near Two-Dimensional Materials in Microgravity 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Outlook: Beyond Conventional Materials Science → Functional Intelligent Materials The phenomena discussed in this paper are in principle possible within the present level of our theoretical understanding and current NASA technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Looking beyond the near-term we can explore some exciting and transformational trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' It is quite possible that the above effects can be further “designed” and engineered, in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Given the wide variety of currently known 2D materials, it is potentially feasible to construct materials, using artificial intelligence [52] with specific properties, optimized in such a way that their interaction with atoms has the correct strength for a given desired functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' This could provide unprecedented control over the fundamental van der Waals / Casimir-Polder force, never previously achieved in a theoretical or laboratory setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' The NASA fundamental physics program can play a decisive role in this process through its unique ability to provide an accessible microgravity laboratory able to probe the quantum effects of atoms near 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} +page_content=' Acknowledgements This work was supported, in part, under NASA grant number 80NSSC19M0143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfnvh8/content/2301.00494v1.pdf'} 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One prominent example is the rainbow +chain where lattice sites symmetric about the center form entangled Bell pairs due to an effective +long-range coupling from the strong inhomogeneity of the coupling strength. This manuscript gen- +eralizes the rainbow chain to higher dimensional space on lattices with Hausdorff dimension one +and enlarged local Hilbert space keeping the Hamiltonian frustration free. The effective Hamilto- +nian from the Schrieffer-Wolf transformation is given by a stacking of layers of k-simplices with +0-dimensional (fully-connected) antiferromagnetic Hamiltonians, which can be diagonalized analyt- +ically with Young operators. The original lattice can be obtained from proliferating disinclination +defects in a regular k-dimensional cubical lattice, which introduces curvature at the center of the +lattice. The model interpolates between the SYK model and the free-fermionic XX spin chain, and +hence might be potentially useful in understanding black hole physics and holography. +I. +INTRODUCTION +Entanglement is an invaluable tool in understanding the structure of phases in many-body systems. A prototypical +idea to visualize the entanglement is to think of the degrees of freedom as forming singlets of Bell pairs that carry units +of entanglement entropy. When the singlets are between neighboring sites, such a picture leads to dimer and valence +bond solid in the Majumdar-Ghosh [1] and AKLT chain [2] in one dimensional systems with matrix-product ground +states obeying the area law of entanglement entropy [3] for gapped systems. When the singlets can be formed between +sites arbitrarily far away, as in the Motzkin and Fredkin spin chains, the system can go through an entanglement phase +transition between area law and extensive scaling of entanglement [4–10], with ground states described by holographic +tensor networks [11, 12]. +In two dimensions, this leads to Anderson’s idea of resonating valance bonds [13, 14], +made concrete in the Rokhsar-Kievelson model [15]. They belong to a more general class of vertex or tiling models +with local constraints, where the singlet is formed between different local configurations of a lattice plaquette. The +entanglement entropy of their ground state naturally obeys the area law, as a projected entangled pair state with +the tensor network being the lattice itself [16]. Yet, by decorating such models with a color degree of freedom, it is +possible to make the singlets formed by the coloring instead, while the vertex or tiling configurations facilitate them +to be separated arbitrarily far away across the lattice [17, 18]. The average distance singlets span is again controlled +by the local deformation parameter, resulting in a quantum phase transition between area-law and volumetric scaling +of entanglement entropy between half systems cut in either direction. +All of the above mentioned models share the common feature of being frustration free, making it convenient to write +down a unique ground state that allows exact results analytically. Frustration becomes an obstacle when generalizing +such models to singlet states among more than two sites, either in the form of trimer, n-mer, valance bond solid [19, 20] +in one dimensional chains, or simplex solid states [21] in higher dimensions, unless the local Hibert space is enlarged +to the corresponding dimension. Such extensions not only provide benchmark for relevant cold atom experiments [22– +26], but also prove useful in the understanding of entanglement structure even when frustration is present [27]. In +fact, the n-singlet picture turns out to be revealing in understanding the entanglement structure of multi-component +generalizations to the plain Heisenberg [28] and XXZ spin chains [29, 30] with permutation Hamiltonian, which is not +deliberately cast into projection operators that are frustration free. +An alternative mechanism to generate long-range entangled singlet pairs is by introducing strong inhomogeneity +in the XX spin chain. This was done in the so-called rainbow chain, as the singlets are at fixed locations pairwise +symmetric about the center of the chain, giving a maximal entanglement between left and right half chains [31]. The +inhomogeneity can be interpreted as an underlying spacetime with constant negative curvature in the continuous free- +fermionic version of the model [32], and the conformal field theory that describes the free fermions has a holographic +dual in the Anti-de Sitter space AdS2 [33]. The metric allows more refined structure of the entanglement than the +entropy to be computed, such as the entanglement Hamiltonian and the entanglement contour [34]. In Ref. [35], a +∗ zhao.zhang@su.se +arXiv:2301.04170v1 [quant-ph] 10 Jan 2023 + +2 +first attempt at generalizing the mechanism to two dimension was made with an anisotropic quasi-two dimensional +model inhomogeneous in one direction but translationally invariant in the other. +In this manuscript, a straightforward isotropic generalization is realized in the spirit of simplex singlet state by +enlarging the dimensionality of the local Hilbert space. Although the lattice lives in a two-dimensional manifold, +its Hausdorff dimension is still one. However, the location of lattice sites sheds light upon an interpretation of the +inhomogeneity of coupling strength as a natural result of its exponential decay over distance, which is absent in the +1D case. The rest of the paper is organized as follows. In Sec. II, the 2D generalization to rainbow chain is defined on +the floral lattice, and the strong disorder renormalizaiton group (RG) procedure is carried out to show the effective +Hamiltonian and its ground state. In Sec. III, the lattice geometry is briefly discussed to show the positive curvature +near the center and how it can be obtained from square lattice by proliferating disclination defects. In Sec. IV, the +model is further generalized to three and higher dimensions outlining the analogous RG transformation. Finally, a +conclusion is given in Sec. V with a few possible future directions. +II. +INHOMOGENEOUS XX MODEL ON THE FLORAL LATTICE +The SU(3) generalization to the XX spin chain is given in terms of the ladder operators corresponding to the two +simple roots, which can be mapped to two species of free fermions [36, 37]. However, here we are dealing with a +few-body problem with exact diagonalization at each order of the Dasgupta-Ma renormalization [31, 38], and the +integrability of the Hamiltonian is not necessary. Moreover, to guarantee the Hamiltonian to be invariant in the RG +procedure, it needs to respect the S3 permutation symmetry by including all three pairs of ladder operators +e1 = +� +� +0 1 0 +0 0 0 +0 0 0 +� +� , +e2 = +� +� +0 0 0 +0 0 1 +0 0 0 +� +� , +e3 = +� +� +0 0 0 +0 0 0 +1 0 0 +� +� , +(1) +and their conjugates f a = ea†, a = 1, 2, 3. The local Hamiltonian between neighboring site i and j is given by +hi,j = +3 +� +a=1 +(ea +i f a +j + f a +i ea +j ) ≡ 2 +7 +� +a=1 +a̸=3 +λa +i λa +j , +(2) +where among the Gell-Mann generators of SU(3), λ3 and λ8 spanning the Cartan subalgebra are excluded in the +sum. We can denote the three components of the local Hilbert space C3 by colors red (R), green (G) and blue (B). +The Hamiltonian consists of kinetic terms that exchanges colors between neighboring pairs of different colors, and its +lowest energy eigenstate is simply the antisymmetrization of whatever states the two neighboring sites are in. +AB6HicbVBNS8NAEJ3Ur1q/qh69LC2 +Cp5KIqMeCF48t2A9oQ9lsJ+3azSbsboQS ++gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8I +BFcG9f9dgobm1vbO8Xd0t7+weFR+fikre +NUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoL +J3dzvPKHSPJYPZpqgH9GR5CFn1Fip6Q3K +VbfmLkDWiZeTKuRoDMpf/WHM0gilYJq3 +fPcxPgZVYzgbNSP9WYUDahI+xZKmE2s +8Wh87IuVWGJIyVLWnIQv09kdFI62kU2M6 +ImrFe9ebif14vNeGtn3GZpAYlWy4KU0FM +TOZfkyFXyIyYWkKZ4vZWwsZUWZsNiUbg +rf68jpX9a869pV86par+RxFOEMKnABHt +xAHe6hAS1gPAMr/DmPDovzrvzsWwtOPn +MKfyB8/kDc+uMng=1 +AB6HicbVDLTgJBEOzF+IL9ehlAjH +xRHYJUY8kXjxCIo8ENmR2aGBkdnYzM2tC +NnyBFw8a49VP8ubfOMAeFKyk0pVd7q7g +lhwbVz328ltbe/s7uX3CweHR8cnxdOzto +4SxbDFIhGpbkA1Ci6xZbgR2I0V0jAQ2Am 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+RsCN7y6ukeV7xLivV+2q5VsriyMJlO +AMPLiCGtxCHRrAYADP8ApvjnBenHfnY9G +ac7KZY/gD5/MHweONXQ=J3 +FIG. 1. (a) Nested triangle lattice with inhomogeneous coupling strength between neighboring sites that decays geometrically +with distance. (d) The effective long-range coupling between lattice cites with the same radius. + +3 +We begin by considering a lattice of size 6, as shown in the two innermost layers in Fig. 1 (a), with the Hamiltonian +H1 =H0 + J1V1, +(3) +H0 =h1,2 + h2,3 + h3,1, +(4) +V1 =h3,4 + h4,2 + h1,5 + h5,3 + h2,6 + h6,3. +(5) +Let J1 = +√ +6α +1 +2 , then in the case of α ≪ 1, it can be transformed into a low energy effective Hamiltonian in the +ground state subspace of the zeroth order Hamiltonian H0, with the standard Schrieffer-Wolff transformation [39, 40]. +Since the number of sites is the same as the dimensionality of the local Hilbert space, H0 is frustration-free, so its +unique ground state is the simultaneous lowest energy eigenstate of each hi,j, i.e. the fully antisymmetrized state with +energy −3, +|p0⟩ = |A⟩123 ≡ +1 +√ +6 +� +σ∈S3 +sgn(σ)|σ(RGB)⟩123, +(6) +where sgn(σ) denotes the signature of the permutation σ, and the configuration of site i is labeled by the i’th letter +denoting its color in the ket-vector1. Even without any explicit calculation, one can already see from the symmetry +and the absence of diagonal terms of the Hamiltonian, that by construction, the effective Hamiltonian at the next +order will be of the same form, up to an overall constant energy shift by the identity operator. In appendix B, one +matrix element of diagonal and off-diagonal terms are evaluated explicitly to fix the coefficients, giving +Heff +1 += P0H0P0 + αP0(h4,5 + h5,6 + h6,4 − 6)P0 + O(α +3 +2 ), +(7) +where P0 = |p0⟩⟨p0|⊗1456 projects onto the ground state subspace of H0. The ground state of this effective Hamiltonian +is +|p1⟩ = |A⟩123 ⊗ |A⟩456. +(8) +For a N-layer system in the lattice shown in Fig. 1 (a) with exponentially decaying Jn = +√ +6αn− 1 +2 , and Hamiltonian +HN = H0 + +N−1 +� +n=1 +Jn(h3n,3n+1 + h3n+1,3n−1 + h3n,3n+2 + h3n+2,3n−2 + h3n+3,3n−1 + h3n+3,3n−2), +(9) +iterating the RG procedure recursively gives the effective Hamiltonian for a lattice with N layers +Heff +N = PN−1 +N−1 +� +n=0 +αn � +h3n+1,3n+2 + h3n+2,3(n+1) + h3(n+1),3n+1 − 6α +� +PN−1 + O(αN− 1 +2 ), +(10) +where Pn = |pn⟩⟨pn|, with +|pn⟩ = +n +� +m=1 +|A⟩3m−2,3m−1,3m, +(11) +projects onto the ground state subspace of Heff +n . +The ground state |pN⟩ is a maximally entangled state in two +dimensional space generalizing the rainbow state. When cut along the dashed line in Fig. 2 separating the even and +odd sites into two subsystems, lattice sites of the same radius are entangled within each other giving contributions +in units of log 3 each to the EE, making the total EE N log 3. An explicit calculation of the bipartite EE is given in +Appendix C. +III. +LATTICE GEOMETRY AND EE SCALING +Before discussing the scaling of the ground state EE, a closer examination of the geometry of the lattice is in order. +Since the number of lattice sites grows as 3N with the number of layers N, the Hausdorff dimension of the lattice +1 The diagonalization of H0 is carried out in Appendix A. + +4 +AB6Hic +bVBNS8NAEJ3Ur1q/qh69LC2C +p5KIqMeCF48t2A9oQ9lsJ+3 +azSbsboQS+gu8eFDEqz/Jm/ +GbZuDtj4YeLw3w8y8IBFcG9f +9dgobm1vbO8Xd0t7+weFR+f +ikreNUMWyxWMSqG1CNgktsGW +4EdhOFNAoEdoLJ3dzvPKHSPJ +YPZpqgH9GR5CFn1Fip6Q3KVb +fmLkDWiZeTKuRoDMpf/WHM0 +gilYJq3fPcxPgZVYzgbNSP +9WYUDahI+xZKmE2s8Wh87Iu +VWGJIyVLWnIQv09kdFI62kU +2M6ImrFe9ebif14vNeGtn3GZ +pAYlWy4KU0FMTOZfkyFXyIyY +WkKZ4vZWwsZUWZsNiUbgrf6 +8jpX9a869pV86par+RxFOE +MKnABHtxAHe6hAS1gPAMr/D +mPDovzrvzsWwtOPnMKfyB8/k +Dc+uMng=1 +AB6HicbVDLTgJ +BEOzF+IL9ehlAjHxRHYJUY8kXjxCIo8ENmR2aGBkdnYz +M2tCNnyBFw8a49VP8ubfOMAeFKyk0pVd7q7glhwbVz3 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+e/s7uX3CweHR8cnxdOzto4SxbDFIhGpbkA1 +Ci6xZbgR2I0V0jAQ2Amdwu/84RK80g+mFmM +fkjHko84o8ZKzdqgWHYr7hJk3gZKUOGxqD +41R9GLAlRGiao1j3PjY2fUmU4Ezgv9BONMWV +TOsaepZKGqP10eicXFplSEaRsiUNWaq/J1I +aj0LA9sZUjPR695C/M/rJWZU81Mu48SgZK +tFo0QE5HF12TIFTIjZpZQpri9lbAJVZQZm0 +3BhuCtv7xJ2tcV76ZSbVbL9VIWRx4uoARX4 +MEt1OEeGtACBgjP8ApvzqPz4rw7H6vWnJPNn +MfOJ8/foeMpQ=8 +FIG. 2. (a) The three faces at the center A0,1,2,3 have three edges and introduce positive curvature. (b) The lattice can be +obtained from a square lattice by cutting out the shaded regions and identifying the lattice sites along the two rays of the +same cut-out region. (c) The lattice put on a conical manifold with a smooth apex, in which the interaction strength Ji decays +exponentially with the distance on the manifold. The lattice bonds are geodesics between vertices. The purple lines mark the +cut between the two entangled subsystems. +is one. Next observe that the interactions are short ranged, as the coupling strength decays exponentially with the +distance between neighboring sites, with the exception of the innermost layer. The exception can be eliminated by +putting the lattice on a conical surface with a smooth apex as in Fig. 2 (c). Another way to look at this is to notice that +the vertices in the lattice are all quad-valent. Hence the lattice can be obtained from proliferating disclination defects +on a square lattice which introduces positive curvature only in the four central faces, see Fig. 2 (b). The apex angle of +the cone decreases as the parameter α gets smaller, in such way that the coupling strength between neighboring sites +among the first 6 lattice sites decay exponentially with the distance along the geodesics of the surface, with α → 1 +6 +corresponding to the flat surface where the perturbation theory fails. Outside the two innermost layers, the spatial +lattice is flat, but an nontrivial spacetime metric can nonetheless be derived along the lines of Ref. [33], as the spatial +distance doubles in each layer. +In this two dimensional manifold where the lattice reside, there are two ways to divide the system that give +completely different EE scaling behavior. A cut along the angular direction in Fig. 2 (a), or equivalently with a +conic cross-section in Fig. 3 (c), which separates the system into two concentric subsystems gives zero EE, as the +ground state is a product state of each layer. A cut along any of the radial direction in Fig. 2 (a) or with a triangular +cross-section gives an EE that scales with the number of lattice sites in each subsystem. Notice that despite the +number of bonds between neighboring site also scales linearly with the subsystem size, this is different from the area +law of EE, as the coupling strength forms a geometric series accounting for effectively a bounded constant number of +bonds across the cut. +IV. +HIGHER DIMENSIONAL GENERALIZTION TO NESTED K-SIMPLICES +The Hamiltonian (9) can be generalized to be Sk+1 symmetric in the case of nested k-simplex lattice in Fig. 3 (a), +with sites labeled by linearly increasing integers from 1 to kN in an N-layer system. Each layer consists of k + 1 +regular k-simplices, each of which is spanned by k vertices of the the previous layer, and an additional one outside +such that they form a regular polytope. The Hamiltonian can be written as +H(k) +N += +k+1 +� +i,j=1 +i̸=j +h(k) +i,j + +� +(k + 1)! +N−1 +� +n=1 +α +2n−1 +2 +� + +i∈[(k+1)(n−1)+1,(k+1)n] +j∈[(k+1)n+1,(k+1)(n+1)] +h(k) +i,j , +(12) + +5 +AB6nicb +VBNS8NAEJ34WetX1aOXxSLUS0mkqMeCF48V7Qe0o +Wy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MP +N6bYWZekAiujet+O2vrG5tb24Wd4u7e/sFh6ei4p +eNUMWyWMSqE1CNgktsGm4EdhKFNAoEtoPx7cxvP +6HSPJaPZpKgH9Gh5CFn1FjpoUIv+qWyW3XnIKvEy +0kZcjT6pa/eIGZphNIwQbXuem5i/Iwqw5nAabGXa +kwoG9Mhdi2VNELtZ/NTp+TcKgMSxsqWNGSu/p7Ia +KT1JApsZ0TNSC97M/E/r5ua8MbPuExSg5ItFoWpIC +Yms7/JgCtkRkwsoUxeythI6oMzadog3BW35lb +Quq95VtXZfK9creRwFOIUzqIAH1CHO2hAExgM4R +le4c0Rzovz7nwsWtecfOYE/sD5/AGDx405(a) +AB6nicbVBNS8NAEJ34WetX1aOXxSLUS0mkqMeCF48V7Qe0oWy2k3bpZhN2N0IJ/QleP +Cji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O2vrG5tb24Wd4u7e/sFh6ei4peNUMWyWMSqE1CNgktsG +m4EdhKFNAoEtoPx7cxvP6HSPJaPZpKgH9Gh5CFn1FjpoRJc9Etlt+rOQVaJl5My5Gj0S1+9QczSCKVhg +mrd9dzE+BlVhjOB02Iv1ZhQNqZD7FoqaYTaz+anTsm5VQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xj +c14Y2fcZmkBiVbLApTQUxMZn+TAVfIjJhYQpni9lbCRlRZmw6RuCt/zyKmldVr2rau2+Vq5X8jgKcA +pnUAEPrqEOd9CAJjAYwjO8wpsjnBfn3flYtK45+cwJ/IHz+QOFTI06(b) +FIG. 3. (a) Nested tetrahedron (3-simplex) lattice with inhomogeneous coupling strength between neighboring sites that decays +geometrically with distance. (d) The effective long-range coupling between lattice cites with the same radius. +where < i, j > denotes a pair of nearest neighboring sites i and j, and +h(k) +i,j = +k+1 +� +a,b=1 +a̸=b +eab +i eba +j , +(13) +with (eab)cd = δa +c δb +d being the standard basis of (k + 1) × (k + 1) matrices. +Following the same renormalization procedure as in Sec. II and Appendix B, as outlined in Appendix D, in the +α ≪ 1 limit, it becomes the effective Hamiltonian +˜H(k) +N += P (k) +N−1 +N−1 +� +n=0 +αn +(n+1)(k+1) +� +i,j=n(k+1)+1 +i̸=j +� +h(k) +i,j − (k + 1)!α +� +P (k) +N−1 + O(αN+ 1 +2 ), +(14) +where P (k) +n += +���p(k) +n +� � +p(k) +n +���, and +���p(k) +n +� += +n−1 +� +m=0 +� +� +1 +� +(k + 1)! +� +σ∈Sk+1 +sgn(σ) +k+1 +� +a=1 +|cσa⟩(k+1)m+a +� +� . +(15) +The ground state of this effective Hamiltonian is +���p(k) +N +� +, and it has similar EE scaling behavior of N log(k + 1) as +the two dimensional model, when the two subsystems are divided by a (k − 1)-dimensional hyperplane that passes +through the center of the lattice. +V. +CONCLUSIONS +In this paper, I reported a higher dimensional generalization to the highly entangled rainbow chain, with spatially +decaying density of lattice sites in the radial direction, naturally revealing the AdS spacetime metric therein. The +dimensionality of the local Hilbert space grows with the dimensionality of the spatial lattice, in order for the effective +Hamiltonian to remain frustration free at each order of the perturbation theory. However, it would be interesting to +see how much of the result rely on the local degrees of freedom, either with analytical or numerical approaches. +Some aspects of the model bear resemblances with the SYK model [41, 42], namely the large degrees of freedom and +the fully connected interaction at each layer of the lattice. To some extent, it provides an intermediate setting both +between a uniform and a random coupling strength, and between a lattice model with local interaction only between +neighbors and a zero-dimensional model where every site interact with each other. At the moment it is not clear how +such a model might be useful for understanding black hole physics or metal without quasiparticles. However, recently + +6 +there have also been similar models constructed more specifically towards realizing AdS/CFT duality on a discrete +holographic lattice [43, 44]. +Although one can easily perform a Jordan-Wigner transformation on the lattices of the models discussed here, the +outcome does not satisfy a fermionic anti-commutation relation, nor a parafermionic one [45]. Therefore, much of the +recent finding in the rainbow chain about depletion away from half-filling does not apply here directly [46]. However, +one might be able to construct from scratch another model with similar features with multiple flavors of free fermions +with correlated hopping interaction. +Finally, other variants and deformations of the lattice could also be worth exploring. For instance, one can define +the same Hamiltonian on the dual lattice of Fig. 1 (a) and Fig. 3 (a), which generalizes a version of the rainbow +chain symmetric about a lattice site instead of a bond, as was also discussed in Ref. 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Hamermesh, Group Theory and Its Application to Physical Problems, Addison Wesley Series in Physics (Dover Publi- +cations, 1989). +Appendix A: Diagonalization of Sk+1 symmetric Hamiltonians in one k-simplex +This appendix details the diagonalization the Sk+1 symmetric Hamiltonian in a k-simplex using the example of +k = 2 and 3. First for a Hamiltonian written as a sum of permutation, or more specifically, transposition operators, +and then show how removing the diagonal terms in the Cartan subalgebra modifies the spectrum without changing +the eigenvectors. +Since the Hamiltonian does not change the total number of sites in each configuration, but simply rearranges them, +the Krylov subspaces of the fragmented Hilbert space are spanned by elements of the permutation group Sk+1 acting +on one particular configuration with a fixed ordering. Thus an arbitrary choice of the product configuration maps the +Krylov subspace to the group algebra of Sk+1, and the eigenstates of the Hamiltonian, which is an element of the group +algebra, becomes the resolution of the unit element into primitive idempotents into minimal left ideals [47]. This is +the same problem as finding the wave-functions of identical particles of a certain symmetry type and the minimal left +ideals are precisely the irreducible representations of the group algebra given by Young operators corresponding to +Young Tableaux. The Hamiltonian being the sum of all the group elements of the same conjugacy class, namely those +of the cycle type of a single transposition, commutes with all of the group elements, as a result of the rearrangement +theorem. Therefore within each irreducible representation of the symmetric group, it has the same eigenvalue. +The Young operator corresponding to a Young tableau τ is defined as Yτ = QτPτ, where the symmetrizer Pτ is the +product of sums of all the permutations within each row of τ and the antisymmetrizer Qτ is the product of algebraic +sums of all the permutations within each column of τ with a sign from the parity of the permutation. The outcome +of acting the Hamiltonian on Yτ can be evaluated separately in three parts. First, for terms acting on sites in the +same column c of τ, +� +i̸=j∈c +hi,jYτ = +� +c′̸=c +Qc′ +� +i̸=j∈c +(i, j) +� +q∈Sc +sgn(q)q +� +r∈τ +Pr += − +� +c′̸=c +Qc′ +� +i̸=j∈c +� +q′∈Sc +sgn(q′)q′ � +r∈τ +Pr += − +�lc +2 +� +Yτ, +where (i, j) denotes the transposition between site i and j, lc denotes the length of the column c, and the substitution + +8 +q′ = (i, j) ◦ q is made in the second line. Likewise , for terms acting on sites in the same row r, we have +� +i̸=j∈r +hi,jYτ = +� +c∈τ +Qc +� +r′̸=r +Pr′ +� +i̸=j∈r +(i, j) +� +p∈Sr +p += +� +c∈τ +Qc +� +r′̸=r +Pr′ +� +i̸=j∈r +� +p′∈Sr +p′ += +�lr +2 +� +Yτ, +with lr denoting the length of row r, and p′ = (i, j)◦p. The remaining terms can be grouped into sums of transpositions +between each site in one column c and a given site in a different column to show that they sum to zero: +� +j∈c +hi,jYτ = +� +c′̸=c +Qc′ +� +j∈c +(i, j) +� +q∈Sc +sgn(q)q +� +r∈τ +Pr. +Since the first sum on the r.h.s. is symmetric with respect to sites j in column c, while the second sum is antisymmetric, +their product is identically zero. Hence we have shown that +H(k)Yτ = 1 +2 +�� +r∈τ +lr(lr − 1) − +� +c∈τ +lc(lc − 1) +� +Yτ. +(A1) +Thus Yτ acting on a certain classical configuration gives the eigenstation of the quantum Hamiltonian of symmetry +type τ. Since Young tableaux provide a irreducible representation of the symmetry group Sk+1, we have found all +the eigenstates forming a complete basis. +For instance, in the SU(3) model, the ground state is given by the Young tableau +1 +2 +3 +. It has energy −3 and +is unique because there is only one choice of assigning three different colors to the three rows. The excited states +corresponding to standard Young tableaux 1 2 +3 +and 1 3 +2 +can act nontrivially on configurations with at least colors. +In the Krylov subspace that contain three colors, since site 2 and 3 are neither symmetrized nor antisymmetrized, there +are 2 different ways to assign colors to the boxes for tableau, namely by acting the corresponding Young operators +on the vectors |RGB⟩ and |RBG⟩. And in the two color subspace, there are 3 × 2 = 6 choices of coloring for the two +rows as they cannot be in the same color. In total they give 16 degenerate excited states of energy 0. The highest +energy excited state is given by the tableau 1 2 3 . It has energy +3 and can act on any color configuration, giving +a total of 1 + +�3 +2 +� ++ 3 = 10 degenerate states. +In the SU(4) model, the unique ground state is given by the Young tableau +1 +2 +3 +4 +, with energy −6. The first excited +states come from +1 2 +3 +4 +, +1 3 +2 +4 +, and +1 4 +2 +3 +, with energy −2 and degeneracy 3× +� +3 + ×4 × +�3 +2 +�� += 45. The second excited +states of energy 0 are from 1 2 +3 4 , and 1 3 +2 4 , with degeneracy 2 × +� +3 × 2 + 4 × 2 + +�4 +2 +�� += 40. The third excited states +of energy +2 come from 1 2 3 +4 +, 1 2 4 +3 +, and 1 3 4 +2 +, with degeneracy 3 × +� +3 + 4 × 2 × 3 + +�4 +2 +� +× 3 +� += 135. And the +highest energy eigenstates correspond to 1 2 3 4 , with energy +6 and degeneracy 1 + 4 × 3 + +�4 +2 +� ++ 4 × +�3 +2 +� ++ 4 = 35. +Note that the eigenstates generated by the Young operators although linearly independent, are not orthonormal. In +the next appendix, we use a graphical notation to evaluate the matrix elements of the Hamiltonian in an overcomplete +basis of dimers and simplex singlets with fixed eigenvalue. +Now the Hamiltonian terms in (2) deviate from the permutation operator by omitting all the diagonal terms from +the Cartan subalgebra of SU(k + 1). Yet, since all the eigenstates above are superpositions permutations of the same +set of configurations, each of which has the same eigenvalue of the difference between the two Hamiltonians +δH := H(perm) − H(od) = +k=1 +� +i,j=1 +i̸=j +k+1 +� +a=1 +eaa +i eaa +j , +(A2) + +9 +the eigenvectors remain the same after removing the diagonal terms. So we just need to evaluate the eigenvalues +of δH and modify the eigenvalues accordingly. +Since δH simply counts the number of pairs in the same state, +this correction is just − � +ni≥2 +�ni +2 +� +, for ci appearing ni number of times in each configuration. This will spit the +degeneracy of eigenstates corresponding to the same Young tableaux but with different color content. But for our +purpose of performing the strong disorder renormalization group in the next appendix, the only relevant information +is the eigenvalue of eigenstates corresponding to Young diagrams +and +with only one color appearing twice, +now have energies −1 and −3 respectively, as opposed to the eigenvalues 0 and −2 for the Hamiltonian with diagonal +terms included. +Appendix B: Strong disorder renormalization group with Schrieffer-Wolff transformation +The analysis in Appendix A concludes that Hilbert space of H0 is fragmented into Krylov subspaces of fixed color +content. There is a unique 6-dimensional sector with all three colors that contain the antisymmetrized ground state, +six 3-dimensional sectors with two colors, and three 1-dimensional sectors with all three sites in the same color. Only +the antisymmetric subspace in the sectors involving only two colors are relevant for the second order perturbation as +the ground state |p0⟩ is antisymmetric and the action of H0 changes only one color among the three sites. +To evaluate the non-vanishing off-diagonal elements of ⟨c′ +4c′ +5c′ +6| ⊗ ⟨qk|V1|p0⟩ ⊗ |c4c5c6⟩ for excited state |qk⟩ of H0 +consisting of two different colors and one of c4, c5, or c6, we just need to compute for instance +h3,4|p0⟩ ⊗ |R⟩4 = 1 +√ +6(|RGR⟩ − |GRR⟩) ⊗ |B⟩4 + 1 +√ +6(|BRR⟩ − |RBR⟩) ⊗ |G⟩4, +(B1) +h4,2|p0⟩ ⊗ |R⟩4 = 1 +√ +6(|GRR⟩ − |RRG⟩) ⊗ |B⟩4 + 1 +√ +6(|RRB⟩ − |BRR⟩) ⊗ |G⟩4, +(B2) +so that +V1|p0⟩ ⊗ |R⟩4 = +1 +√ +6(|RGR⟩ − |RRG⟩) ⊗ |B⟩4 + 1 +√ +6(|RRB⟩ − |RBR⟩) ⊗ |G⟩4, +(B3) +with both of the excited states of H0 of energy −1. To get the diagonal elements of Heff +1 , the color on site 4 must be +changed back to R, with another action of h3,4 or h4,2, by the Hermiticity of which, +⟨R|4 ⊗ ⟨p0|V1 +1 +√ +2(|RGR⟩ − |RRG⟩) ⊗ |B⟩4 ≡ ⟨B|4 ⊗ 1 +√ +2(⟨RGR| − ⟨RRG|)V1|p0⟩ ⊗ |R⟩4 = +1 +√ +3. +(B4) +The off-diagonal element that switches the color between site 4 and 5 is obtained by evaluating, for example, +(h1,5 + h5,3) 1 +√ +2(|RGR⟩ − |RRG⟩) ⊗ |BB⟩45 = +1 +√ +2(|BGR⟩ − |BRG⟩ + |RGB⟩) ⊗ |BR⟩45 − 1 +√ +2|RRBBG⟩. +(B5) +Hence, +⟨BR|45 ⊗ ⟨p0|V1| 1 +√ +2(|RGR⟩ − |RRG⟩) ⊗ |BB⟩45 = − 1 +2 +√ +3. +(B6) +Due to the S3 symmetry of the model, the effective Hamiltonian after the renormalization in the Shrieffer-Wolff +transformation [39, 40] is given by +⟨c4c5c6| ⊗ ⟨p0|V eff +1 |p0⟩ ⊗ |c4c5c6⟩ = +6 +� +i=4 +� +c′ +i̸=ci +⟨ci| ⊗ ⟨p0|V1|qk⟩ ⊗ |c′ +i⟩⟨c′ +i| ⊗ ⟨qk|V1|p0⟩ ⊗ |ci⟩ +E0 − Ek += −1, +(B7) +for the 33 = 27 diagonal elements; and +⟨c′c|ij ⊗ ⟨p0|V eff +1 |p0⟩ ⊗ |cc′⟩ij = +� +c′′=c,c′ +⟨c′c|ij ⊗ ⟨p0|V1|qk⟩ ⊗ |c′′c′′⟩ij⟨c′′c′′|ij ⊗ ⟨qk|V1|p0⟩ ⊗ |cc′⟩ij +E0 − Ek += 1 +6, +(B8) +for the 9 non-vanishing off-diagonal elements. The SU(3) structure modifies the Dasgupta-Ma rule of the renormalized +coupling strength to ˜J1 = +J2 +1 +6J0 = α, giving the effective Hamiltonian (7). + +10 +Appendix C: Entanglement entropy +The Schmidt decomposition of the ground state (11) can be written as +|GS⟩ = +N +� +n=1 +� +1 +√ +3 +� +cn +|cn⟩3n−1 ⊗ |¯cn⟩3n−2,3n +� +(C1) +with the normalized wavefunction of the even subsystem +|¯cn⟩ = +1 +√ +2 +� +a,b=R,G,B +ϵabcn |a⟩3n−2 ⊗ |b⟩3n , +(C2) +where ϵabcn is the Levi-Civita symbol. Taking the partial trace over the subsystem of even sites, we get the reduced +density matrix +ρo ≡ tre |GS⟩ ⟨GS| = 1 +3N +� +cn +N +� +n=1 +|cn⟩3n−1 ⟨cn|3n−1 +(C3) +with the constant Schmidt coefficient 1/3N. Hence the bipartite entanglement entropy between even and odd subsys- +tems is +SN = +� +{cn}=R,G,B +1 +3N log 1 +3N = N log 3. +(C4) +Appendix D: Diagramatics of overcomplete k-simplex singlet basis +The explicit evaluation of matrix elements in Appendix B can be simplified with a graphical notation. Assigning +an ordering to the color degree of freedom c1 < c2 < · · · , we can denote the antisymmetrized singlet state between +two neighboring sites by a directed bond +���� +AB6nicbVBN +S8NAEJ3Ur1q/qh69LJaCp5JIU +Y8FLx4r2g9oQ9lsJ+3SzSbsboQ +S+hO8eFDEq7/Im/GbZuDtj4Y +eLw3w8y8IBFcG9f9dgobm1vbO8 +Xd0t7+weFR+fikreNUMWyxWMS +qG1CNgktsGW4EdhOFNAoEdoLJ7 +dzvPKHSPJaPZpqgH9GR5CFn1F +jpgQ28Qbni1twFyDrxclKBHM1 +B+as/jFkaoTRMUK17npsYP6PKc +CZwVuqnGhPKJnSEPUsljVD72e +LUGalaZUjCWNmShizU3xMZjbSe +RoHtjKgZ61VvLv7n9VIT3vgZl +0lqULlojAVxMRk/jcZcoXMiKk +lClubyVsTBVlxqZTsiF4qy+v +k/Zlzbuq1e/rlUY1j6MIZ3AOF ++DBNTgDprQAgYjeIZXeHOE8+K +8Ox/L1oKTz5zCHzifP+XYjXc= +c1 +AB6nicbVBNS8NAEJ34WetX1 +aOXxVLwVJS1GPBi8eK9gPaUDbSbt0swm7G6GE/gQvHhTx6i/y +5r9x2+agrQ8GHu/NMDMvSATXxnW/nY3Nre2d3cJecf/g8Oi4dH +La1nGqGLZYLGLVDahGwSW2DcCu4lCGgUCO8Hkdu53nlBpHstHM +03Qj+hI8pAzaqz0wAa1QansVt0FyDrxclKGHM1B6as/jFkaoTRM +UK17npsYP6PKcCZwVuynGhPKJnSEPUsljVD72eLUGalYZUjCWNm +ShizU3xMZjbSeRoHtjKgZ61VvLv7n9VIT3vgZl0lqULlojAVxM +Rk/jcZcoXMiKklClubyVsTBVlxqZTtCF4qy+vk3at6l1V6/f1c +qOSx1GAc7iAS/DgGhpwB01oAYMRPMrvDnCeXHenY9l64aTz5zB +HzifP+dcjXg=c2� +ij += |c1c2⟩ij − |c2c1⟩ij, +(D1) +where the sign is positive when the color indices increase following the direction of the arrow and negative when +against, such that +���� +AB6nicbVBN +S8NAEJ3Ur1q/qh69LJaCp5JIU +Y8FLx4r2g9oQ9lsJ+3SzSbsboQ +S+hO8eFDEq7/Im/GbZuDtj4Y +eLw3w8y8IBFcG9f9dgobm1vbO8 +Xd0t7+weFR+fikreNUMWyxWMS +qG1CNgktsGW4EdhOFNAoEdoLJ7 +dzvPKHSPJaPZpqgH9GR5CFn1F +jpgQ28Qbni1twFyDrxclKBHM1 +B+as/jFkaoTRMUK17npsYP6PKc +CZwVuqnGhPKJnSEPUsljVD72e +LUGalaZUjCWNmShizU3xMZjbSe +RoHtjKgZ61VvLv7n9VIT3vgZl +0lqULlojAVxMRk/jcZcoXMiKk +lClubyVsTBVlxqZTsiF4qy+v +k/Zlzbuq1e/rlUY1j6MIZ3AOF ++DBNTgDprQAgYjeIZXeHOE8+K +8Ox/L1oKTz5zCHzifP+XYjXc= +c1 +AB6nicbVBNS8NAEJ34WetX1 +aOXxVLwVJS1GPBi8eK9gPaUDbSbt0swm7G6GE/gQvHhTx6i/y +5r9x2+agrQ8GHu/NMDMvSATXxnW/nY3Nre2d3cJecf/g8Oi4dH +La1nGqGLZYLGLVDahGwSW2DcCu4lCGgUCO8Hkdu53nlBpHstHM 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|cσ2⟩j ⊗ |cσ3⟩k ⊗ |cσ4⟩l . +(D6) +Then the analogous relation need for the perturbation calculation becomes +(hj,m + hk,m + hl,m) +��������� +AB6n +icbVBNS8NAEJ3Ur1q/qh69L +JaCp5JIUY8FLx4r2g9oQ9l +sJ+3SzSbsboQS+hO8eFDEq7 +/Im/GbZuDtj4YeLw3w8y8I +BFcG9f9dgobm1vbO8Xd0t7+ +weFR+fikreNUMWyxWMSqG1C +NgktsGW4EdhOFNAoEdoLJ7d +zvPKHSPJaPZpqgH9GR5CFn1 +FjpgQ28Qbni1twFyDrxclKB +HM1B+as/jFkaoTRMUK17nps +YP6PKcCZwVuqnGhPKJnSEP +UsljVD72eLUGalaZUjCWNmS +hizU3xMZjbSeRoHtjKgZ61V +vLv7n9VIT3vgZl0lqULloj +AVxMRk/jcZcoXMiKklClub +yVsTBVlxqZTsiF4qy+vk/Zl +zbuq1e/rlUY1j6MIZ3AOF+D +BNTgDprQAgYjeIZXeHOE8+ +K8Ox/L1oKTz5zCHzifP+XYj +Xc=c1 +AB6nicbVBN +S8NAEJ34WetX1aOXxVLwVJS1 +GPBi8eK9gPaUDbSbt0swm7G6G +E/gQvHhTx6i/y5r9x2+agrQ8G +Hu/NMDMvSATXxnW/nY3Nre2d3c +Jecf/g8Oi4dHLa1nGqGLZYLGL +VDahGwSW2DcCu4lCGgUCO8Hkd +u53nlBpHstHM03Qj+hI8pAzaq +z0wAa1QansVt0FyDrxclKGHM1 +B6as/jFkaoTRMUK17npsYP6PKc +CZwVuynGhPKJnSEPUsljVD72e +LUGalYZUjCWNmShizU3xMZjbSe +RoHtjKgZ61VvLv7n9VIT3vgZl +0lqULlojAVxMRk/jcZcoXMiKk 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a/xNE2T4oBgHgl3EQf3Qj1/content/tmp_files/load_file.txt b/xNE2T4oBgHgl3EQf3Qj1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..91fb2242434de019db584589651531775266c13d --- /dev/null +++ b/xNE2T4oBgHgl3EQf3Qj1/content/tmp_files/load_file.txt @@ -0,0 +1,2942 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf,len=2941 +page_content='Entanglement blossom in a simplex matryoshka Zhao Zhang1, ∗ 1SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136, Trieste, Italy (Dated: January 12, 2023) Exotic entanglement entropy scaling properties usually come with interesting entanglement struc- tures in real space and novel metrics of the spacetime lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' One prominent example is the rainbow chain where lattice sites symmetric about the center form entangled Bell pairs due to an effective long-range coupling from the strong inhomogeneity of the coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' This manuscript gen- eralizes the rainbow chain to higher dimensional space on lattices with Hausdorff dimension one and enlarged local Hilbert space keeping the Hamiltonian frustration free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' The effective Hamilto- nian from the Schrieffer-Wolf transformation is given by a stacking of layers of k-simplices with 0-dimensional (fully-connected) antiferromagnetic Hamiltonians, which can be diagonalized analyt- ically with Young operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' The original lattice can be obtained from proliferating disinclination defects in a regular k-dimensional cubical lattice, which introduces curvature at the center of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' The model interpolates between the SYK model and the free-fermionic XX spin chain, and hence might be potentially useful in understanding black hole physics and holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' INTRODUCTION Entanglement is an invaluable tool in understanding the structure of phases in many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' A prototypical idea to visualize the entanglement is to think of the degrees of freedom as forming singlets of Bell pairs that carry units of entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' When the singlets are between neighboring sites, such a picture leads to dimer and valence bond solid in the Majumdar-Ghosh [1] and AKLT chain [2] in one dimensional systems with matrix-product ground states obeying the area law of entanglement entropy [3] for gapped systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' When the singlets can be formed between sites arbitrarily far away, as in the Motzkin and Fredkin spin chains, the system can go through an entanglement phase transition between area law and extensive scaling of entanglement [4–10], with ground states described by holographic tensor networks [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' In two dimensions, this leads to Anderson’s idea of resonating valance bonds [13, 14], made concrete in the Rokhsar-Kievelson model [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' They belong to a more general class of vertex or tiling models with local constraints, where the singlet is formed between different local configurations of a lattice plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' The entanglement entropy of their ground state naturally obeys the area law, as a projected entangled pair state with the tensor network being the lattice itself [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' Yet, by decorating such models with a color degree of freedom, it is possible to make the singlets formed by the coloring instead, while the vertex or tiling configurations facilitate them to be separated arbitrarily far away across the lattice [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' The average distance singlets span is again controlled by the local deformation parameter, resulting in a quantum phase transition between area-law and volumetric scaling of entanglement entropy between half systems cut in either direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' All of the above mentioned models share the common feature of being frustration free, making it convenient to write down a unique ground state that allows exact results analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' Frustration becomes an obstacle when generalizing such models to singlet states among more than two sites, either in the form of trimer, n-mer, valance bond solid [19, 20] in one dimensional chains, or simplex solid states [21] in higher dimensions, unless the local Hibert space is enlarged to the corresponding dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' Such extensions not only provide benchmark for relevant cold atom experiments [22– 26], but also prove useful in the understanding of entanglement structure even when frustration is present [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' In fact, the n-singlet picture turns out to be revealing in understanding the entanglement structure of multi-component generalizations to the plain Heisenberg [28] and XXZ spin chains [29, 30] with permutation Hamiltonian, which is not deliberately cast into projection operators that are frustration free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' An alternative mechanism to generate long-range entangled singlet pairs is by introducing strong inhomogeneity in the XX spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' This was done in the so-called rainbow chain, as the singlets are at fixed locations pairwise symmetric about the center of the chain, giving a maximal entanglement between left and right half chains [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' The inhomogeneity can be interpreted as an underlying spacetime with constant negative curvature in the continuous free- fermionic version of the model [32], and the conformal field theory that describes the free fermions has a holographic dual in the Anti-de Sitter space AdS2 [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' The metric allows more refined structure of the entanglement than the entropy to be computed, such as the entanglement Hamiltonian and the entanglement contour [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' [35], a ∗ zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content='zhang@su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content='se arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content='04170v1 [quant-ph] 10 Jan 2023 2 first attempt at generalizing the mechanism to two dimension was made with an anisotropic quasi-two dimensional model inhomogeneous in one direction but translationally invariant in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' In this manuscript, a straightforward isotropic generalization is realized in the spirit of simplex singlet state by enlarging the dimensionality of the local Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' Although the lattice lives in a two-dimensional manifold, its Hausdorff dimension is still one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' However, the location of lattice sites sheds light upon an interpretation of the inhomogeneity of coupling strength as a natural result of its exponential decay over distance, which is absent in the 1D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' II, the 2D generalization to rainbow chain is defined on the floral lattice, and the strong disorder renormalizaiton group (RG) procedure is carried out to show the effective Hamiltonian and its ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' III, the lattice geometry is briefly discussed to show the positive curvature near the center and how it can be obtained from square lattice by proliferating disclination defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' IV, the model is further generalized to three and higher dimensions outlining the analogous RG transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' Finally, a conclusion is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' V with a few possible future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' INHOMOGENEOUS XX MODEL ON THE FLORAL LATTICE The SU(3) generalization to the XX spin chain is given in terms of the ladder operators corresponding to the two simple roots, which can be mapped to two species of free fermions [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' However, here we are dealing with a few-body problem with exact diagonalization at each order of the Dasgupta-Ma renormalization [31, 38], and the integrability of the Hamiltonian is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' Moreover, to guarantee the Hamiltonian to be invariant in the RG procedure, it needs to respect the S3 permutation symmetry by including all three pairs of ladder operators e1 = � � 0 1 0 0 0 0 0 0 0 � � , e2 = � � 0 0 0 0 0 1 0 0 0 � � , e3 = � � 0 0 0 0 0 0 1 0 0 � � , (1) and their conjugates f a = ea†, a = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' The local Hamiltonian between neighboring site i and j is given by hi,j = 3 � a=1 (ea i f a j + f a i ea j ) ≡ 2 7 � a=1 a̸=3 λa i λa j , (2) where among the Gell-Mann generators of SU(3), λ3 and λ8 spanning the Cartan subalgebra are excluded in the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' We can denote the three components of the local Hilbert space C3 by colors red (R), green (G) and blue (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' The Hamiltonian consists of kinetic terms that exchanges colors between neighboring pairs of different colors, and its lowest energy eigenstate is simply the antisymmetrization of whatever states the two neighboring sites are in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQf3Qj1/content/2301.04170v1.pdf'} +page_content='